Topical Review

First exit times of harmonically trappedparticles: a didactic review

Denis S Grebenkov

Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS–EcolePolytechnique, 91128 Palaiseau, FranceSt. Petersburg National Research University of Information Technologies, Mechanicsand Optics, 197101 St. Petersburg, Russia

E-mail: denis.grebenkov@polytechnique.edu

Received 21 June 2014, revised 13 September 2014Accepted for publication 14 October 2014Published 9 December 2014

AbstractWe revise the classical problem of characterizing first exit times of aharmonically trapped particle whose motion is described by a one- ormultidimensional Ornstein–Uhlenbeck process. We start by recalling themain derivation steps of a propagator using Langevin and Fokker–Planckequations. The mean exit time, the moment-generating function and thesurvival probability are then expressed through confluent hypergeometricfunctions and thoroughly analyzed. We also present a rapidly convergingseries representation of confluent hypergeometric functions that is parti-cularly well suited for numerical computation of eigenvalues and eigen-functions of the governing Fokker–Planck operator. We discuss severalapplications of first exit times, such as the detection of time intervals duringwhich motor proteins exert a constant force onto a tracer in optical tweezerssingle-particle tracking experiments; adhesion bond dissociation undermechanical stress; characterization of active periods of trend-following andmean-reverting strategies in algorithmic trading on stock markets; relationto the distribution of first crossing times of a moving boundary by Brow-nian motion. Some extensions are described, including diffusion underquadratic double-well potential and anomalous diffusion.

Keywords: first exit time, harmonic potential, Ornstein–Uhlenbeck process,confluent hypergeometric function, Fokker–Planck equation, optical tweezers,survival probabilityPACS numbers: 02.50.Ey, 05.10.Gg, 05.40.-a, 02.30.Gp

(Some figures may appear in colour only in the online journal)

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 48 (2015) 013001 (44pp) doi:10.1088/1751-8113/48/1/013001

1751-8113/15/013001+44$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

1. Introduction

First passage time (FPT) distributions have found numerous applications in applied mathe-matics, physics, biology and finance [1–3]. The FPT can characterize the time needed for ananimal to find food; the time for an enzyme to localize a specific DNA sequence and toinitiate a biochemical reaction; the time to exit from a confining domain (e.g., a maze); or thetime to buy or sell an asset when its price deviation from the mean exceeds a prescribedthreshold. The FPT distribution has been studied for a variety of diffusive processes, rangingfrom ordinary diffusion (Brownian motion) to continuous-time random walks (CTRWs) [4–11], fractional Brownian motion [12–15], Lévy flights [16–18], surface-mediated diffusion[19–22] and other intermittent processes [23, 24], diffusion in scale-invariant media [25, 26],trapped diffusion [27], thermally driven oscillators [28], Ornstein–Uhlenbeck process [29–38], and many others [1, 2, 39–48].

In this review, we revise the classical problem of characterizing the first exit time (FET)distribution of a multidimensional Ornstein–Uhlenbeck process from a ball [49–51]. Theprobability distribution can be found through the inverse Fourier (resp. Laplace) transform ofthe characteristic (resp. moment-generating) function for which explicit representations interms of special functions are well known [30, 39]. Although the problem is formally solved,the solution involves confluent hypergeometric functions and thus requires subtle asymptoticmethods and computational hints. The aim of the review is to provide a didactic self-con-sistent description of theoretical, numerical and practical aspects of this problem.

First, we recall the main derivation steps of the FET distribution, from the Langevinequation (section 2.1), through forward and backward Fokker–Planck (FP) equations(section 2.2 and 2.3), to spectral decompositions based on the eigenvalues and eigenfunctionsof the FP operator (section 2.4). This general formalism is then applied to describe the firstexit times of harmonically trapped particles in one dimension: the mean exit time(section 2.5), the survival probability (section 2.6) and the moment-generating function(section 2.7). In particular, we analyze the asymptotic behavior of the mean exit time andeigenvalues in different limits (e.g., strong trapping potential, large constant force, etc).Extensions to the radial Ornstein–Uhlenbeck process in higher-dimensional cases for bothinterior and exterior problems are presented in section 2.8 and 2.9, respectively. Althoughmost of these results are classical, their systematic self-contained presentation and numericalillustrations are missing.

Section 3 starts from the summary of computational hints for computing confluenthypergeometric functions, while technical details are reported in appendix B. We then discussthree applications: (i) calibration of optical tweezers’ stiffness in single-particle trackingexperiments and the detection of eventual constant forces exerted on a tracer by motorproteins (section 3.2), (ii) adhesion bond dissociation under mechanical stress (section 3.3),and (iii) distribution of triggering times of trend-following strategies in algorithmic trading onstock markets (section 3.4). We also illustrate a direct relation to the distribution of first-crossing times of a moving boundary by Brownian motion (section 3.5). Finally, we presentseveral extensions of the spectral approach, including diffusion under quadratic double-wellpotential (section 3.6) and anomalous diffusion (section 3.7). Many technical details aresummarized in the appendices.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

2

2. First exit time distribution

We first recall the standard theoretical description of harmonically trapped particles byLangevin and Fokker–Planck equations [46, 52, 53]. We start with the one-dimensionalOrnstein–Uhlenbeck process and then discuss straightforward extensions to higherdimensions.

2.1. Langevin equation

We consider a diffusing particle of mass m trapped by a harmonic potential of strength k andpulled by a constant force F0. The thermal bath surrounding the particle results in its sto-chastic trajectory, which can be described using a Langevin equation [52]

γ ξ= − + +mX t X t F X t t¨ ( ) ˙ ( ) ( ( )) ( ), (1)

where γ− X t˙ ( ) is the viscous Stokes force (γ being the drag constant),= − +F X t kX t F( ( )) ( ) 0 includes the externally applied Hookean and constant forces, and

ξ t( ) is the thermal driving force with Gaussian distribution such that ξ⟨ ⟩ =t( ) 0 andξ ξ γδ⟨ ′ ⟩ = − ′t t k T t t( ) ( ) 2 ( )B , with ≃ −k 1.38 · 10B

23 J K being the Boltzmann constant, Tthe absolute temperature (in degrees Kelvin), δ t( ) the Dirac distribution, and ⟨…⟩ denotingthe ensemble average or expectation. In the overdamped limit (m = 0), one gets the first-orderstochastic differential equation

γξ

γξ

γ= + = − + =( )X t F X t t

kx X t

tX x˙ ( )

1[ ( ( )) ( )] ˆ ( )

( ), (0) , (2)0

where =x F kˆ 0 is the stationary position (mean value), and x0 is the starting position. TheLangevin equation can also be written in a conventional (dimensionless) stochastic form[39, 40]

μ σ= + =( ) ( )X X t t X t W X xd , d , d , , (3)t t t t 0 0

where Wt is the standard Wiener process (Brownian motion), μ x t( , ) and σ x t( , ) are the driftand volatility, which in general can depend on x and t. In our case, the volatility is constant,while the drift is a linear function of x, μ θ= −x t x x( , ) ( ˆ ) , i.e.

θ σ= − + =( )X x X t W X xd ˆ d d , , (4)t t t 0 0

where θ δ γ= k , and σ δ= D2 , with δ being a time scale, and γ=D k TB the diffusioncoefficient. This stochastic differential equation defines an Ornstein–Uhlenbeck (OU)process, with mean x̂ , variance σ2 and rate θ. An integral representation of equation (4) reads

∫σ= + − + ′′

θ θ θ− − −X x e x e e dWˆ(1 ) (5)tt t

t

t tt0

0

( )

One can see that Xt is a Gaussain process with mean ⟨ ⟩ = + −θ θ− −X x e x eˆ(1 )tt t

0 and

covariance ⟨ ⟩ − ⟨ ⟩⟨ ⟩ = −σθ

θ θ′ ′

− − ′ − + ′X X X X e e( )t t t tt t t t

2| | ( )2

. The discrete version of equation (4)with a fixed time step δ is known as an auto-regressive model AR(1):

δ γ δ γ δ ξ= − + +−X k X F D(1 ) 2 , (6)n n n1 0

where ξn are standard iid Gaussian variables with mean zero and unit variance. This discretescheme can be used for numerical generation of stochastic trajectories. An extension of theabove stochastic description to multidimensional processes is straightforward.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

3

2.2. Forward Fokker–Planck equation

The Langevin equation (2) expresses the displacement δX t˙ ( ) over a short time step δ in termsof the current position X(t). In other words, the distribution of the next position is fullydetermined by the current position, the so-called Markov property. Such a Markov processcan be characterized by a propagator or a transition density, i.e., the conditional probabilitydensity p x t x t( , | , )0 0 of finding the particle at x at time t, given that it was at x0 at earlier timet0. The propagator can be seen as a ‘fraction’ of paths from x0 to x among all paths started atx0 (of duration −t t0), which formally writes as the average of the Dirac distributionδ −X t x( ( ) ) over all random paths started from x0: δ= ⟨ − ⟩ =p x t x t X t x( , | , ) ( ( ) ) X t x0 0 ( )0 0.The Markov property implies the Chapman–Kolmogorov (or Smoluchowski) equation

∫= ′ ′ ′ ′ ′ < ′ <−∞

∞( ) ( ) ( )p x t x t x p x t x t p x t x t t t t, , d ( , , ) , , , (7)0 0 0 0 0

which expresses a simple fact that any continuous path from =X t x( )0 0 to =X t x( )can be split at any intermediate time ′t into two independent paths, from x0 to ′x , and from ′xto x.

As a function of the arrival state (x and t), the propagator satisfies the forward Fokker–Planck (FP) equation [52, 53]. We reproduce the derivation of this equation from [54], whichrelies on the evaluation of the integral

∫ δ= + −−∞

∞ ⎡⎣ ⎤⎦( ) ( )I x h x p x t x t p x t x td ( ) , , , ,0 0 0 0

for any smooth function h(x) with compact support. One has

∫ ∫∫ ∫

∫ ∫

δ

δ

δ

= ′ + ′ ′

− ′ ′ ′ + ′

= ′ + ′ ′ − ′

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

( )

( )

( )

I x h x x p x t x t p x t x t

x h x p x t x t x p x t x t

x x p x t x t p x t x t h x h x

d ( ) d ( , , ) , ,

d ( ) , , d ( , , )

d d ( , , ) , , [ ( ) ( )],

0 0

0 0

0 0

where the first term was represented using equation (7), while the normalization of theprobability density δ+ ′p x t x t( , | , ) allowed one to add the integral over x in the secondterm. Expanding h(x) into a Taylor series around ′x and then exchanging the integrationvariables x and ′x , one gets

∫ ∫∑ δ= ′ ′ + ′ −−∞

∞

=

∞

−∞

∞⎛⎝⎜

⎞⎠⎟( )I x p x t x t

xh x

nx p x t x t x xd , ,

d

d( )

1

!d ( , , )( ) .

n

n

nn

0 0

1

Finally, integrating each term by parts n times, dividing by δ and taking the limit δ → 0 yield

∫ ∫ ∑∂

∂= −

−∞

∞

−∞

∞

=

∞

( )( ) ( )x h xp x t x t

tx h x

xD x p x t x td ( )

, ,d ( ) ( 1)

d

d( ) , , ,

n

nn

nn0 0

1

( )0 0

where the left-hand side is the limit of δI as δ → 0, and

∫δδ= ′ ′ + ′ −

δ→ −∞

∞D x

nx p x t x t x x( )

1

!lim

1d ( , , )( ) . (8)n n( )

0

Since the above integral relation is satisfied for the arbitrary function h(x), one deduces the so-called Kramers–Moyal expansion:

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

4

∑∂

∂= −

=

∞

( )( ) ( )p x t x t

t xD x p x t x t

, ,( 1)

d

d( ) , , . (9)

n

nn

nn0 0

1

( )0 0

Here, we assumed that the process is time homogeneous, i.e., p x t x t( , | , )0 0 is invariant undertime shift: = + ′ + ′p x t x t p x t t x t t( , | , ) ( , | , )0 0 0 0 , which implies the time-independence ofD x( )n( ) .

The density δ′ +p x t x t( , | , ) in equation (8) characterizes the displacement between=X t x( ) and δ+ = ′X t x( ) , which can be written as δ ξ+ − ≃ +δ

γX t X t F x t( ) ( ) [ ( ) ( )] for

small δ according to the Langevin equation (2). After discretization in units of δ, the thermalforce ξ t( ) becomes a Gaussian variable with mean zero and variance γ δk T2 B . As a con-sequence, the displacement ′ −x x is also a Gaussian variable with mean δ γ F x( ) ( ) andvariance δ γ γ δk T( ) 22

B , i.e.,

δπ δ

δ γδ

′ + = − ′ − −⎛⎝⎜

⎞⎠⎟p x t x t

D

x x F x

D( , , )

1

4exp

( ( ) )

4

2

for small δ. Substituting this density into equation (8) and evaluating Gaussian integrals, onegets γ=D x F x( ) ( )(1) , =D D(2) , and =D 0n( ) for >n 2 that yields the forward Fokker–Planck equation

γ∂∂

= = − ∂ + ∂ ( ) ( )tp x t x t p x t x t

F xD, , , , ,

( ), (10)x x x x0 0 0 0

2

where x is the Fokker–Planck operator acting on the arrival point x. This equation iscompleted by the initial condition δ= −p x t x t x x( , | , ) ( )0 0 0 0 at =t t0, with a fixed startingpoint x0. Note that the forward FP equation can be seen as the probability conservation law,

∂∂

= −∂( ) ( )tp x t x t J x t x t, , , , ,x0 0 0 0

where = − ∂γ

J x t x t p x t x t D p x t x t( , | , ) ( , | , ) ( , | , )F xx0 0

( )0 0 0 0 is the probability flux. Setting

J = 0, one can solves the first-order differential equation to retrieve the equilibrium solution=p x Zw x( ) ( )eq , where Z is the normalization factor, and

∫= ′ ′ = − = − +⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟w x dx

F x

k T

V x

k T

kx

k T

F x

k T( ) exp

( )exp

( )exp

2, (11)

x

0 B B

2

B

0

B

where ∫= − ′ ′V x dx F x( ) ( )x

0is the potential associated to the force F(x). This is the standard

Boltzmann–Gibbs equilibrium distribution.When the FP operator x has a discrete spectrum, the probability density admits the

spectral decomposition

∑= λ

=

∞− −( )p x t x t v x v x w x e, , ( ) ( ) ˜ ( ) (12)

n

n nt t

0 0

0

0 0( )n 0

over the eigenvalues λn and eigenfunctions vn(x) of x:

λ+ = = … v x v x n( ) ( ) 0 ( 0, 1, 2, ) (13)x n n n

(eventually with appropriate boundary conditions; see below). The weight =w x w x˜ ( ) 1 ( )ensures the orthogonality of eigenfunctions:

∫ δ=x w x v x v xd ˜ ( ) ( ) ( ) , (14)m n m n,

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

5

while the closure (or completeness) relation reads

∑ δ= −=

∞

v x v x w x x x( ) ( ) ˜ ( ) ( ). (15)n

n n

0

0 0 0

This relation implies the initial condition δ= −p x t x t x x( , | , ) ( )0 0 0 0 . As for the Langevinequation, an extension to the multidimensional case is straightforward. In particular, thederivative ∂x is replaced by the gradient operator, while ∂x2 becomes the Laplaceoperator [53, 55].

2.3. Backward Fokker–Planck equation

The forward FP equation describes the evolution of the probability density p x t x t( , | , )0 0

from a given initial state (here, the starting point x0 at time t0). Alternatively, if the particle isfound at the arrival point x at time t (or, more generally, in a prescribed subset of states), onecan interpret p x t x t( , | , )0 0 as the conditional probability density that the particle is startedfrom x0 at time t0 knowing that it arrived at x at later time t. As a function of x0 and t0, thisprobability density satisfies the backward Fokker–Planck (or Kolmogorov) equation [53]:

− ∂∂

= ( ) ( )t

p x t x t p x t x t, , , , , (16)x0

0 0 * 0 00

where the backward FP operator * is adjoint to the forward FP operator (i.e.,= f g f g( , ) ( , * ) for any two functions f and g from an appropriate functional space).

Equation (10) implies

γ γ= ∂ + ∂ = − ∂ + ∂ ( )

F xD

kx x D

( )ˆ . (17)x x x x x

* 0 20

20 0 0 0 0

Note that this operator acts on the starting point x0 while the minus sign in front of the timederivative reflects the backward time direction. Equation (16) is easily obtained bydifferentiating the Champan–Kolmogorov equation (7) with respect to the intermediate time′t .

The eigenvalues of both forward and backward FP operators are identical, while theeigenfunctions un(x) of the backward FP operator * are simply =u x v x w x( ) ( ) ( )n n . As aconsequence, one can rewrite the spectral decomposition (12) as

∑= λ

=

∞− −( )p x t x t u x u x w x e, , ( ) ( ) ( ) , (18)

n

n nt t

0 0

0

0( )n 0

with the weight w x( ) from equation (11). The eigenfunctions un(x) are as well orthogonal:

∫ δ=x w x u x u xd ( ) ( ) ( ) , (19)m n m n,

while the closure (or completeness) relation reads

∑ δ= −=

∞

u x u x w x x x( ) ( ) ( ) ( ). (20)n

n n

0

0 0

This relation implies the terminal condition δ= −p x t x t x x( , | , ) ( )0 0 at =t t0 . In contrast toequation (12), the weight w x( ) in the spectral representation (18) depends on the fixed arrivalpoint x, while the backward FP operator x

*0acts on eigenfunctions u x( )n 0 .

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

6

When there is no force term, the operator is self-adjoint, = *, and the probabilitydensity is invariant under time reversal: =p x t x t p x t x t( , | , ) ( , | , )0 0 0 0 . This property doesnot hold in the presence of force.

Finally, the backward FP equation is closely related to the Feynman–Kac formula fordetermining distributions of various Wiener functionals [56–60]. For instance, we alreadymentioned that the probability density p x t x t( , | , )0 0 can be understood as the conditionalexpectation: δ= ⟨ − ⟩ =p x t x t X t x( , | , ) ( ( ) ) X t x0 0 ( )0 0. More generally, for given functionsψ x( )0 , f x t( , )0 0 and U x t( , )0 0 , the conditional expectation

∫

∫ ∫

ψ= − ′ ′ ′

+ ′ ′ ′ − ″ ″ ″′

=

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( )

( )

( )

( )

u x t t U X t t X t

t f X t t t U X t t

( , ) exp d , ( ( ))

d ( ( ), )exp d , (21)

t

t

t

t

t

t

X t x

0 0

( )

0

0 00 0

satisfies the backward FP equation

− ∂∂

= − +t

u x t u x t U x t u x t f x t( , ) ( , ) ( , ) ( , ) ( , ), (22)x0

0 0 * 0 0 0 0 0 0 0 00

subject to the terminal condition ψ=u x t x( , ) ( )0 0 at a later time >t t0.

2.4. First exit times

In this review, we study the random variable τ = > >t X t Linf { 0 : | ( )| }, i.e., the first exittime of the process X(t) from an interval −L L[ , ] when started from x0 at =t 00 . Thecumulative distribution function of τ is related to the survival probability

τ= >S x t t( , ) { }0 up to time t of a particle that started from x0. The notion of survival isassociated with the disappearing of the particle that hit either endpoint, due to chemicalreaction, permeation, adsorption, relaxation, annihilation, transformation or any other‘killing’ mechanism. The survival probability S x t( , )0 can be expressed through theprobability density p x t x( , | , 0)0 of moving from x0 to x in time t without visiting theendpoints ±L during this motion. Alternatively, p x t x( , | , 0)0 can be seen as the conditionalprobability density of starting from point x0 at time =t 00 under condition to be at x at timet. This condition includes the survival up to time t, i.e., not visiting the endpoints ±L. Theprobability density p x t x( , | , 0)0 satisfies the backward FP equation with Dirichlet boundarycondition at = ±x L0 : ± =p x t L( , | , 0) 0. This condition simply states that a particle startedfrom either endpoint has immediately hit this endpoint, i.e. not survived. Note that thiscondition is preserved during all intermediate times ′t due to the Chapman–Kolmogorovequation (7).

Since the survival probability S x t( , )0 ignores the actual position x at time t, one justneeds to average the density p x t x( , | , 0)0 over x:

∫ ∫∑= = λ− =

∞−

−( )S x t x p x t x u x e x u x w x( , ) d , , 0 ( ) d ( ) ( ), (23)

L

L

n

nt

L

L

n0 0

0

0n

where the spectral decomposition (18) was used. The eigenfunctions un(x) of the backward FPoperator should satisfy the Dirichlet boundary condition at = ±x L0 : ± =u L( ) 0n .Equation (23) also implies the backward FP equation

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

7

∂∂

= S x t

tS x t

( , )( , ), (24)x

0 * 00

which is completed by the initial condition =S x( , 0) 10 (the particle exists at the beginning)and Dirichlet boundary condition ± =S L t( , ) 0 (the process is stopped upon the first arrival ateither endpoint of the confining interval −L L[ , ]). Since the process is homogeneous in time,p x t x t( , | , )0 0 depends on t t– 0 and thus ∂ ∂ = −∂ ∂p t p t/ /0 that allows one to write the left-hand side of the backward FP equation (24) with the plus sign. Note that the characterizationof first passage times through the backward FP equation goes back to the seminal work in1933 by Pontryagin et al [61]. Similar equations emerge in quantum mechanics when onesearches for eigenstates of a particle trapped by a short-range harmonic potential [62] (seealso appendix D for the quantum harmonic oscillator).

The FET probability density is = − ∂∂

q x t( , )S x t

t0( , )0 , while the moment-generating func-

tion is given by its Laplace transform:

∫= ≡τ−∞

−e t e q x t q x sd ( , ) ˜( , ), (25)s st

00 0

with tilde denoting Laplace-transformed quantities. The Laplace transform of equation (24)yields the equation − = − s S x s( ) ˜( , ) 1x

*00

with Dirichlet boundary conditions. Since

= −q x s sS x s˜( , ) 1 ˜( , )0 0 , one gets

− =( )s q x s˜( , ) 0, (26)x* 00

with Dirichlet boundary condition ± =q L s˜ ( , ) 1.Finally, the moments τ⟨ ⟩m

x0 can be found in one of these standard ways:

(i) from the moment-generating function,

τ = − ∂∂→ s

q x s( 1) lim ˜( , ); (27)mx

m

s

m

m000

(ii) from the spectral representation of the survival probability:

∫∑τ λ==

∞−

−m u x x u x w x! ( ) d ( ) ( ); (28)m

xn

n nm

L

L

n

0

00

(iii) from recurrence partial differential equations (PDEs)

τ τ= − − m , (29)xm

xm

x* 1

0 0 0

with Dirichlet boundary conditions [30].

In what follows, we focus on the mean exit time τ⟨ ⟩ ,x0 the moment-generating functionq x s˜( , )0 , and the survival probability S x t( , )0 for harmonically trapped particles.

2.5. Mean exit time

The mean exit times of diffusive processes were studied particularly well because of theirpractical importance and simpler mathematical analysis (see [1, 2, 63–65] and referencestherein). In fact, the mean exit time,

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

8

∫ ∫τ = =∞ ∞

t tq x t t S x td ( , ) d ( , ), (30)x0

00

00

satisfies the simpler equation than the time-dependent PDE (16):

τ =− 1, (31)x x*

0 0

with Dirichlet boundary conditions at = ±x L0 . The double integration and imposedboundary conditions yield [66]1

∫ ∫ ∫

∫ ∫ ∫

τ = ′ ′

× − ′ ′

−

−

−

− −

⎪⎪

⎧⎨⎩

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎫⎬⎭

D

x

w x

x

w xx w x

x

w x

x

w xx w x

1 d

( )

d

( )d ( )

d

( )

d

( )d ( ) . (32)

xL

L

L

L x

L

x

L

x x

1

0

0

0

0 0

Substituting w(x) from equation (11), one gets

∫ ∫

τ πκ

κ φ κ φ

κ φ κ φ=

− + +

− + +

× −κ φ

κ φ

κ φ

κ φ

− −

−

− −

−

⎪⎪

⎧⎨⎩

⎫⎬⎭

( )( ) ( )( ) ( )

( ) ( )

( )

L

D

i x L i

i i

z e z z e z

2

erf erf (1 )

erf (1 ) erf (1 )

d erf ( ) d erf ( ) , (33)

x

zx L

z

2 0

1

(1 )

1

0

2 0 2

where zerf ( ) is the error function, and κ and φ are two dimensionless parameterscharacterizing the trapping harmonic potential and the pulling constant force, respectively

κ φ≡ ≡ =kL

k T

x

L

F

kL2,

ˆ. (34)

2

B

0

Throughout the paper, we consider φ ⩾ 0, while all the results for φ < 0 can be obtained byreplacing φ φ→ − and → −x x0 0. For large κ or φ, one can use an equivalent representation(A.1) provided in appendix A.1.

Several limiting cases are of interest:

• When φ = 0 (i.e., =F 00 ), equation (33) is reduced to

∫τ πκ

=κ

κL

Dz e z

2d erf ( ). (35)x

x L

z2

00

2

• In the limit →k 0, one gets a simpler expression

τη

= − − −−

η η

η η

− −

−

⎛⎝⎜

⎞⎠⎟

L

Dx L

e e

e e1 2 , (36)x

x L2

00

0

where η = F L k T( )0 B is another dimensionless parameter. If =F 00 , one retrieves theclassical result for Brownian motion:

τ = −( )( )L

Dx L

21 . (37)x

2

02

0

1 In [66], the minus sign in front of U(z) in the second integral in the numerator of the first term in equation (7.7) ismissing.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

9

• For small κ, the Taylor expansion of equation (33) yields

τ κφ

κ≃−

+− +

+⎛

⎝⎜⎜

⎞

⎠⎟⎟( )( ) ( )L x

D

x L x LO

21

1 2

3. (38)x

202

0 02

20

We emphasize that the limits κ → 0 and →k 0 are not equivalent because in the latter case,φ → ∞ according to equation (34).• In the opposite limit of large κ, four cases can be distinguished (see appendix A.1):

τ

πκ

φ

πκ φ

φ

κκ

φ

κφ

φφ

≃

=

−< <

−…

=

−−

>

κ

κ φ−

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

( )L

D

e

e

x L

x L

4( 0),

2 (1 )(0 1),

1

2ln

1

0.375( 1),

1

2ln

1( 1),

(39)x

2

3 2

(1 )

3 2

0

0

0

2

and the exponential growth in the first two relations is valid for any x0 not too close to ±L.Note that the limit of the second asymptotic relation (for φ< <0 1) as φ → 0 is differentfrom the case φ = 0 by a factor of 2. In fact, when φ > 0, it is much more probable to reachthe right endpoint than the left one, and τ⟨ ⟩ x0 characterizes mainly the exit through the rightendpoint at large κ. In turn, when φ = 0, both endpoints are equivalent, which doubles thechances to exit and thus reduces by a factor of 2 the mean exit time. Note that the first tworelations (up to a numerical prefactor) can be obtained by the Kramers theory of escapefrom a potential well [66, 67]. The last relation in equations (39) can be retrieved from thelast line of equation (7.9) of [66].

The behavior of the mean exit time τ⟨ ⟩ x0 as a function of the starting point x0 is illustratedin figure 1. The increase of κ at fixed φ = 0 transforms the spatial profile of the mean exittime from the parabolic shape (35) at κ = 0 to a Π-shape at large κ (figure 1(a)). In otherwords, the dependence on the starting point becomes weak at large κ. At the same time, the

Figure 1.Mean exit time τ⟨ ⟩ z0 as a function of =z x L0 0 : for different κ at fixed φ = 0

(a) and for different φ at fixed κ = 1 (b). The timescale L D2 is set to 1. For plot (a), themean exit time is divided by its maximal value (at =z 00 ) in order to rescale the

curves. Circles indicate the mean exit time − z(1 )L

D2 022

without trapping (k = 0).

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

10

height of the profile rapidly grows with κ according to equation (39). On the other hand, thespatial profile becomes more skewed and sensitive to the starting point as φ increases at fixedκ = 1, while the height is decreasing (figure 1(b)). As expected, the constant force breaks theinitial symmetry of the harmonic potential and facilitates the escape from the trap.

Figure 2 shows how the mean exit time τ⟨ ⟩ 0 from the center varies with κ and φ. Whenthere is no constant force (φ = 0), one observes a rapid (exponential) growth at large κ, inagreement with equation (39) (shown by circles). The presence of a moderate constant force(with φ< <0 1) slows down the increase of the mean exit time. For instance, at φ = 0.5,τ⟨ ⟩ 0 exhibits a broad minimum at intermediate values of κ, but it resumes growing at largervalues of κ. In turn, for φ ⩾ 1, there is no exponential growth with κ, and the mean exit timeslowly decreases, as expected from equation (39). Since the constant force shifts the mini-mum of the harmonic potential from 0 to =x F kˆ 0 , the border value φ = 1 corresponds to theminimum x̂ at the exit position ( =x Lˆ ). For φ < 1, the harmonic potential keeps the particleaway from the exit and thus greatly increases the mean exit time. In turn, for φ > 1, theharmonic potential attracts the particle to x̂ , which is outside the interval −L L[ , ] and thusspeeds up the escape.

Although we considered the FET from a symmetric interval −L L[ , ] for convenience,shifting the coordinate by x̂ allows one to map the original problem to the FET from anonsymmetric interval −a b[ , ] with φ= +a L (1 ) and φ= −b L (1 ), with the starting pointx0 being shifted by φL to vary from −a to b. As a consequence, the choice of the symmetricinterval −L L[ , ] is not restrictive, and all the results can be recast for a general interval −a b[ , ]by shifts.

2.6. Survival probability

The survival probability is fully determined by the eigenvalues and eigenfunctions of thebackward FP operator. The eigenvalue equation (13) reads

γ λ″ − − ′ + =( )Du k x x u u( ) ˆ 0. (40)

Figure 2.Mean exit time τ⟨ ⟩0 as a function of κ for fixed φ (a) and as a function of φ forfixed κ (b). The timescale L D2 is set to 1. On both plots, circles show the exponentialasymptotic relation in equation (39) for large κ and φ⩽ <0 1. On plot (a), crossespresent the asymptotic equation (38) for small κ, to which the next-order term, κ2 452 ,is added. On plot (b), crosses present the logarithmic asymptotic relation inequation (39) for φ > 1.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

11

A general solution to this equation is well known [68]

ακ

κ φ φ ακ

κ φ= − − + − − + −⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟u z c M z c z M z( )

4,

1

2, ( ) ( )

4

1

2,

3

2, ( ) , (41)1

22

2

22

where =z x L is the dimensionless coordinate, λ α= D L2 2, c1 and c2 are arbitrary constants,and

∑= ==

∞

M a b z F a b za z

b n( , , ) ( , , )

!(42)

n

n n

n1 1

0

( )

( )

is the confluent hypergeometric function of the first kind (also known as the Kummerfunction), with =a 1(0) and = + … + − = Γ

Γ+a a a a n( 1) ( 1)n a n

a( ) ( )

( ), where Γ z( ) is the

gamma function. The first and second terms in equation (41) are, respectively, the symmetricand antisymmetric functions with respect to φ.

To shorten notations, we set

ακ

κ≡ −α κ

⎛⎝⎜

⎞⎠⎟m z M z( )

4,

1

2, , (43),

(1)2

2

ακ

κ≡ − +α κ

⎛⎝⎜

⎞⎠⎟m z zM z( )

4

1

2,

3

2, , (44),

(2)2

2

so that

φ φ= − + −α κ α κu z c m z c m z( ) ( ) ( ). (45)1 ,(1)

2 ,(2)

The Dirichlet boundary conditions read

φ φ

φ φ

− − + − − = = −

− + − = =α κ α κ

α κ α κ

( )( )

c m c m x L

c m c m x L

( 1 ) ( 1 ) 0 at ,

(1 ) (1 ) 0 at .

1 ,(1)

2 ,(2)

0

1 ,(1)

2 ,(2)

0

In the special case φ = 1, one gets =c 01 , and the eigenvalues are determined from theequation =α κm (2) 0,

(2) . In general, for φ ≠ 1, one considers the determinant of the underlying2 × 2 matrix:

φ φ φ φ= − − − − − − −α κ φ α κ α κ α κ α κ m m m m( 1 ) (1 ) ( 1 ) (1 ). (46), , ,(1)

,(2)

,(2)

,(1)

n

Setting this determinant to 0 yields the equation on α:

=α κ φ 0, (47), ,n

where αn ( = …n 0, 1, 2, ) denote all positive solutions of this equation (for fixed κ and φ).The eigenfunctions then read

βφ φ= − − −α κ α κ

⎡⎣ ⎤⎦u zL

c m z c m z( ) ( ) ( ) , (48)nn

n n(1)

,(1) (2)

,(2)

n n

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

12

where

φ φ= − = −α κ α κc m c m(1 ), (1 ), (49)n n(1)

,(2) (2)

,(1)

n n

and the normalization constant is

∫β = −φ

φκ

α κ α κ−

− −

−− ⎡⎣ ⎤⎦z e c m z c m zd ( ) ( ) . (50)n

zn n

2

1

1(1)

,(1) (2)

,(2) 2

n n

2

Multiplying equation (40) by w x( ) and integrating from a to b, one obtains

∫ λ= ′ − ′x u x w x

Du a w a u b w bd ( ) ( ) [ ( ) ( ) ( ) ( )]. (51)

a

b

The derivative of the Kummer function can be expressed through Kummer functions, inparticular,

ακ

∂ = −α κ α κ α κ κ−( )m zz

m z m z( )2

( ) ( ) , (52)z ,(1)

2

,(1)

4 ,(1)

2

κ ακ

ακ

∂ = − − + +α κ ακ α κ κ+

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟m z z m z m z( ) 2 1

2( ) 2

2( ), (53)z ,

(2) 22

,(2)

2

4 ,(2)

2

from which one gets explicit formulas for ′u a( )n and ′u b( )n and thus for the integral inequation (51). We get therefore,

∑ φ φ= − − −αα κ α κ

=

∞− ⎡⎣ ⎤⎦( ) ( )S x t w e c m x L c m x L( , ) , (54)

n

nDt L

n n0

0

(1),

(1)0

(2),

(2)0n

n n

2 2

where

β

α= − −

κ−

we

v v[ ( 1) (1)], (55)nn

n

2

2

with

φ φ= ∂ − − ∂ −κφα κ α κ( )v z e c m z c m z( ) ( ) ( ) . (56)z

n z n z2 (1)

,(1) (2)

,(2)

n n

Taking the derivative with respect to time, one obtains the FET probability density

∑ α φ φ= − − −αα κ α κ

=

∞− ⎡⎣ ⎤⎦( ) ( )q x t

D

Lw e c m x L c m x L( , ) . (57)

n

n nDt L

n n0 20

2 (1),

(1)0

(2),

(2)0n

n n

2 2

In the limit κ → 0, functions α κm z( ),(1) and α κm z( ),

(2) approach αzcos ( ) and αzsin ( ),

respectively, so that eigenfunctions from equation (48) become α= −βu z z( ) sin ( (1 ))n L

n ,

while the determinant in equation (46) is reduced to αsin (2 ), from which α π= +n( 1) 2n .In this limit, the dependence on φ vanishes, and one retrieves the classical result for Brownianmotion

∑π

π= −+

+π

=

∞ − +( )S x t

e

nn x L( , ) 2 ( 1)

( 1 2)cos ( 1 2) . (58)

n

nDt n L

0

0

( 1 2)

0

2 2 2

Only symmetric eigenfunctions with α π= +n( 1 2)n contribute to this expression.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

13

For centered harmonic potential (φ = 0), equation (47) is reduced to

=α κ α κm m(1) (1) 0, (59),(1)

,(2)

n n

which determines two sequences of zeros: αn,1 from =α κm (1) 0,(1)

n,1, and αn,2 from

=α κm (1) 0,(2)

n,2. Consequently, one can consider separately two sequences of symmetric and

antisymmetric eigenfunctions: α κm z( ),(1)

n,1and α κm z( ),

(2)n,2

. According to equation (23), integra-tion over arrival points removes all the terms containing antisymmetric eigenfunctions. Thissimpler situation is considered as a particular case in section 2.8.

Figure 3 illustrates the behavior of the probability density q x t( , )0 . For fixed φ = 0, anincrease of κ increases the mean exit time and makes the distribution wider. Note that themost probable FET remains almost constant. The opposite trend appears for variable φ atfixed κ = 1: an increase of φ diminishes the mean exit time and makes the distributionnarrower. This is expected because a strong constant force would drive the particle to one exitand dominate over the stochastic part.

Figure 4 shows the dependence of the survival probability S x t( , )0 on the starting pointx0. At short times, the survival probability is close to 1 independently of x0, except for theclose vicinity of the endpoints. As time increases, S x t( , )0 is progressively attenuated. Thespatial profile is symmetric for centered harmonic potential (φ = 0), and skewed to the left in

Figure 3. FET probability density q t(0, ) for several κ at fixed φ = 0 (a) and forseveral φ at fixed κ = 1 (b). The timescale L D2 is set to 1. The spectral decomposition(57) is truncated after 30 terms.

Figure 4. Survival probability S x t( , )0 as a function of the starting point =z x L0 0 ,with κ = 1, and φ = 0 (a) and φ = 0.9 (b). The timescale L D2 is set to 1. The spectraldecomposition (54) is truncated after 30 terms.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

14

the presence of a positive constant force (φ = 0.9); reaching the right endpoint is moreprobable due to the drift caused by a constant force.

2.7. Moment-generating function

Since any linear combination of functions in equation (45) satisfies equation (26), withα= −s D L2 2, one can easily find the moment-generating function q x s˜( , )0 by imposing the

boundary condition ± =q L s˜ ( , ) 1:

φ φ= − + −α κ φ

α κ φα κ

α κ φ

α κ φα κ ( ) ( )q x s

Am x L

Am x L˜( , ) , (60)0

, ,(1)

, ,,

(1)0

, ,(2)

, ,,

(2)0

n n

where

φ φ

φ φ

= − − − −

= − − − −α κ φ α κ α κ

α κ φ α κ α κ

A m m

A m m

(1 ) ( 1 ),

( 1 ) (1 ),

, ,(1)

,(2)

,(2)

, ,(2)

,(1)

,(1)

and α κ φ , , is defined by equation (46). Setting φ= +a L (1 ) and φ= −b L (1 ), one retrievesthe moment-generating function of the FET of an Ornstein–Uhlenbeck process from aninterval −a b[ , ] reported in [39] (p. 548, 3.0.1), in which equation (60) is written morecompactly in terms of the two-parametric family νS a b( , , ) of parabolic cylinder functions(see appendix B.1). For the symmetric interval −a a[ , ], a similar expression for the moment-generating function was provided in [30].

It is worth noting that the probability density q x t( , )0 could be alternatively found by theinverse Laplace transform of equation (60). For this purpose, one determines the poles sn ofq x s˜( , )0 in the complex plane which are given by zeros αn of α κ φ , , according to

equation (47). In other words, one has α= −s D Ln n2 2, and the residue theorem yields

∑κ φ φ=′

− +′

−α α κ φ

α κ φα κ

α κ φ

α κ φα κ

=

∞−

⎡⎣⎢⎢

⎤⎦⎥⎥( ) ( )q x t

D

Le

Am x L

Am x L( , )

4, (61)

n

Dt L0 2

0

, ,(1)

, ,,

(1)0

, ,(2)

, ,,

(2)0n

n

n

n

n

n

n

2 2

where ′α κ φ , , denotes the derivative of α κ φ , , with respect to α κ= −s (4 )2 . Comparing theabove formula to equation (57), one gets another representation for coefficients wn

κα

=′α κ φ

α κ φwA4

, (62)nn2

, ,(1)

, ,

n

n

where we used the identity = −α κ φ α κ φc A c An n(1)

, ,(2) (2)

, ,(1)

n n, with cn

(1,2) from equation (49). Twoalternative representations (55) and (62) allow one to compute the normalization coefficientsβn without numerical integration in equation (50).

2.8. Higher-dimensional case

In higher dimensions, we consider the FET of a multidimensional Ornstein–Uhlenbeckprocess from a ball of radius L. For centered harmonic potential (i.e., =F 00 ), the derivationfollows the same steps used earlier. In fact, the integration of the probability densityp x t x( , | , 0)0 over the arrival point x in the multidimensional version of equation (23) removesthe angular dependence of the survival probability so that the eigenvalue equation is reducedto the radial part

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

15

γλ∂ + − ∂ − ∂ + =

⎛⎝⎜

⎡⎣⎢

⎤⎦⎥

⎞⎠⎟D

d

r

kru r u r

1( ) ( ) 0, (63)r r r n n n

2

where d is the space dimension. In other words, we consider the FPT of the radial Ornstein–Uhlenbeck process to the level L. In turn, the analysis for noncentered harmonic potentialwith ≠F 00 is much more involved in higher dimensions due to angular dependence, and isbeyond the scope of this review.

Survival probability. A solution of equation (63) is given by the Kummer function, which isregular at r = 0

β ακ

κ= − = …⎛⎝⎜

⎞⎠⎟u r

LM

dr L n( )

4,

2, ( ) ( 0, 1, 2, ), (64)n

n

dn

2

22

where βn is the normalization factor:

∫βα

κκ= −κ− − −

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥z z e M

dzd

4,

2, . (65)n

d z n2

0

11

22

22

The eigenvalues λ α= D Ln n2 2 are determined by the positive zeros αn of the equation

ακ

κ− =⎛⎝⎜

⎞⎠⎟M

d

4,

2, 0. (66)n

2

Repeating the same steps as in section 2.6 yields the spectral representation of the survivalprobability

∑ ακ

κ= −α

=

∞−

⎛⎝⎜

⎞⎠⎟( )S r t w e M

dr L( , )

4,

2, , (67)

n

nDt L n

0

0

2

02

n2 2

where

βκ

ακ

κ= − +κ− ⎛

⎝⎜⎞⎠⎟w

eM

d

2 41,

2, , (68)n

n n2 2

and we used the identity

∫ ακ

κκ

ακ

κ− = − +κκ

− −−⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟z z e M

dz

eM

dd

4,

2,

2 41,

2, . (69)d z n n

0

11

22

22

The FET probability density is then

∑ αα

κκ= −α

=

∞−

⎛⎝⎜

⎞⎠⎟( )q r t

D

Lw e M

dr L( , )

4,

2, . (70)

n

n nDt L n

0 20

22

02

n2 2

In the limit κ → 0, one can use the identity (see appendix B.1)

ακ

κ Γα

α

αα

αα

− = =

==

=κ→

−−

⎛⎝⎜

⎞⎠⎟

⎧⎨⎪⎪

⎩⎪⎪

Md

z dJ z

z

z dJ z d

z

zd

lim4

,2

, ( 2)( )

( 2)

cos ( ) ( 1)( ) ( 2)

sin ( )( 3)

(71)d

d0

22 2 1

2 1

0

to retrieve the classical results for Brownian motion (here J z( )n is the Bessel function of thefirst kind). In particular, one retrieves α π= +n( 1 2)n in one dimension and α π= +n( 1)n

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

16

in three dimensions (with = …n 0, 1, 2, ). For the one-dimensional case, we retrieved onlythe zeros of symmetric eigenfunctions that contribute to the survival probability (cf discussionin section 2.6).

Moment-generating function. The moment-generating function, obeying equation (63) with−s instead of λn, is

κ

κ=

κ

κ

( )( )

q r sM

M˜( , )

, ,

, ,, (72)

sL

D

d r

L

sL

D

d0

4 2

4 2

202

2

2

in agreement with [39] (p. 581, 2.0.1). This function satisfies the boundary condition=q L s˜ ( , ) 1 and is regular at =r 00 . The Laplace inversion of this expression yields another

representation of the probability density

∑κ κ

κ=

−

′ −α

ακ

ακ

=

∞−

( )( )

q r tD

Le

M

M( , )

4, ,

, ,, (73)

n

Dt L

d r

L

d0 2

0

4 2

4 2

n

n

n

2 2

202

2

2

where ′M a b z( , , ) denotes the derivative of M a b z( , , ) with respect to a. Comparing thisrelation to equation (70), the coefficients wn from equation (68) can also be identified as

κ

α κ=

′ − ακ( )

wM

4

; ;. (74)n

nd2

4 2n2

As mentioned above, two expressions (68, 74) for wn can be used to compute thenormalization constants βn without numerical integration in equation (65).

Mean exit time. Following the same steps as in section 2.5, one gets the mean exit time forthe higher-dimensional case

∫ ∫τκ

=κ

κ− − −L

Dr r e r r e

1d d , (75)r

r L

d rr

d r2

1 11

02 2

10

0

12 1

22

where we imposed the Dirichlet boundary condition at =r L0 and the regularity condition at=r 00 . In the limit κ → 0, one retrieves the classical result τ⟨ ⟩ = −L r dD( ) (2 )r

202

0 . In theopposite limit κ ≫ 1, one gets

τ Γκ

κ≃ ≫κ

+L

D

d e( 2)

4( 1), (76)r d

2

1 20

which is applicable for any r0 not too close to L. The behavior of the mean exit time forgeneral spherically symmetric potentials is discussed in [66].

In appendix A.2, the asymptotic behavior of the smallest eigenvalue λ α= D L0 02 2 is

obtained:

λ κΓ

κ≃ ≫κ+

−D

L de

4

( 2)( 1), (77)

d

0 2

1 2

which is just the inverse of the above asymptotic relation for the mean exit time. While thefirst eigenvalue exponentially decays with κ, the other eigenvalues linearly grow with κ (seeappendix A.2):

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

17

λ κ κ≃ ≫D

Ln4 ( 1). (78)n 2

As a consequence, the gap between the lowest eigenvalue λ0 and the next eigenvalue λ1 growslinearly with κ. For λ≫t 1 1, the contribution of all excited eigenmodes becomes negligibleas compared to the lowest mode, and the first exit time follows approximately an exponentiallaw, τ τ> ≃ − ⟨ ⟩t t{ } exp ( ), with the mean τ⟨ ⟩ from equation (76).

We illustrate this behavior for the 3D case in figure 5(a), which presents the first threeeigenvalues λn as functions of κ. Note that the correction term to the asymptotic line for λ3 issignificant even for κ = 10 (see appendix A.2 for details). For comparison, figure 5(b) showsthe first three eigenvalues for the exterior problem discussed in the next subsection.

2.9. Exterior problem

For the exterior problem, when the process is started outside the interval −L L[ , ] (or outsidethe ball of radius L in higher dimensions), the FET is also referred to as the first passage timeto the boundary of this domain: τ = > <t X t Linf { 0 : | ( )| }. While the mean exit time andthe probability distribution can be found in a very similar way (see below), their properties arevery different from the earlier considered interior problem. For the sake of simplicity, we onlyconsider the centered harmonic potential (i.e., =F 00 ), although the noncentered case in onedimension can be treated similarly.

In one dimension, the domain ∪−∞ − ∞L L( , ) ( , ) is split into two disjointed sub-domains so that τ is in fact the first passage time to a single barrier, either at x = L (if startedfrom >x L0 ), or at = −x L (if started from < −x L0 ). This situation is described inappendix C.

Mean exit time. Following the steps of section 2.5, one obtains the mean exit time

∫ ∫τκ

=κ

κ−

∞− −L

Dr r e r r e

1d d , (79)r

r Ld r

r

d r2

1 11

2 21

0

012

1

22

where we imposed the Dirichlet boundary condition at =r L0 and the regularity condition atinfinity.

Figure 5. First three eigenvalues λn of the Fokker–Planck operator as functions of κ , forthe interior problem (a) and the exterior problem (b) in three dimensions. The timescaleL D2 is set to 1. On plot (a), crosses present the asymptotic relation (77) for the firsteigenvalue λ0 while thin solid lines indicate the asymptotic relation κn4 for highereigenvalues ( = …n 1, 2, ). At κ = 0, one retrieves the eigenvalues π +n( 1)2 2 forBrownian motion. On plot (b), thin lines indicate the asymptotic behavior (A.18) atsmall κ .

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

18

For even dimensions d, the change of integration variables yields the explicit formula

∑τκ

Γ κ

Γ= +

−−

=

− −−

⎡

⎣⎢⎢

⎤

⎦⎥⎥( )( )

( )( ) ( )L

Dr L

j jr L

42 ln 1 (80)r

j

d j

d

j2

0

1

12

2

02

d

0

2

(we use the convention that ∑ = ajn

j1 is zero if <n 1). For instance, the mean exit time in twodimensions is particularly simple:

τκ

= =( )L

Dr L d

2ln ( 2). (81)r

2

00

For odd d, repeated integration by parts yields

∫ ∑

∑

∑

τκ

πΓ

Γ

Γ

Γ κ

κ Γ κ

κ Γ κ

= +−

−+

−

+ −

− −

κ

κ

κ

κ

=

−

=

−

=

−

−

−

−

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟⎪⎪

⎧⎨⎪⎪

⎩⎪⎪

⎛

⎝

⎜⎜⎜⎜

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎞

⎠

⎟⎟⎟⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎫⎬⎭

( )

( )

( )

L

Dz e z

d

dj

jz

j

ed

j

e zd

j z

4

12 d erfc ( )

2

2

1

21

2

1

erfc2

erfc2

, (82)

r

zz

j

j

j

j

j d

z

j

j d

2

1

02

1

2

0

102 2

d

d

d

0

0 2

32

12

02

12

where =z r L0 0 , and = −z zerfc ( ) 1 erf ( ). Note that for d = 1, all terms vanish except theintegral.

For large r0 or large κ, the leading asymptotic term isκ

r Lln ( )L

D2 02

for all dimensions (forodd dimensions, this term comes from the integral). When r L0 approaches 1, the mean exittime vanishes as −c r L( 1)0 , where the prefactor c depends on κ and d.

When κ → 0, the mean exit time diverges:

τΓ

κ≃−

− ≠− −( )( )( )L

D dr L d

2( 2)1 ( 2) (83)r

d

d d2

20

2 20

(for d = 2, see equation (81)). This divergence is expected because, for the exterior problem,the mean exit time for Brownian motion is infinite in all dimensions, irrespectively of itsrecurrent or transient character.

Finally, the mean exit time for non-centered harmonic potential (i.e., ≠F 00 ) in onedimension reads for >x L0 as

∫τ πκ

=κ φ

κ φ

−

−

( )

( )L

Dz e z

2d erfc ( ). (84)x

x Lz

2

10

0 2

In the limit of large κ, two asymptotic regimes are distinguished:

(i) when φ < 1, the upper and lower limits of integration go to infinity so that the mean exittime behaves as

τκ

φφ

≃−

−L

D

x L1

2ln

1; (85)x

20

0

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

19

(ii) when φ > 1, the lower limit of integration goes to −∞, and the mean exit timeexponentially diverges as

τ πκ φ

≃−

κ φ−L

D

e

2 ( 1). (86)x

2 ( 1)

3 20

2

Both regimes are similar to that of the interior problem considered in section 2.5.

Probability distribution. The moment-generating function q r s˜( , )0 for the exterior problemsatisfies the same equation (63), with −s instead of λn, as q r s˜( , )0 from equation (72) for theinterior problem. In order to ensure the regularity condition at infinity (as → ∞r0 ), onereplaces M a b z( , , ) by the confluent hypergeometric function of the second kind (also knownas Tricomi function):

ΓΓ

ΓΓ

= −− +

+ − − + −−U a b zb

a bM a b z

b

az M a b b z( , , )

(1 )

( 1)( , , )

( 1)

( )( 1, 2 , ) (87)b1

(for integer b, this relation is undefined but can be extended by continuity; see appendix B.2).For >a 0, the functionU a b z( , , ) vanishes as → ∞z , in contrast to an exponential growth ofM a b z( , , ) according to equations (B.7) and (B.8). In turn, U a b z( , , ) exhibits non-analyticbehavior near z = 0, ≃ + + …Γ

ΓΓ

Γ−

− +− −U a b z z( , , ) b

a b

b

ab(1 )

( 1)

( 1)

( )1 , that limited its use for the

interior problem.The moment-generating function for the exterior problem is then

κ

κ= ⩾

κ

κ

( )( )

( )q r sU

Ur L˜( , )

, ,

, ,, (88)

sL

D

d r

L

sL

D

d0

4 2

4 2

0

202

2

2

in agreement with [39] (p. 581, 2.0.1).Denoting by αn the positive zeros of the equation

ακ

κ− =⎛⎝⎜

⎞⎠⎟U

d

4,

2, 0, (89)n

2

the inverse Laplace transform yields the FET probability density:

∑κ κ

κ=

−

′ −α

ακ

ακ

=

∞−

( )( )

q r tD

Le

U

U( , )

4, ,

, ,, (90)

n

Dt L

d r

L

d0 2

0

4 2

4 2

n

n

n

2 2

202

2

2

where ′U a b z( , , ) is the derivative with respect to a. Its integral over time is the survivalprobability:

∑ ακ

κ= −α

=

∞−

⎛⎝⎜

⎞⎠⎟S r t w e U

d r

L( , )

4,

2, , (91)

n

nDt L n

0

0

202

2n2 2

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

20

with

κ

α κ=

′ − ακ( )

wU

4

, ,. (92)n

nd2

4 2n2

Alternatively, one can use the eigenvalues λ α= D Ln n2 2 and the corresponding

eigenfunctions

β ακ

κ= −⎛⎝⎜

⎞⎠⎟u r

LU

dr L( )

4,

2, ( ) , (93)n

n

dn

2

22

where βn is the normalization factor:

∫βα

κκ= −κ−

∞− −

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥z z e U

dzd

4,

2, . (94)n

d z n2

1

12

2

22

The normalization factors βn diverge as κ → 0.Repeating the same steps as earlier, one retrieves the spectral representation (91) of the

survival probability with

βκ

ακ

κ= − +κ− ⎛

⎝⎜⎞⎠⎟w

eU

d

2 41,

2, . (95)n

n n2 2

The asymptotic behavior of eigenvalues as κ → 0 is discussed in appendix A.3.

2.10. Similarities and distinctions

In spite of the apparent similarities between the interior and the exterior problems, there is asignificant difference in the spectral properties of the two problems. This difference becomesparticularly clear in the limit κ → 0 when the harmonic potential is switched off (see figure 5).For the interior problem, the spectrum remains discrete and continuously approaches to thespectrum of the radial Laplacian. In this limit, one retrieves the classical results for Brownianmotion (e.g., α π→ +n( 1) 2n as κ → 0 in one dimension). In turn, the Laplace operator forthe exterior problem has a continuum spectrum so that the continuous transition from discreteto continuum spectrum as κ → 0 is prohibited. In particular, all eigenvalues λn vanish asκ → 0 (see appendix A.3). In other words, the spectral properties for infinitely small κ > 0and κ = 0 are drastically different. One can see that the asymptotic behavior of the eigen-values λn is quite different for the interior and the exterior problems.

3. Discussion

In this section, we discuss computational hints for confluent hypergeometric functions(section 3.1), three applications in biophysics and finance (section 3.2, 3.3 and 3.4), relationto the distribution of first crossing times of a moving boundary by Brownian motion(section 3.5), diffusion under quadratic double-well potential (section 3.6), and furtherextensions (section 3.7).

3.1. Computational hints

The probability distribution of first exit times involves confluent hypergeometric functionsM a b z( , , ) (for interior problem) and U a b z( , , ) (for exterior problem). For instance, the

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

21

eigenvalues of the Fokker–Planck operator are obtained through zeros αn of the equation

κ− =ακ

M ( , , ) 0d

4 2n2

or similar. As a consequence, one needs to compute these functions for

large α κ=a| | (4 )n2 . Although the series in the definition (42) of M a b z( , , ) converges for all

z, numerical summation becomes inaccurate for large a| |, and other representations of con-fluent hypergeometric functions are needed. In appendix B.2, we discuss an efficientnumerical scheme for rapid and accurate computation of M a b z( , , ) for large a| | and mod-erate z that relies on the expansion (B.14). Moreover, we show that this scheme is appropriatefor computing the derivative of M a b z( , , ) with respect to a that appears in equation (73) orsimilar after the inverse Laplace transform.

For noninteger b, the Tricomi function U a b z( , , ) is expressed through M a b z( , , ) byequation (87) that allows one to apply the same numerical scheme for the exterior problem inodd dimensions d. Although the Tricomi function for integer b can be obtained by con-tinuation, the derivation of its rapidly converging representation is more subtle. In practice,one can compute U a b z( , , ) for an integer b by the extrapolation of a sequence εU a b z( , , )computed for noninteger εb approaching b as ε → 0.

In the case of large z and moderate a| |, one can use integral representations of M a b z( , , )and U a b z( , , ) (see appendix B.2). The same algorithms can also be applied to computeparabolic cylinder function νD z( ) and Whittaker functions (see appendix B.1). The MATLABcode for computing both Kummer and Tricomi functions is available2.

3.2. Single-particle tracking

The Langevin equation (2) can describe the thermal motion of a small tracer in a viscousmedium. The Hookean force−kX t( ) incorporates the harmonic potential of an optical tweezerwhich is used to trap the tracer in a specific region of the medium [69]. Optical trappingstrongly diminishes the region accessible to the tracer and thus enables reduction of the fieldof view and increase of the acquisition rate up to few MHz [70–74]. At the same time,trapping affects the intrinsic dynamics of the tracer and may screen or fully remove itsfeatures at long times. The choice of the stiffness k is therefore a compromise between the riskof losing the tracer from the field of view (too small k) and risk of suppressing importantdynamical features (too large k). The FET statistics can then be used for estimating theappropriate stiffness due to a quantitative characterization of escape events. For instance, onecan choose the stiffness to ensure that the mean exit time strongly exceeds the duration ofexperiment, or that the escape probability is below a prescribed threshold.

Another interesting option consists of detecting events in which a constant force isapplied to the tracer. In living cells, such events can mimic the action of motor proteins thatattach to the tracer and pull it in one direction [75–79]. The presence of a constant forcefacilitates the escape from the optical trap, while higher fraction of escape events (as com-pared to the case without constant force) can be an indicator of such active transportmechanisms.

Originally, the idea of fast escape in the case of comparable Hookean and external forceswas used to estimate the force generated by a single protein motor [75]. A ‘trap and escape’experiment consisted in trapping a single organelle moving along microtubules at strongstiffness and then gradually reducing it until the organelle escapes the trap. Repeating suchmeasurements, one can estimate the ‘escape power’ kL as a measure of the driving force F0

when φ = ∼F kL( ) 10 , where L is the size of the trap. In this way, the driving force generatedby a single (presumably dynein-like) motor was estimated to be 2.6 pN [75]. This

2 See http://pmc.polytechnique.fr/pagesperso/dg/confluent/confluent.html

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

22

approximate but direct way of force measuring relies on the drastic change in the mean exittime behavior at φ = 1 according to equation (39).

Interestingly, these mechanisms can even be detected from a single trajectory. Whenthere is no constant force, the mean-square displacement (MSD) of a trapped tracer, Δ⟨ ⟩X( )2 ,is known to approach the constant level k T k2 B [52, 80, 81]. In other words, the square rootof the long-time asymptotic MSD determines the typical size τ= =ℓ k T k D2 2k kB of theconfining region due to optical trapping, with τ γ= kk . Setting =L qℓk , one gets κ = q2, i.e.,the dimensionless parameter κ can be interpreted as the squared ratio between the exit distanceL and the characteristic size of the trap ℓk. The above analysis showed that a tracer can rapidlyreach levels below or slightly above ℓk. However, significantly longer explorations areextremely improbable. In fact, according to equation (76), the mean exit time for κ ≫ 1 is

τ τ Γκ

κ≃ ≫κd e( 2)

2( 1), (96)k d0 2

where we set κ τ= =L qℓ D2k k . For large enough t (i.e., τ≫t k), the contributions of allexcited eigenstates vanish, and the survival probability exhibits a mono-exponential decay:

τ≃ − ⟨ ⟩S t t(0, ) exp ( )0 , where we replaced the smallest eigenvalue λ0 by τ⟨ ⟩1 0 for κ ≫ 1according to equation (77), while ≈w 10 as shown in appendix A.2. In the intermediateregime τ τ≪ ≪ ⟨ ⟩tk 0, the survival probability therefore remains close to 1.

A constant force F0 pulling the tracer from the optical trap strongly affects the mean exittime and the survival probability. The dimensionless parameter φ from equation (34) is theratio between the new stationary position x̂ of the trajectory and the exit level κ=L ℓk:

φκ

τκ

= = =F

kL

x

ℓ

F D

k T

ˆ 2

2. (97)

k

k0 0

B

For large φ, the mean exit time can be approximated according to equation (39) asτ τ τ φ⟨ ⟩ ≃ ≃φ

φ −lnk k0 1

, i.e., it becomes smaller than τk, and much smaller than the meanexit time from equation (96) without force. As expected, exit events would be observed muchmore often in the presence of strong constant force.

For a long acquired trajectory, one can characterize how often different levels arereached. Strong deviations from the expected statistics (given by the survival probability)

Figure 6. Two simulated trajectories of a spherical tracer submerged in water under theoptical trapping: (a) no constant force; (b) constant force =F 0.20 pN is appliedbetween 0.3 and 0.5 s (indicated by vertical lines). Horizontal dotted lines indicate thetypical trapping size ± =ℓ 91k nm, while dashed red line shows the stationary level x̂(equal to 0 for zero force and 200 nm for =F 0.20 pN). The other parameters areprovided in the text.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

23

would suggest the presence of a constant force. To illustrate this idea, we simulate the thermalmotion of a spherical tracer of radius μ=a 1 m submerged in water and trapped by an opticaltweezer with a typical stiffness constant = −k 10 6 Nm−1 [73, 74]. The Stokes relation impliesγ π η= ≈ −a6 1.88 · 100

8 kgs−1, from which the diffusion coefficient is

γ= ≈ −D k T 2.20 · 10B13 m2s−1 at T = 300 K (with η ≈ −100

3 kgm−1s−1 being the waterviscosity). The characteristic trapping time is τ γ= ≈k 18.8k ms, while the confinementlength is = ≈ℓ k T k2 91k B nm. Figure 6(a) shows one simulated trajectory of the tracer.According to equation (33), the mean exit times from the intervals −ℓ ℓ( , )k k and − ℓ ℓ( 2 , 2 )k k

are 27.2 ms and 517 ms, respectively. For a generated sample of duration 1 s, one observesmultiple crossings of levels ±ℓk and only a few crossings of levels ± ℓ2 k . For comparison, wegenerated another trajectory for which a constant force =F 0.20 pN (yielding φ κ= 2.20 )is applied between 0.3 s and 0.5 s (figure 6(b)). Since motor proteins exert forces that aretypically tenfold higher [75, 76], their effect is expected to be much stronger and thus easier todetect. The constant force reduces the mean exit times to 9.8 ms and 28 ms, i.e., by factors 2.8and 18, respectively. As expected, once the constant force is applied, the tracer tends to reachthe new stationary level =x̂ 200 nm so that the trajectory crosses the level ℓk and remainsabove this level for whole duration of the forced period. Once the force is switched off, thetrajectory returns to its initial regime with zero mean. One can see that the use of FETstatistics presents a promising perspective for the design and analysis of single-particletracking experiments, while Bayesian techniques can be further applied to get more reliableresults [82, 83]. Note that the FPT statistics have also been suggested as robust estimators ofdiffusion characteristics [84] (see also [26]). Another method relying on the time evolution ofthe tracer probability distribution was proposed for simultaneously extracting the restoring-force constant and diffusion coefficient [85].

At the same time, we emphasize that this perspective needs further analysis. First, wefocused on normal diffusion in a harmonic potential while numerous single-particle trackingexperiments evidenced anomalous diffusion in living cells and polymer solutions [73, 86–89].Several theoretical models have been developed to describe anomalous processes such ascontinuous-time random walks (CTRW), fractional Brownian motion (fBm), and generalizedLangevin equation [5, 6, 78, 79]. While an extension of the presented results is ratherstraightforward for CTRW (section 3.7), the FPT problems for non-Markovian fBm orgeneralized Langevin equation are challenging due to the lack of an equivalent Fokker–Planck formulation. Second, the quadratic profile is an accurate approximation for opticaltrapping potential only for moderate deviations from the center of the laser beam [69], whilethe spatial profile can be more complicated for strong deviations. In other words, an accuratedescription of the tracer escape may require more sophisticated analysis. Finally, the inferenceof constant forces from a single trajectory may present some statistical challenges becausedifferent escape events can be correlated.

3.3. Adhesion bond dissociation under mechanical stress

We briefly mention another biophysical example of bond dissociation. Adhesion betweencells or of cells to surfaces is mediated by weak noncovalent interactions. While a reversiblebond between two molecules can break spontaneously (due to thermal fluctuations), anexternal force is needed to rupture the multiple bonds linking two cells together [90]. Thedynamics of bond rupture can be seen as the first exit time problem in which exit or escapeoccurs when the intermolecular distance exceeds an effective interaction radius. Bell sug-gested application of the kinetic theory of the strength of solids to describe the lifetime of abond (i.e., the mean exit time) as

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

24

= −⎡⎣ ⎤⎦( ) ( )t t E r F k Texp , (98)b 0 b b 0 B

where Eb is the bond energy, rb is the range of the minimum of the binding free energy, F0 isthe applied external force per bond, and t0 is the lifetime at the critical force E rb b at which theminimum of the free energy vanishes [90]. This relation became a canonical description ofadhesion bond dissociation under force.

If the binding potential can be approximated as quadratic, then the lifetime of a bond isprecisely the mean exit time τ⟨ ⟩ of a harmonically trapped particle. In that case, the secondasymptotic relation in equation (39) implies the quadratic dependence on the force,τ⟨ ⟩ ∼ κ φ−e (1 )2

, where φ = F kr( )0 b , and κ = =E k T kr k T( ) (2 )b B b2

B . In other words,equation (98) is retrieved only for weak forces when the quadratic term φ2 can be neglected.However, in the regime where the bond is most likely to break, the applied force is large, andthe mean exit time may have completely different asymptotics (see, e.g., the last line ofequations (39) for φ > 1). This discrepancy was already outlined in [91], in which the casesof a harmonic potential and an inverse power law attraction were discussed, and in [92],which presented a molecular dynamics study of unbinding and the related analysis of first exittimes. Other effects such as the dependence of the bond strength and survival time on theloading rate, were investigated both theoretically and experimentally (see [91–96] andreferences therein).

3.4. Algorithmic trading

Algorithmic trading is another field for applications of FETs. In algorithmic trading, a set oftrading rules is developed in order to anticipate the next price variation of an asset from itsearlier (historical) prices [97]. Although the next price is random (and thus unpredictable),one aims to catch some global or local trends which can be induced by collective behavior ofmultiple traders or macroeconomic tendencies [98–100]. Many trading strategies rely on theexponential moving average p̄n of the earlier prices pk [101–104]

∑λ λ= −=

∞

−p p¯ (1 ) , (99)nk

kn k

0

where λ< ⩽0 1 characterizes how fast the exponential weights of more distant prices decay.The difference between the current price pn and the ‘anticipated’ average price p̄n,

∑δ λ λ≡ − = − − = −=

∞

− −p p r r p p¯ (1 ) (1 ) , ( ), (100)n n nk

kn k n n n

01

can be seen as an indicator of a new trend. For independent Gaussian price variations rn,writing δ λ δ λ= − + −+ +r(1 ) (1 )n n n1 1, one retrieves equation (5) for a discrete version ofan Ornstein–Uhlenbeck process, where μ λ λ= −x̂ (1 ) is related to the mean price variation

μ, θ λ= − −ln (1 ), and σ σ= λ θ

λ

−

− −0

(1 ) 2

1 (1 )2is proportional to the standard deviation (volatility)

σ0 of price variations.The indicator δn can be used in both mean-reverting and trend-following strategies. In the

mean-reverting frame, if δn exceeds a prescribed threshold L, this is a trigger to sell the asset atits actual (high) price, in anticipation of its return to the expected (lower) level p̄n in the nearfuture. Similarly, the event δ < −Ln triggers buying the asset. In the opposite trend-followingframe, the condition δ > Ln is interpreted as the beginning of a strong trend, and is thus thesignal to buy the asset at its actual price, in anticipation of its further growth (similarly forδ < −Ln ). In other words, the same condition δ > Ln (or δ < −Ln ) can be interpreted

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

25

differently depending on the empirical knowledge on the asset behavior. Whatever thestrategy is used, the statistics of crossings of the prescribed levels ±L is precisely the FETproblem. Theoretical results in section 2 can help to characterize durations between buyingand selling moments. In particular, the choice of the threshold L is a compromise betweenexecution of too frequent buying/selling transactions (i.e., higher transaction costs) at small Land missing intermediate trends (i.e. smaller profits) at large L. We also note that Ornstein–Uhlenbeck processes often appear in finance to model, e.g., interest rates (Vasicek model) andcurrency exchange rates [105, 106]. A general frame of using eigenfunctions for pricingoptions is discussed in [107].

3.5. First crossing of a moving boundary by Brownian motion

The first exit time problem can be extended to time-evolving domains [108–112]. Forinstance, one can investigate the first passage time of Brownian motion to a time-dependentbarrier L(t), τ = > =t X t L tinf { 0 : ( ) ( )}, or the first exit time from a symmetric ‘envelope’−L t L t[ ( ), ( )], τ = > =t X t L tinf { 0 : | ( )| ( )}. Although the survival probability S x t( , )0

satisfies the standard diffusion equation with the Dirichlet boundary condition, the boundaryL(t) evolves with time. For a smooth L(t), setting =S x t v z t( , ) ( , )0 with a new space variable

=z x L t( )0 yields

∂∂

= ∂ − ′ ∂v z t

t

D

L tv z t

L t

L tz v z t

( , )

( )( , )

( )

( )( , ), (101)z z2

2

with Dirichlet boundary condition ± =v t( 1, ) 0 at two fixed endpoints (here we focus on theexit time). Setting a new time variable =T L t Lln ( ( ) (0)), the above equation can also bewritten as the backward Fokker–Planck equation with time-dependent diffusion coefficient

= =′ ′

−

−D T D D( )L t L t

e

L L L L e

1

( ) ( ) (0) ( ( (0) ))

T

T1and a centered harmonic potential:

∂∂

= ∂ − ∂v z T

TD T v z T z v z T

( , )( ) ( , ) ( , ). (102)z z

2

In higher dimensions, the second derivative ∂2z is simply replaced by the radial Laplaceoperator ∂ + ∂−

rd

r r2 1 .

In general, the above equation does not admit explicit solutions. A notable exception isthe case of square-root boundaries, which have been thoroughly investigated [113–118]. Infact, when = +L t b t t( ) 2 ( )0 (with >b 0 and >t 00 ), one has ′ =L t L t b( ) ( ) so that D(T)is independent of T (or t). In other words, one retrieves the backward Fokker–Planck problem(17, 24) with =x̂ 0, γ =k 1, L = 1, and D replaced by D b. Its exact solution is given byequation (73) for the d-dimensional case:

∑ν

ν=

−

′ −ν

=

∞−

( )( )

( )q z T eM

M, 2

, ,

, ,, (103)

n

Tn

d bz

D

nd b

D

0

0

2 2 2

2 2

n

02

where = =z r L r bt(0) 20 0 0 0 denotes the rescaled starting point r0, and ν α κ= (4 )n n2 are

zeros of ν− =( )M , , 0d b

D2 2. Changing back T to t, one gets

∑ν

ν= +

−

′ −ν

=

∞− − ( )

( )( )p z t

tt t

M

M( , )

11

, ,

, ,. (104)

n

nd bz

D

nd b

D

00 0

01 2 2

2 2

n

02

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

26

This expression in a slightly different form was provided for d = 1 in [118]. Note also that theprobability << < W t c{sup (| | ) }t T t1 admits a similar expansion [119].

In addition, equation (72) yields

τ νν

ν+ = = − =

−

−ν ν ν− ( )

( )( )t t e q z

M

M( ) ˜ , 2

, ,

, ,, (105)T

d bz

D

d b

D

0 02

02 2

2 2

02

from which one retrieves

τν

+ =−

=νν

( )( )t

t

Mz( )

, ,at 0 , (106)

d b

D

00

2 2

0

which was reported for d = 1 in [114]. Note that the νth moment exists under the conditionν ν<R{ } 0, as is clearly seen from equation (104). In the special case b = D, the square-root

boundary = +L t b t t( ) 2 ( )0 grows in the same way as the root mean square of Brownian

motion ⟨ ⟩ =W Dt2t2 . Since ν = 10 at b = D, the mean exit time is infinite. More

generally, the mean exit time is infinite for broader envelopes ( ⩾b D) and finite for narrowerenvelopes ( <b D), as expected. The shift t0 plays a minor role of a time scale.

3.6. Quadratic double-well potential

The above spectral approach can be extended to more complicated trapping potentials. As anexample, we briefly describe diffusion under double-well (or bistable) piecewise quadraticpotential:

=+ ⩽

− + ⩾

⎧⎨⎪⎪

⎩⎪⎪

V xk x x x

k x x v x( )

1

2( ) , 0,

1

2( ) , 0,

(107)1 1

2

2 22

0

where two minima are located at −x1 and x2 (with >x 01 and >x 02 ), k1 and k2 are twospring constants, and = −v k x k x( )0

1

2 1 12

2 22 is a constant ensuring the continuity of the

potential at x = 0. The resulting Langevin equation remains linear, in contrast to other bistablepotentials such as a quartic potential (e.g., = + +V x ax bx cx( ) 4 2 ). The diffusive dynamicsunder double-well potentials were thoroughly investigated by using general theoretical tools(e.g. Kramers’ theory [66, 67] or WKB approximation [120–122]) and exactly solvablemodels (see [123–129] and references therein).

For each semi-axis, an eigenfunction satisfies equation (40) with the proper ki. However,neither the Kummer, nor the Tricomi functions are appropriate to represent the solution in thiscase. In fact, the Kummer function M a z( , 1 2, )2 rapidly grows at infinity, while the Tricomifunction U a z( , 1 2, )2 behaves as − + …π

Γπ

Γ+z| |

a a( 1 2)

2

( )for small z, i.e., its derivative is

discontinuous at 0. A convenient representation can still be obtained as a linear combinationof two Kummer functions, in which the rapid growth of these functions is compensated. Thisis precisely the case of parabolic cylinder functions νD z( ) and −νD z( ) which vanish as → ∞z(resp., → −∞z ) but rapidly grow as → −∞z (resp. → ∞z ) unless ν is a non-negativeinteger [see equations (B.3), (D.2) and (D.3)]. An eigenfunction can therefore be written as

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

27

κ

κ=

− + ⩽

− ⩾

κν

κν

+

−

⎧⎨⎪⎩⎪

( )( )

( )

( )

( )

( )u x

c e D x x x

c e D x x x( )

2 1 , 0,

2 1 , 0,(108)

x x

x x

11 2

1 1

21 2

2 2

1 12

1

2 22

2

where κ = k x k T(2 )i i i2

B , ν λ κ= x D(2 )i i i2 , and λ, c1, c2 are determined by normalization and

two interface conditions at x = 0. The continuity of the eigenfunction at x = 0 can be satisfiedby choosing

β κ β κ= − = −κν

κν( ) ( )c e D c e D2 , 2 ,1

22 2

21

22

11

where β is a normalization constant.The second interface condition is deduced from the orthogonality of eigenfunctions with

two weights w1,2 from equation (11) for positive and negative semi-axes,

κ= − ±⎡⎣ ⎤⎦( )( )w x x x( ) exp 1 1 ,i i i2

where plus (resp., minus) corresponds to i = 1 (resp., i = 2). The orthogonality imposes theinterface condition

′ − ′ =− +( )( )Dw u Dw u(0) 0 (0) 0 0, (109)1 2

where the same diffusion coefficient D is assumed on both sides. Since =w (0) 1i , oneretrieves the standard flux continuity equation, ′ = ′− +u u(0 ) (0 ), yielding an equationdetermining the eigenvalues λ:

κ κ κ κ κ

κ κ κ κ κ

− − + −

+ − − + − =

ν ν ν

ν ν ν

+

+

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

( ) ( ) ( )( ) ( ) ( )

x D D D

x D D D

2 2 2 2 2

2 2 2 2 2 0,

1 1 2 2 2 1 2

2 2 1 1 1 1 1

1 2 2

2 1 1

where we used the identity = −ν ν ν∂∂ +D z D z D z( ) ( ) ( )z

z

2 1 , and λ appears in ν λ κ= x D(2 )i i i2 .

The smallest eigenvalue λ = 0 corresponds to the steady state.The normalization constant β is found according to

∫

∫

βκ

κ

κ

κ

=−

+−

κ κ ν

κν

ν

κν

− +−

∞

−

∞

⎡

⎣⎢⎢

⎤

⎦⎥⎥

( )

( )

ex D

z D z

x Dz D z

2

2d ( )

2

2d ( ) , (110)

21

22

1 2

2

22

1

2 2

2

1 22

11

1

22

in which both integrals can be partly computed by using the identity [130]

∫ π ψ ψ

Γ ν=

− −

−ν

ν ν∞ −( ) ( )z D zd ( )

2 ( ), (111)

0

23 2

12 2

where ψ Γ Γ= ′z z z( ) ( ) ( ) is the digamma function. The lowest eigenfunction correspondingto λ = 00 , is constant, β=u x( )0 0, with

β π κ

κ

κ

κ=

++

+κ κ−

⎡

⎣⎢⎢

⎤

⎦⎥⎥

( ) ( )( ) ( )x e x e

2

1 erf 1 erf.0

2 1 1

1

2 2

2

1 2

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

28

As a consequence, one retrieves the equilibrium Boltzmann–Gibbs distribution,β= ∞ = =p x p x x u x u x w x w x( ) ( , | , 0) ( ) ( ) ( ) ( )eq 0 0 0 0 0

2 .Figure 7 illustrates the evolution of the probability density p x t x t( , | , )0 0 for diffusion

under quadratic double-well potential with two minima at ±1 (i.e., = =x x 11 2 ) and twodifferent dimensionless strengths: κ = 21 and κ = 12 . One can see how the initial Diracdistribution, concentrated at =x 20 (figure 7(a)) or = −x 20 (figure 7(b)) and shown by adashed vertical line, is progressively transformed into the equilibrium distribution p x( )eq(shown by a black solid line). Other diffusion characteristics can be deduced from theprobability density.

3.7. Further extensions

The spectral approach is a general tool for computing FETs and other first-passage quantities.We briefly mention four straightforward extensions.

(i) In one dimension, one can easily derive the splitting probability H x( )0 , i.e., theprobability to exit from one endpoint (e.g., =x L0 ) before the other ( = −x L0 ). Thesplitting probability is governed by the stationary equation = H x( ) 0x

*00

so that H x( )0

is given by a general solution in equation (45) with α = 0. Two constants c1 and c2 areset by boundary conditions =H L( ) 1 and − =H L( ) 0, from which

κ φ κ φ

κ φ κ φ=

− + +

− + +( )( ) ( )

( ) ( )H x

i x L i

i i( )

erf erf (1 )

erf (1 ) erf (1 ), (112)0

0

where we used =M b z(0, , ) 1 and = πM z(1 2, 3 2, ) i z

i z

erf ( )

2. Note that this

expression can be recognized in equation (33) for the mean exit time.(ii) The Dirichlet boundary conditions, ± =q L t( , ) 0, were imposed on the FET probability

density at both endpoints in order to stop the process whenever it exits from the interval.One can consider other boundary value problems, e.g., with one reflecting endpoint orone/two semi-reflecting points. In this case, the Dirichlet boundary condition at one orboth endpoints is replaced by Neumann or Robin boundary conditions [46]. For instance,

Figure 7. Evolution of the probability density p x t x t( , | , )0 0 for diffusion underquadratic double-well potential (sketched by black dotted line) with two minima at ±1(i.e. = =x x 11 2 ), and κ = 21 , κ = 12 . Dashed vertical line indicates the starting point at

=t 00 : =x 20 (a) and = −x 20 (b). Symbols represent normalized histograms ofarrival positions obtained by Monte Carlo simulations of an adapted version ofequation (5) (with time step δ = −10 3 and 105 sample trajectories), while lines show thespectral decomposition (18) with 50 terms. We set D=1.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

29

the condition =∂∂

q x t( , ) 0x 0

0at = −x L0 describes the reflecting barrier at −L. The

Robin boundary condition, + =∂∂

q x t hq x t( , ) ( , ) 0x 0 0

0, allows one to consider partial

absorptions/reflections for modeling various transport mechanisms on the boundary andto switch continuously between Neumann (pure reflections) and Dirichlet (pureabsorptions) cases by varying h from 0 to infinity [131–137]. The solution can beobtained in the same way.

(iii) The first passage time to a single barrier can be deduced from the first exit time from aninterval by sending one endpoint to infinity (see appendix C).

(iv) A straightforward extension of the spectral approach allows one to deduce FETs ofcontinuous-time random walks (CTRW) [5, 6]. In this model, long stalling periodsbetween moves result in anomalous subdiffusion, when the mean-square displacementevolves sublinearly with time: ⟨ − ⟩ ≃ α

αX t X D t( ( ) (0)) 22 , with the exponent α< <0 1and the generalized diffusion coefficient αD . The same derivations can be formallyrepeated for the fractional Fokker–Planck equation governing the survival probability ofCTRWs. In practice, it is sufficient to replace s D by α

αs D in the Laplace domain that inthe time domain yields the replacement of exponential functions λ− texp ( )n by Mittag–Leffler functions λ−α α

αE D t D( )n in spectral decompositions such as equation (23) orsimilar. As expected for CTRWs, the mean exit time diverges due to long stalling periodswhile the survival probability exhibits a power law decay α−t at long times instead of theexponential decay for normal diffusion.

Conclusion

We revised the classical problem of finding first exit times for harmonically trapped particles.Although the explicit formulas for the moment-generating function ⟨ ⟩τ−e s can be found instandard textbooks (e.g. [39]), the computation of the probability density and the survivalprobability through the inverse Laplace transform requires substantial analysis of confluenthypergeometric functions. For didactic purposes, we reproduced the main derivation stepsand resulting spectral decompositions that involve the eigenvalues and eigenfunctions of thegoverning Fokker–Planck operator. We also provided explicit formulas for the mean exit timeand discussed its asymptotic behavior in different limits. We considered the general case ofnoncentered harmonic potential in one dimension (Ornstein–Uhlenbeck process with nonzeromean) and the centered harmonic potential in higher dimensions (radial Ornstein–Uhlenbeckprocess). Both interior and exterior problems were analyzed.

After revising this classical problem, we discussed some practical issues. First, wedescribed a rapidly converging series representation of confluent hypergeometric functionswhich is particularly well suited for rapid numerical computation of eigenvalues and eigen-functions of the governing Fokker–Planck operator. Second, we showed how the mean exittime and the survival probability can be used for the analysis of single-particle trackingexperiments with optically trapped tracers. The derived formulas allow one to choose theappropriate value of the optical tweezers’ stiffness and to detect in acquired trajectories theactive periods with nonzero force exerted by motor proteins. Third, we mentioned the relationof the first exit time problem to the dynamics of bond dissociation under mechanical stress,which plays an important role in cell adhesion and motility. Fourth, we considered anapplication of FETs for algorithmic trading in stock markets in which buying or sellingsignals are triggered when the difference between the current and anticipated prices exceeds aprescribed threshold. In a first approximation, these events correspond to exits of an Ornstein–

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

30

Uhlenbeck process from an interval so that the FET statistics can be used to estimate strategy-holding periods and to choose the appropriate threshold that ensures the desired transactionrate. Fifth, we mentioned the relation to the distribution of first crossing times of a movingboundary by Brownian motion. Finally, we discussed several extensions of the spectralapproach, including diffusion under quadratic double-well potential and anomalous diffusion.

Acknowledgments

The author acknowledges partial support by the grant ANR-13-JSV5-0006-01 of the FrenchNational Research Agency.

Appendix A. Limit of large κ

A.1. Mean exit time

For large κ or φ, equation (33) is not appropriate for numerical computation of the mean exittime because integrals and error functions are exponentially large. Re-arranging these terms,one can rewrite equation (33) as

∫ ∫

τ πκ

κ φκ φ

κ φκ φ

=+

−+

+−+

× −

κφ κ

κφ

κ φ

κ φ

κ φ

κ φ

− + − −

−

−

+

−

+

⎧

⎨⎪⎪

⎩⎪⎪

⎫⎬⎭

( )( )

( )( )

( )( )

( ) ( )

L

D

eD z

D

eD

D

z e z z e z

2

1( )

(1 )

1(1 )

(1 )

d erfc ( ) d erfc ( ) , (A.1)

z

z z

z

z

z

22 1 1 0

4

1

(1 ) (1 )

0

0 02

2

0

2

where =z x L0 0 , and D(x) is the Dawson function:

∫= −D x e t e( ) d , (A.2)xx

t

0

2 2

which is related to the error function of an imaginary argument as

π=ix

ie D xerf ( )

2( ). (A.3)x2

For large x, the Dawson function decays as

≃ + + + …D xx x x

( )1

2

1

4

3

8. (A.4)

3 5

The relation (A.1) allows one to compute the mean exit time in the limit of large κ and/orφ. In fact, since the Dawson function vanishes for large argument, the ratio in front of the firstintegral in equation (A.1) becomes exponentially close to 1 so that

∫τ πκ

≃κ φ

κ φ

−

−

( )

L

Dz e z

2d erfc ( ). (A.5)z

zz

2

1

( )

0

0 2

Three situations can be considered separately.

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

31

(i) If φ > 1, the upper and lower limits of the above integral are positive and large, so that

τκ

φφ

κ≃−−

≫L

D

z1

2ln

1( 1), (A.6)z

20

0

where we used the asymptotic relation

∫ π≃ ≫z e z

b aa bd erfc ( )

ln ( )( , 1). (A.7)

a

bz2

Note that equation (A.6) is accurate already for φ ≳ 2 (and κ ⩾ 1).(ii) If φ< <0 1 but κ → ∞, the lower limit goes to −∞, and the integral exponentially

diverges:

τ πκ φ

κ≃−

≫κ φ−L

D

e

2 (1 )( 1) (A.8)0

2 (1 )

3 2

2

(here, the starting point is set to 0, but the result holds for all z0 not too close to 1). Thisrelation is valid for any φ< <0 1. Setting formally φ = 0, one gets the relation that istwice larger than the asymptotic equation (39) derived for φ = 0. The missing factor 2can be retrieved from the ratio in front of the first integral in equation (A.1). Thedifference between the cases φ = 0 and φ > 0 (small but strictly positive) can also beexplained by the following argument. For the nonsymmetric case (φ > 0), the rightendpoint =x L0 is closer to the minimum position x̂ than the left endpoint = −x L0 .When κ is large, the probability of large deviations from x̂ rapidly decays with distance,so that the probability of exiting through the left endpoint is exponentially smaller thanthat from the right endpoint. In other words, the above relation essentially describes themean exit time from the right endpoint. In turn, when φ = 0 (and thus =x̂ 0), bothendpoints are equivalent, which doubles the chances to exit and thus twice reduces themean exit time.

(iii) In the marginal case φ = 1, the integral in equation (A.5) grows logarithmically with κ.One can split the integral by an intermediate point ≫z̄ 1 so that

∫ ∫ πκ

+ ≃−κ − ( )( )

z e z z e zz

c zd erfc ( ) d erfc ( )

1ln

1

( ¯),

zz

z

zz

0

¯

¯

1 02 0 2

where

∫π≡ − → … → ∞⎜ ⎟⎛⎝

⎞⎠c z z z e z z( ¯) ¯ exp d erfc ( ) 0.375 ( ¯ ).

zz

0

¯ 2

We get therefore

τκ

κκ≃

−…

≫( )L

D

z1

2ln

1

0.375( 1). (A.9)z

2 00

This asymptotic relation is accurate starting from κ − ≳z(1 ) 20 .

A.2. Eigenvalues (interior problem)

For large κ, we search for positive solutions αn of the equation κ− =ακ( )M b, , 0

4n2

in the

form: α κ ε= −n(4 )n2 , where ε is a small parameter, and = …n 0, 1, 2, One gets then

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

32

∑

∑

ε κ κ

ε κ ε

= − + ≃ − − + … − + −

+ − − + − +

=

= +

∞

( )

M n bn n n j

b j

nn j

b jO

0 ( ; ; )( )( 1) ( 1)

!

( 1) !( 1) !

!,

j

n j

j

n

j n

j

j

0( )

1( )

2

from which the small parameter ε can be determined as

ε ≃ −−

S

n S( 1) !, (A.10)

n1

2

where S1 and S2 denote two above sums. The second sum can be written as

∑ ∑κ Γ κΓ

= − + − =+ + + + … + += +

∞

=

∞ + +S

n j

b jb

b j n j j n( 1) !

!( )

( 1 )( 1) ( 1 ).

j n

j

jj

j n

2

1( )

0

1

This expression can be obtained by integrating +n 1 times the Mittag–Leffler functionκ+ +E ( )b n1, 1 which asymptotically behaves as κ κ κ≃ +κ

+ +− −E e O( ) (1 (1 ))b n

b n1, 1 as κ ≫ 1.

Since the integration does not change the leading term, one concludes that

Γ κ κ κ≃ + ≫κ− −S b e O( ) (1 (1 )) ( 1).b n2

Keeping the highest-order term in the first sum, κ≃ −S b( 1)n n n1

( ), one gets

ε κΓ

≃ −+

κ+

−n b n

e! ( )

,b n2

from which we obtain the asymptotic behavior of the positive solution αn as κ ≫ 1:

α κ κΓ

≃ ++

= …κ+ −⎡

⎣⎢⎤⎦⎥n

e

n b nn4

! ( )( 0, 1, 2, ). (A.11)n

b n2

2

In particular, the smallest solution α0 exponentially decays with κ,

α κΓ

κ≃ ≫κ+

−b

e4

( )( 1), (A.12)

b

02

1

while the other eigenvalues grow linearly with κ:

α κ κ≃ ≫ = …n n4 ( 1, 1, 2, ), (A.13)n2

and the first-order correction ε decays exponentially fast. This asymptotic behavior can berelated to equidistant energy levels of a quantum harmonic oscillator (see appendix D).

Since α0 rapidly vanishes, the first eigenfunction approaches the unity:

κ− →ακ( )zM , , 1d

4 220

2

. As a consequence, the normalization constant is simply

β κ Γ≈ d2 ( 2)d02 2 so that ≃w 10 , because Γ=b z b zM(1, , ) ( )E ( )b1, .

A.3. Eigenvalues (exterior problem)

For the exterior problem, we consider the asymptotic behavior of solutions of

κ− =ακ( )U b, , 0

4

2

as κ → 0. For noninteger b, one can use equation (87) to write in the

lowest order in κ

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

33

ακ

κ Γ

Γ

Γ

Γκ= − ≃ −

− −+ −

−ακ

ακ

−⎛⎝⎜

⎞⎠⎟ ( ) ( )

U bb

b

b0

4, ,

(1 )

1

( 1). (A.14)b

2

4 4

12 2

For <b 1, κ −b1 is a small parameter so that the first term has to be small. Settingε− − = − +α

κb n1

4

2

(with = …n 0, 1, 2, ) one gets

ε ΓΓ Γ

κ= − −− − −

− −b

n b n b( 1)

( 1)

! ( 1 ) (1 ), (A.15)n b1 1

from which

α κ ΓΓ Γ

κ≃ − + + − −− − −

+ …−⎛⎝⎜

⎞⎠⎟b n

b

n b n b4 1

( 1) ( 1)

! ( 1 ) (1 ). (A.16)n

nb2 1

In turn, if >b 1, κ −b1 is a large parameter so that the second term has to be small. Settingε− = − +α

κn

4

2

, one gets

ε ΓΓ Γ

κ= − −− − −

− −b

n b n b( 1)

(1 )

! (1 ) ( 1), (A.17)n b1 1

from which

α κ ΓΓ Γ

κ≃ + − −− − −

+ …−⎛⎝⎜

⎞⎠⎟n

b

n b n b4

( 1) (1 )

! (1 ) ( 1). (A.18)n

nb2 1

For integer b, the analysis is more subtle and is beyond the scope of this paper. We justchecked numerically that α κ∝ b

02 as κ → 0 for b = 1 and b = 2 that corresponds to

dimensions d = 2 and d = 4.

Appendix B. Confluent hypergeometric functions

For the sake of completeness, we summarize selected relations between special functionsoften used to describe first passage times of Ornstein–Uhlenbeck processes (see [68] fordetails). After that, we describe a rapidly converging representation of confluent hypergeo-metric functions.

B.1. Relations

The Kummer confluent hypergeometric function =M a b z F a b z( , , ) ( ; ; )1 1 defined inequation (42) satisfies the Kummer equation:

″ + − ′ − =zy b z y ay( ) 0. (B.1)

For =b 1 2, this equation is also related to Weberʼs equation

″ − + =( )y z c y4 0, (B.2)2

which has two independent solutions: +−e M c z( 2 1 4, 1 2, 2)z 4 22(even) and

+−ze M c z( 2 3 4, 3 2, 2)z 4 22(odd). These solutions are often expressed through the

parabolic cylinder function νD z( ), which satisfies equation (B.2) with ν = − −c 1 2:

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

34

πν Γ ν

πν

πν Γ ν

πν

=

+

−

+

+

− +

ν ν

ν

−−

− +−

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠ ⎛

⎝⎜⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠ ⎛

⎝⎜⎞⎠⎟

D z e Mz

e zMz

( )cos

2

1

2

2 2,

1

2,

2

sin2

2

2

2 2

1

2,

3

2,

2(B.3)

z

z

24

2

( 1) 24

2

2

2

ν= −ν −⎛⎝⎜

⎞⎠⎟e U

z2

2,

1

2,

2(B.4)z2 4

22

(the last relation is only valid for ⩾R z{ } 0).The confluent hypergeometric functions M a b z( , , ) andU a b z( , , ) are also related to the

Whittaker functions M z( )a b, and W z( )a b, [68]

= + − +

= + − +

− +

− +

M z e z M b a b z

W z e z U b a b z

( ) (1 2 , 1 2 , ),

( ) (1 2 , 1 2 , ).

a bz b

a bz b

,2 1 2

,2 1 2

The following relations help to analyze the Brownian motion limit [39]

κκ Γ+ = +

κ→⎜ ⎟⎛⎝

⎞⎠

( )M

ab x b

I xa

xalim

4, 1, 2 ( 1)

( ), (B.5)b b

b0 2

κ Γκ κ

κ+ =κ→

−⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )aU

ab x

K xa

x alim

4 4, 1, 2

( ), (B.6)b b b

b0

12

where νI z( ) and νK z( ) are the modified Bessel functions of the first and second kind,respectively. The asymptotic expansions for large z| | (and fixed a and b) are [68] (section13.5):

∑

∑

ΓΓ

ΓΓ

≃ − − +

+−

+ −−

+π

−

=

−−

± −

=

−−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

( )

( )

M a b ze z b

a

b a a

n zO z

e z b

b a

a a b

n zO z

( , , )( )

( )

( ) (1 )

!

( )

( )

( ) (1 )

! ( ), (B.7)

z a b

n

N n n

nN

ia a

n

N n n

nN

0

1 ( ) ( )

0

1 ( ) ( )

1

1

2

2

∑ π≃ + −−

+ <−

=

−−

⎛⎝⎜⎜

⎞⎠⎟⎟( )U a b z z

a a b

n zO z z( , , )

( ) (1 )

! ( )( arg( ) 3 2), (B.8)a

n

N n n

nN

0

1 ( ) ( )

where the upper (resp., lower) sign in the second line is taken if π π− < <z2 arg( ) 3 2 (resp.,π π− < ⩽ −z3 2 arg( ) 2), and N, N1, and N2 are truncation orders.

B.2. Computation

Series representations. The computation of the Kummer function M a b z( , , ) by directseries summation in equation (42) is not convenient for large a| |. For this case, two equivalentrepresentations were proposed:

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

35

(i)

∑Γ=−

−−

=

∞ − +

− +

( )( )

M a b z b e A zJ z b a

z b a( , , ) ( ) 2

(2 4 )

(2 4 ), (B.9)z b

n

nn

b n

b n2 1

0

1

1

where the coefficients An are defined by

= = = = − + + −− −A A A b nA n b A a b A1, 0, 2, ( 2 ) (2 )n n n0 1 2 2 3

(see [68], section 13.3.7, and [138], section 4.8). Note that the coefficients An depend ona and grow with a| |.

(ii)

∑Γ=−

−−

=

∞ − +

− +

( )( )

M a b z b e p b zJ z b a

z b a( , , ) ( ) 2 ( , )

(2 4 )

(2 4 ), (B.10)z b

nn

b n

b n2 1

0

1

1

where p b z( , )n are the Buchholz polynomials in b and z (see [139], section 7.4). Thesepolynomials are less explicit than the coefficients An, but they are independent of a.Consequently, this representation is particularly convenient for large a| |.

The recurrence relations for the Buchholz polynomials were derived in [140]:

∑==

−⎜ ⎟⎛⎝

⎞⎠p b z

iz

n

n

kf b g z( , )

( )

! 2( ) ( ), (B.11)n

n

k

n

k n k0

[ 2]

2

where the polynomials fk(b) and gk(z) are defined recursively by

∑= − − −−

==

− −−

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟f b

b k

j

B

k jf b f b( )

21

2 1

2

4( ), ( ) 1, (B.12)k

j

k k jk j

j0

12( )

0

∑= − −+

==

− ++

− −

⎛⎝⎜

⎞⎠⎟g z

iz k

j

B

jg z g z( )

4

1

2

4

1( ), ( ) 1, (B.13)k

j

k jj

k j0

[( 1) 2] 12( 1)

2 1 0

and B2j are the Bernoulli numbers. Using the recurrence relations between Bessel functions,= +ν

ν ν ν− +J x J x J x( ) ( ) ( )x

21 1 , one can express

= +− + −J x P x J x Q x J x( ) (1 ) ( ) (1 ) ( ),b n n b n b1 1

where the polynomials Pn(z) and Qn(z) are defined recursively

= = = − + −= = = − + −

+ −

+ −

P z P z P z b n zP z P z

Q z Q z Q z b n zQ z Q z

( ) 1, ( ) 0, ( ) 2( 1 ) ( ) ( ),

( ) 0, ( ) 1, ( ) 2( 1 ) ( ) ( ).n n n

n n n

0 1 1 1

0 1 1 1

We get therefore the following expansion that rapidly converges for large x and moderate z:

∑= +=

∞

−

⎡⎣⎢

⎤⎦⎥M a b z e p b z F x

P x

xG x

Q x

x( , , ) ( , ) ( )

(1 )( )

(1 ), (B.14)z

nn b

nn b

n

n2

01

where = −x z b a(2 4 ) , and

Γ Γ= =− −−

− −F x b x J x G x b x J x( ) ( )2 ( ), ( ) ( )2 ( ). (B.15)bb b

b bb b

b1 1

11

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

36

In particular, for =b d 2, one has

−

d F x G x

x x xJ x J x x

x x x x x x

( ) ( )1 cos ( ) sin ( )2 ( ) ( )

3 sin ( ) (sin ( ) cos ( )) (B.16)

b b

0 1

3

The above recursive relations allow one to compute rapidly the polynomials p b z( , )n , P x(1 )n

and Q x(1 )n . The series (B.14) can be truncated after 5–10 terms when az| | is large enoughand z is not too large (see [140] for several examples).

According to equation (87), one can apply this method to compute the Tricomi confluenthypergeometric function U a b z( , , ) for noninteger b. Other series expansions for U a b z( , , )are discussed in [141, 142]. For integer b, one can substitute ε= +εb b into equation (87)and then take the limit ε → 0. This extension by continuity yields [68]

∑

∑

Γ

Γ

ψ ψ ψ

= −− − +

+ − − +−

+ + − + − + = …

=

− − +

=

∞⎪

⎪

⎫⎬⎭

U a b zb a b

M a b z z

b

a

a b z

k b

a z

k ba k k b k b

( , , )( 1)

( 1) ! ( 1){ ( , , ) ln

( 2)!

( )

( 1)

! (2 )

!( ( ) (1 ) ( )) , 1, 2, ,

b

k

b k k b

k

k

k k

k

0

2 ( ) 1

( )

0

( )

( )

where ψ Γ Γ= ′z z z( ) ( ) ( ) is the digamma function, and the intermediate sum is omitted forb=1. In practice, one can apply the above numerical scheme to rapidly compute εU a b z( , , )through εM a b z( , , ) with several noninteger εb approaching the integer b, and then toextrapolate them in the limit →εb b.

Taking the derivative of equation (B.10) with respect to a and using the relation′ = −ν

νν ν+J x J x J x( ) ( ) ( )

x 1 , one obtains

∑Γ∂∂

=−

−=

∞ +

+

( )( )a

M a b z b e z p b zJ z b a

z b a( , , ) ( ) 2 ( , )

(2 4 )

(2 4 )(B.17)z b

nn

b n

b n2

0

or, equivalently,

∑∂∂

= +=

∞+

++⎡

⎣⎢⎤⎦⎥a

M a b z ze p b z F xP x

xG x

Q x

x( , , ) 2 ( , ) ( )

(1 )( )

(1 ), (B.18)z

nn b

n

n bn

n2

0

1

1

1

with = −x z b a(2 4 ) . This expression allows one to rapidly compute the coefficients wn inthe spectral representation of the survival probability. Similar relations can be derived for∂∂

U a b z( , , )a

using equation (87) for noninteger b. Finally, one can also apply these formulas

for computing the parabolic cylinder function νD z( ) and its derivativeν ν∂∂

D z( ), which are usedto characterize the first passage time to a single barrier (appendix C).

Integral representations. The above scheme is convenient for large a| | and moderate z| |.However, if a| | is moderate while z| | is large, the numerical convergence of the above series isslowed down due to a rapid growth of Buchholz polynomials with z. In addition, thecomputation of the Tricomi function U a b z( , , ) as a linear combination (87) of two largeKummer functions can result in significant round-off errors at large z. In this case, one can

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

37

apply a different technique that relies on integral representations of confluent hypergeometricfunctions.

For the Kummer function M a b z( , , ), one can use the following integral representationfor − >R b a{ } 03

∫ΓΓ

=−

∞− −

−

−− ( )M a b z

e z b

b at e t J zt( , , )

( )

( )d 2 . (B.19)

zt a

b0

1

bb

12 1

2

This representation is convenient for computing eigenvalues and eigenfunctions becauseα κ= − <a (4 ) 02 and = >b d 2 0.

The Tricomi function U a b z( , , ) has an integral representation for positive a [68]

∫Γ= + > >

∞− − − − R RU a b z

at e t t a z( , , )

1

( )d (1 ) ( { } 0, { } 0). (B.20)zt a b a

0

1 1

When a is negative, one can use the recurrence relation to increase a:

− + − − + + − + =U a b z b a z U a b z a a b U a b z( 1, , ) ( 2 ) ( , , ) ( 1 ) ( 1, , ) 0. (B.21)

Applying this relation repeatedly, one gets

= + + + +U a b z p a b z U a n b z q a b z U a n b z( , , ) ( , , ) ( , , ) ( , , ) ( 1, , ), (B.22)n n

where the polynomials p a b z( , , )n and q a b z( , , )n can be rapidly computed throughrecurrence relations:

= − − + − == − + + + − =− −

−

p a b z q a b z b a n z p a b z p

q a b z a n a n b p a b z q

( , , ) ( , , ) ( 2( ) ) ( , , ), 1,

( , , ) ( )( 1 ) ( , , ), 0.n n n

n n

1 1 0

1 0

Choosing n such that + >a n 0, one can express U a b z( , , ) in terms of +U a n b z( , , ) and+ +U a n b z( 1, , ) which are found by numerical integration of equation (B.20). If z is too

large, it is convenient to divide each recurrence relation by z and to consider them aspolynomials of z1 . The resulting value can be compared with the asymptoticexpansion (B.8).

Appendix C. First passage time to a single barrier

The first passage times (one-barrier problem) for harmonically trapped particles have attractedmore attention than the first exit times (two-barrier problem) [29, 32–36]. In general, the firstpassage time τℓ to a single barrier at >ℓ 0 in one dimension can be found following the stepsfrom section 2.6. In practice, these results can be deduced from the FET statistics. If thestarting point x0 lies on the right to ℓ (i.e., >x ℓ0 ), this problem is equivalent to the exteriorproblem to reach the interval −ℓ ℓ[ , ] from outside (see section 2.9). In turn, if < <x ℓ0 0 , theFPT to a single barrier ℓ can be deduced from the FET from the interval −a ℓ[ , ] in the limit

→ ∞a .In order to illustrate this point, we focus on the moment-generating function q x s˜( , )0

given by equation (60), with φ= −ℓ L (1 ) and φ= +a L (1 ). Setting φ ε− =1 , we con-sider the limit ε → 0, for which ε= → ∞L ℓ and φ ε− = →1 0, so that → ∞a while ℓ iskept fixed. The asymptotic behavior of functions α κm ,

(1,2) from equation (43) as ε → 0 can beeasily found:

3 See http://dlmf.nist.gov/13.16.E3

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

38

φ

φ ε

φ ΓΓ

ε

φ ΓΓ

ε

− ≃

− ≃ +

− − ≃

− − ≃ −+

α κ

α κ

α κε

α κε

−

−

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

( )

m M a y

m M a y

ma

y e

ma

y e

(1 ) ,1

2, ,

(1 )1

2,

3

2, ,

( 1 )(1 2)

( )4 ,

( 1 ) 2(3 2)

( 1 2)4 ,

a y

a y

,(1)

,(2)

,(1) 2 1 2 4

,(2) 2 1 4

2

2

where we replaced α and κ by −Ds L2 and γkL D(2 )2 , introduced short notations γ=a s k(2 )and γ=y kℓ D(2 )2 , and used the asymptotic relation (B.7) for the last two functions.Substituting the above expressions into equation (60), one deduces in the limit ε → 0

=+ +

+ +

ΓΓ

ΓΓ

+

+

( ) ( )( ) ( )

q x sM a y y M a y

M a y y M a y˜( , )

, , 2 1 2, ,

, , 2 1 2, ,, (C.1)

a

a

a

a

0

12 0

( 1 2)

( ) 032 0

12

( 1 2)

( )32

where γ=y kx D(2 )0 02 . Using equation (B.3), one can alternatively write the moment-

generating function as

γ=

− −

−⩽ ⩽

γ γ

γ γ

−

−

⎜ ⎟

⎜ ⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )( )q x s

k x ℓ

D

D x

D ℓx ℓ˜( , ) exp

40 . (C.2)

s kk

D

s kk

D

002 2 0

0

Note also that equation (88) for the exterior case >x ℓ0 can also be written in terms of theparabolic cylinder function νD z( ) according to equation (B.4):

γ=

−>

γ γ

γ γ

−

−

⎜ ⎟

⎜ ⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )( )q x s

k x ℓ

D

D x

D ℓx ℓ˜( , ) exp

4, (C.3)

s kk

D

s kk

D

002 2 0

0

in agreement with [39] (see also [32, 42]).The inverse Laplace transform yields the probability density p t( )x a, [35, 42]

∑γ γ

= −− ±

′ ±

ν γ

ν γ

ν γ

=

∞−

⎜ ⎟

⎜ ⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )q x t

k k x ℓ

D

D x

D ℓe( , ) exp

4, (C.4)

n

k

D

k

D

kt0

02 2

1

0n

n

n

where ν ν< < < <0 ... ...n1 are the zeros of the function γ±νD ℓ k D( ( ) ), and ′νD z( )n

is thederivative of νD z( ) with respect to ν, evaluated at point ν ν= n [42] (p.154). The signs plusand minus correspond to >x ℓ0 and <x ℓ0 , respectively. Both νD z( ) and ′νD z( ) can berapidly evaluated by the numerical scheme presented in appendix B.2.

In the special case =ℓ 0, the FET probability density gets a simple explicit form:

πγ

γ γ γ γ= − +

γ−⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟q x t

x

D

k

kt

kx

D

e

kt

kt( , )

4 sinh ( )exp

4 sinh ( ) 2. (C.5)

kt

00

3 202

In the limit →k 0, one retrieves the classical formula for the FPT of Brownian motion at theorigin

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

39

π= − =

⎛⎝⎜

⎞⎠⎟q x t

x

Dt

x

Dtk( , )

4exp

4( 0). (C.6)0

0

3

02

Appendix D. Quantum harmonic oscillator

The eigenvalue problem (40) with =b 1 2 is closely related to eigenstates of a quantumharmonic oscillator of mass m and frequency ω [120]. In fact, the eigenstates ψn and energies

En of the Hamiltonian = + ωH p

m

m x

m

ˆ

2 2

2 2 2

satisfy the time-independent Schrödinger equation

ω ψ ψ− ∂ + =⎡⎣⎢

⎤⎦⎥

m

m xx E x

2 2( ) ( ), (D.1)x

22

2 2

where = − ∂p iˆ x is the momentum operator, and ℏ is the (reduced) Planck constant. Interms of the dimensionless coordinate ω= z x m2 , the above Schrödinger equation isreduced to Weberʼs equation (B.2), with = −

ωc E . Setting ψ = −z e u z( ) ˜ ( )z 42yields

″ − ′ − + =u zu c u˜ ˜ ( 1 2) ˜ 0, from which the rescaling γ= −( )u x u k D x x( ) ˜ ( ) ( ˆ) implies

equation (40), with λ = − +γ

c( 1 2)k . Consequently, the energies of the quantum oscillator

and the eigenvalues of the FP operator are simply related as: λ = −γ ω( )k E 1

2.

If no boundary condition is imposed, the non-normalized eigenstate is simplyψ ω= ν x D x m( ) ( 2 ), where νD z( ) is the parabolic cylinder function (see appendix B.1),and ν = − −c 1 2. One can check that

ν ν≃ − − + ≫νν− −⎡⎣⎢

⎤⎦⎥( )D z e z

zO z z( ) 1

( 1)

2( 1), (D.2)z 4

242

ν ν

πΓ ν

ν ν

≃ − − +

−−

+ + + + ≪ −

νν

π ν ν

− −

− − −

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( )

( )

D z e zz

O z

e e zz

O z z

( ) 1( 1)

2

2

( )1

( 1)( 2)

2( 1). (D.3)

z

i z

42

4

4 12

4

2

2

In order to eliminate the unphysical rapid growth of the eigenstate as → −∞z , one needs toimpose ν = n with = …n 0, 1, 2, to remove the last term, from which one retrieves thequantized energies of the quantum harmonic oscillator: ω= +E n( 1 2)n , while theeigenfunctions become expressed through the Hermite polynomials Hn(z)

ψ ωπ

ω ω= − ⎜ ⎟⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

xn

m m xH

mx( )

1

2 !exp

2,n n

n

1 4 2

where the usual normalization prefactor is included, and we used= − −D z e H z( ) 2 ( 2 )n

n zn

2 42. Since imposing no boundary condition corresponds to barriers

at distance → ∞L , we retrieve the asymptotic behavior λ ≃γnn

k or, equivalently,

α λ κ= ≃ n2nL

D n2 2

as κ → ∞. Note that the prefactor κ2 is twice smaller than that ofequation (A.13) because the latter relation accounts only for symmetric eigenfunctions thatcontribute to the survival probability.

Imposing the Dirichlet boundary condition at = ±x L corresponds to setting infinitepotential outside the interval −L L[ , ] (and keeping the harmonic potential inside). The

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

40

eigenvalue problem for a quantum oscillator in such potential is equivalent to the analysis ofthe first exit time distribution in section 2.6.

References

[1] Redner S 2001 A Guide to First Passage Processes (Cambridge: Cambridge University Press)[2] Metzler R, Oshanin G and Redner S (ed) 2014 First-Passage Phenomena and Their Applications

(Singapore: World Scientific)[3] Bénichou O and Voituriez R 2014 Phys. Rep. 539 225–84[4] Bouchaud J-P and Georges A 1990 Phys. Rep. 195 127–293[5] Metzler R and Klafter J 2000 Phys. Rep. 339 1–77[6] Metzler R and Klafter J 2004 J. Phys. A: Math. Gen 37 161–208[7] Condamin S, Bénichou O and Klafter J 2007 Phys. Rev. Lett. 98 250602[8] Yuste S B and Lindenberg K 2007 Phys. Rev. E 76 051114[9] Yuste S B, Oshanin G, Lindenberg K, Bénichou O and Klafter J 2008 Phys. Rev. E 78 021105[10] Grebenkov D S 2010 J. Chem. Phys. 132 034104[11] Grebenkov D S 2010 Phys. Rev. E 81 021128[12] Molchan G M 1999 Commun. Math. Phys. 205 97–111[13] Likthman A E and Marques C M 2006 Europhys. Lett. 75 971–7[14] Jeon J-H, Chechkin A V and Metzler R 2011 Europhys. Lett. 94 20008[15] Sanders L P and Ambjörnsson T 2012 J. Chem. Phys. 136 175103[16] Koren T, Lomholt M A, Chechkin A V, Klafter J and Metzler R 2007 Phys. Rev. Lett. 99 160602[17] Koren T, Klafter J and Magdziarz M 2007 Phys. Rev. E 76 031129[18] Tejedor V, Bénichou O, Metzler R and Voituriez R 2011 J. Phys. A 44 255003[19] Bénichou O, Grebenkov D S, Levitz P, Loverdo C and Voituriez R 2010 Phys. Rev. Lett. 105

150606[20] Bénichou O, Grebenkov D S, Levitz P, Loverdo C and Voituriez R 2011 J. Stat. Phys. 142

657–85[21] Rupprecht J-F, Bénichou O, Grebenkov D S and Voituriez R 2012 J. Stat. Phys. 147 891–918[22] Rupprecht J-F, Bénichou O, Grebenkov D S and Voituriez R 2012 Phys. Rev. E 86 041135[23] Oshanin G, Lindenberg K, Wio H S and Burlatsky S 2009 J. Phys. A 42 434008[24] Bénichou O, Loverdo C, Moreau M and Voituriez R 2011 Rev. Mod. Phys. 83 81–130[25] Condamin S, Bénichou O, Tejedor V, Voituriez R and Klafter J 2007 Nature 450 77–80[26] Condamin S, Tejedor V, Voituriez R, Bénichou O and Klafter J 2008 Proc. Natl Acad. Sci. USA

105 5675–80[27] Holcman D, Marchewka A and Schuss Z 2005 Phys. Rev. E 72 031910[28] Crandall S H 1970 J. Sound. Vibr. 12 285–99[29] Siegert A J F 1951 Phys. Rev. 81 617–23[30] Darling D A and Siegert A J F 1953 Ann. Math. Statist. 24 624–39[31] Lindenberg K, Shuler K E, Freeman J and Lie T J 1975 J. Stat. Phys. 12 217–51[32] Ricciardi L M and Sato S 1988 J. Appl. Prob. 25 43–57[33] Leblanc B, Renault O and Scaillet O 2000 Finance Stochast. 4 109–11[34] Göing-Jaeschke A and Yor M 2003 Finance Stochast. 7 413–5[35] Alili L, Patie P and Pedersen J L 2005 Stoch. Models 21 967–80[36] Yi C 2010 Quant. Finance 10 957–60[37] Spendier K, Sugaya S and Kenkre V M 2013 Phys. Rev. E 88 062142[38] Toenjes R, Sokolov I M and Postnikov E B 2013 Phys. Rev. Lett. 110 150602[39] Borodin A N and Salminen P 1996 Handbook of Brownian Motion: Facts and Formulae (Basel-

Boston: Birkhauser Verlag)[40] Revuz D and Yor M 1999 Continuous Martingales and Brownian Motion 3rd edn (Berlin:

Springer)[41] Itô K and McKean H P 1996 Diffusion Processes and Their Sample Paths 2nd edn (Berlin:

Springer)[42] Jeanblanc M, Yor M and Chesney M 2009 Mathematical Methods For Financial Markets

(Berlin: Springer-Verlag)[43] Crank J 1975 The Mathematics of Diffusion 2nd edn (Oxford: Clarendon)[44] Carslaw H S and Jaeger J C 1959 Conduction of Heat Solids 2nd edn (Oxford: Clarendon)

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

41

[45] Hughes B D 1995 Random Walks and Random Environments (Oxford: Clarendon)[46] Gardiner C 2004 Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences

3rd edn (Berlin: Springer)[47] Doob J L 1984 Classical Potential Theory and Its Probabilistic Counterpart (Berlin: Springer)[48] van Kampen N G 2007 Stochastic Processes in Physics and Chemistry 3rd edn (Amsterdam:

Elsevier)[49] Uhlenbeck G E and Ornstein L S 1930 Phys. Rev. 36 823[50] Wang M C and Uhlenbeck G E 1945 Rev. Mod. Phys. 17 323[51] Ricciardi L M and Sato S 1990 Lectures in Applied Mathematics and Informatics ed

L M Ricciardi (Manchester: Manchester University Press)[52] Coffey W T, Kalmykov Y P and Waldron J T 2004 The Langevin equation: With Applications to

Stochastic Problems in Physics, Chemistry and Electrical Engineering (Series in ContemporaryChemical Physics vol 14) 2nd edn (Singapore: World Scientific Publishing)

[53] Risken H 1996 The Fokker–Planck Equation: Methods of Solution and Applications 3rd edn(Berlin: Springer)

[54] Kolpas A, Moehlis J and Kevrekidis I G 2007 Proc. Nat. Acad. Sci. 104 5931–5[55] Grebenkov D S and Nguyen B-T 2013 SIAM Rev. 55 601–67[56] Kac M 1949 Trans. Am. Math. Soc. 65 1–13[57] Kac M 1951 Proc. 2nd Berkeley Symp. Math. Stat. Prob ed J Neyman (Berkeley: University of

California Press) pp 189–215[58] Freidlin M 1985 Functional Integration and Partial Differential equations (Annals of

Mathematics Studies) (Princeton, NJ: Princeton University Press)[59] Simon B 1979 Functional Integration and Quantum Physics (London: Academic)[60] Bass R F 1998 Diffusions and Elliptic Operators (New York: Springer)[61] Pontryagin L, Andronov A and Witt A 1933 Zh. Eksp. Teor. Fiz 3 172

Pontryagin L, Andronov A and Witt A 1989 Reprinted in Noise in Nonlinear Dynamical Systemsed F Moss and P V E McClintock Vol 1 (Cambridge: Cambridge University Press) p 329

[62] L B Castro and A S de Castro 2013 J. Math. Chem. 51 265–77[63] Weiss G H 1981 J. Stat. Phys. 24 587–94[64] Szabo A, Schulten K and Schulten Z 1980 J. Chem. Phys. 72 4350–7[65] Weiss G H 1986 J. Stat. Phys. 42 3–36[66] Hänggi P, Talkner P and Borkovec M 1990 Rev. Mod. Phys. 62 251–341 (http://journals.aps.org/

rmp/abstract/10.1103/RevModPhys.62.251)[67] Kramers H A 1940 Physica 7 284[68] Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions (New York: Dover)[69] Rohrbach A, Tischer C, Neumayer D, Florin E-L and Stelzer E H K 2004 Rev. Sci. Instrum.

75 2197[70] Lukić B, Jeney S, Sviben Z, Kulik A J, Florin E-L and Forró L 2007 Phys. Rev. E 76 011112[71] Franosch T, Grimm M, Belushkin M, Mor F M, Foffi G, Forró L and Jeney S 2011 Nature

478 85[72] Huang R, Chavez I, Taute K M, Lukić B, Jeney S, Raizen M G and Florin E-L 2011 Nat. Phys.

7 576[73] Bertseva E, Grebenkov D S, Schmidhauser P, Gribkova S, Jeney S and Forró L 2012 Eur. Phys.

J. E 35 63[74] Grebenkov D S, Vahabi M, Bertseva E, Forró L and Jeney S 2013 Phys. Rev. E 88 040701(R)[75] Ashkin A, Schütze K, Dziedzic J M, Euteneuer U and Schliwa M 1990 Nature 348 346–8[76] Kuo S C and Sheetz M P 1993 Science 260 232–4[77] Brangwynne C P, Koenderink G H, MacKintosh F C and Weitz D A 2009 Trends Cell. Biol. 19

423–7[78] Sokolov I M 2012 Soft Matter 8 9043–52[79] Bressloff P C and Newby J M 2013 Rev. Mod. Phys. 85 135–96[80] Desposito M A and Vinales A D 2009 Phys. Rev. E 80 021111[81] Grebenkov D S 2011 Phys. Rev. E 83 061117[82] Bal G and Chou T 2004 Inv. Problems 20 1053–65[83] Masson J-B, Casanova D, Türkcan S, Voisinne G, Popoff M R, Vergassola M and Alexandrou A

2009 Phys. Rev. Lett. 102 048103[84] Kenwright D A, Harrison A W, Waigh T A, Woodman P G and Allan V J 2012 Phys. Rev. E 86

031910

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

42

[85] Lindner M, Nir G, Vivante A, Young I T and Garini Y 2013 Phys. Rev. E 87 022716[86] Golding I and Cox E C 2006 Phys. Rev. Lett. 96 098102[87] Szymanski J and Weiss M 2009 Phys. Rev. Lett. 103 038102[88] Metzler R, Tejedor V, Jeon J-H, He Y, Deng W H, Burov S and Barkai E 2009 Acta Phys. Pol. B

40 1315[89] Jeon J-H, Tejedor V, Burov S, Barkai E, Selhuber-Unkel C, Berg-Sørensen K, Oddershede L and

Metzler R 2011 Phys. Rev. Lett. 106 048103[90] Bell G I 1978 Science 200 618–27[91] Evans E and Ritchie K 1997 Biophys. J. 72 1541–55[92] Izrailev S, Stepaniants S, Balsera M, Oono Y and Schulten K 1997 Biophys. J. 72 1568–81[93] Merkel R, Nassoy P, Leung A, Ritchie K and Evans E 1999 Nature 397 50–53[94] Heymann B and Grubmüller H 2000 Phys. Rev. Lett. 84 6126–29[95] Evans E 2001 Ann. Rev. Biophys. Biomol. Struct. 30 105–28[96] Butt H-J, Cappella B and Kappl M 2005 Surf. Sci. Rep. 59 1–152[97] Box G, Jenkins G M and Reinsel G C 1994 Time Series Analysis: Forecasting and Control 3rd

edn (Englewood Cliffs, NJ: Prentice-Hall)[98] Covel M W 2009 Trend Following (Updated Edition): Learn to Make Millions in Up or Down

Markets (New Jersey: Pearson Education)[99] Clenow A F 2013 Following the Trend: Diversified Managed Futures Trading (Chichester:

Wiley & Sons)[100] Bouchaud J-P and Potters M 2003 Theory of Financial Risk and Derivative Pricing: From

Statistical Physics to Risk Management (Cambridge: Cambridge University Press)[101] Chan L K C, Jegadeesh N and Lakonishok J 1996 J. Finance 51 1681–13[102] Moskowitz T J, Ooi Y H and Pedersen L H 2012 J. Finan. Econ. 104 228–50[103] Asness C S, Moskowitz T J and Pedersen L H 2013 J Finance 68 929–85[104] Grebenkov D S and Serror J 2014 Physica A 394 288–303[105] Hull J C 2003 Options Futures and Other Derivatives (Upper Saddle River, NJ: Prentice-Hall)[106] Vasicek O 1977 J. Financ. Econom. 5 177–88[107] Davydov D and Linetsky V 2003 Oper. Res. 51 185–09[108] Tuckwell H C and Wan F Y M 1984 J. Appl. Prob. 21 695709[109] Durbin J 1985 J. Appl. Prob. 22 99–122[110] Lerche H R 1986 Boundary crossing of Brownian motion (Lecture Notes in Statistics vol 40)

(Berlin: Springer-Verlag)[111] Wang L and Pötzelberger K 1997 J. Appl. Prob. 34 54–65[112] Kahale N 2008 Ann. Appl. Probab. 18 1424–40[113] Breiman L 1966 Proc 5th Berkeley Symp Math Statist Prob. 2 9–16[114] Shepp L A 1967 Ann. Math. Statist. 38 1912–4[115] Novikov A A 1971 Theory Prob. Appl. 16 449–56[116] Sato S 1977 J. Appl. Prob. 14 53–70[117] Salminen P 1988 Adv. Appl. Prob. 20 411–26[118] Novikov A, Frishling V and Kordzakhia N 1999 J. Appl. Prob. 36 1019–30[119] de Long D M 1981 Commun. Stat. Theory Meth. 10 2197–13[120] Merzbacher E 1998 Quantum Mechanics 3rd edn (New York: Wiley & Sons)[121] Griffiths D J 2004 Introduction to Quantum Mechanics 2nd edn (Englewood Cliffs, NJ:

Prentice-Hall)[122] Caroli B, Caroli C and Roulet B 1979 J. Stat. Phys. 21 415–37[123] van Kampen N G 1977 J. Stat. Phys. 17 71–88[124] Mörsch M, Risken H and Vollmer H D 1979 Z. Phys. B 32 245–52[125] Hongler M O and Zheng W M 1982 J. Stat. Phys. 29 317–27[126] Voigtlaender K and Risken H 1985 J. Stat. Phys. 40 397–29[127] Jung P and Risken H 1985 Z. Phys. B: Conden. Mat 61 367–79[128] Ivlev B I 1988 Sov. Phys. JETP 68 1486[129] Kalmykov Y P, Coffey W T and Titov S V 2006 Phys. Rev. E 74 011105[130] Gradshteyn I S and Ryzhik I M 1980 Table of Integrals, Series, and Products (New York:

Academic)[131] Collins F C and Kimball G E 1949 J. Colloid Sci. 4 425–37[132] Wilemski G and Fixman M 1973 J. Chem. Phys. 58 4009[133] Sano H and Tachiya M 1979 J. Chem. Phys. 71 1276–82

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

43

[134] Tachiya M 1983 Radiat. Phys. Chem. 21 167–75[135] Sapoval B 1994 Phys. Rev. Lett. 73 3314–6[136] Grebenkov D S 2006 Focus on Probability Theory ed L R Velle (Hauppage, NY: NOVA Science

Publishers) pp 135–69[137] Singer A, Schuss Z, Osipov A and Holcman D 2008 SIAM J. Appl. Math. 68 844–68[138] Luke Y L 1969 The Special Functions and their Approximations (New York: Academic)[139] Buchholz H 1969 The Confluent Hypergeometric Function (Heidelberg: Springer-Verlag)[140] Abad J and Sesma J 1995 Mathematica J 5 74–76[141] Abad J and Sesma J 1997 J. Comput. Appl. Math. 78 97–101 (www.mathematica-journal.com/

issue/v5i4/article/abad/74-76abad.mj.pdf)[142] Temme N M 1983 Numer. Math. 41 63–82

J. Phys. A: Math. Theor. 48 (2015) 013001 D S Grebenkov

44