Stochastic Partial Differential Equations: Theory and ... · Stochastic Partial Di erential...

Stochastic Partial Differential Equations:Theory and Numerical Simulations

Adam Q. JaffeAdvised by: Lenya Ryzhik

May 19, 2019

A.Q. Jaffe Stochastic PDE May 19, 2019 1 / 24

Motivation

Why study Stochastic Partial Differential Equations (SPDE)?

In many (PDE) models with unknown parameters, it makes senseencode unknowns by randomness

Examples:

Wave propagation in atmosphereFluid dynamicsEpidemiology

Some Issues:

Lots of technical detailsHard to simulate in a computer


Stochastic Ordinary Differential Equations


SDEs from ODEs

General ODE written as dUtdt = µ(t,Ut)

t ∈ [0,T ]

U0 ∈ R(1)

We have existence, uniqueness, and continuity under very mildassumptions on µ

Lots of numerical methods to solve these; any approach will(essentially) work

Can we add simple random “noise” to a given ODE?

dUt

dt= µ(t,Ut) + σ(t,Ut)W (t) (2)


Brownian Motion and White Noise

Bt : 0 ≤ t ≤ T or simply Bt (3)

A random function

Important properties: Bt is...

almost surely continuous in talmost surely nowhere differentiable in tLimiting object of a random walk (CLT)

“The derivative of Brownian motion is white noise.”

W (t) =dBt

dt(4)


Brownian Motion and White Noise


SDEs from ODEs

The equationdUt

dt= µ(t,Ut) + σ(t,Ut)

dBt

dt(5)

in integral form is

Ut − U0 =

∫ t

0µ(s,Us)ds +

∫ t

0σ(t,Ut)

dBs

dsds (6)

Then reimagine the integral as∫ t

0σ(t,Ut)

dBs

dsds =

∫ t

0σ(t,Ut)dBs (7)

Can we interpret this in a Riemann-Stieljes-type sense, where Bt actsas an “integrator”?


Stochastic Integral

For each sample path of Bt and partition 0 = t1 < · · · < tN = T ,write the Riemann-Stieljes sum∫ T

0σ(t,Ut)dBt ≈

N∑j=0

σ(t∗j ,Ut∗j)(Btj+1 − Btj ) (8)

How to choose t∗j ∈ [tj , tj+1)? Surprisingly, this choice matters!

t∗j = tj leads to the Ito integral,∫ T

0 σ(t,Ut)dBt

t∗j = 12 (tj + tj+1) leads to the Stratonovich integral,

∫ T0 σ(t,Ut) dBt


Ito versus Stratonovich Calculus

Ito Calculus

Does not “see into the future”Is a martingaleThe usual chain rule failsNumerical methods are harder

Stratonovich Calculus

Does not “see into the future”Is not a martingaleUsual chain rule holdsNumerical methods are easier (Can use numerical methods for ODEs)


SDE Existence and Uniqueness

A general SDE is written as

Ut − U0 =

∫ t

0µ(s,Us)ds +

∫ t

0σ(s,Us)()dBs . (9)

Or, by the shorthand:

dUt = µ(t,Ut)dt + σ(t,Ut)()dBt (10)

Can interpret in the Ito or Stratonovich sense

We want a solution defined on t ∈ [0,T ], with a possibly randominitial condition U0.

Theorem

If µ, σ are Lipschitz continuous and satisfy a linear growth bound, thenalmost surely there exists a unique continuous function on [0,T ] whichsatisfies the SDE.


SDE Example 1

Consider the SDE dUt = UtdBt on [0,T ] with deterministic initialcondition U0 = 1.The Ito solution is Ut = U0 exp(Bt − t

2 )The Stratonovich solution is Ut = U0 exp(Bt)


SDE Example 2

Consider the SDE dUt = exp(t − U2t )dt + tanh(Ut)dBt on [0,T ] with

deterministic initial condition U0 = 1.

No explicit solutions exist!

Need to resort to numerical simulations


Stochastic Partial Differential Equations


SPDEs from PDEs

Consider the heat equation∂U∂t = ∂2U

∂x2 + g(t, x)

(t, x) ∈ [0,T ]× RU(0, x) = φ(x)

(11)

Its explicit solutions are given by

U(t, x) =

∫RG (t, x − y)φ(y)dy +

∫ t

0

∫RG (t − s, x − y)g(s, y)dyds

(12)where G is the usual heat kernel:

G (t, x) =1√4πt

exp

(−|x |

2

4t

)(13)


SPDEs from PDEs

Consider the heat equation with g(t, x) = 0, and initial conditionφ(x) = 1[1,3](x).


SPDEs from PDEs

Lesson from SDEs: Easiest to work in integral form. Write

∂U

∂t=∂2U

∂x2+ f (U(t, x))W (t, x) (14)

as

U(t, x) =

∫RG (t, x − y)φ(y)dy (15)

+

∫ t

0

∫RG (t − s, x − y)f (U(s, y))W (s, y)dyds (16)

Let’s come up with a precise definition of∫ t

0

∫RG (t−s, x−y)W (s, y)dyds =

∫ t

0

∫RG (t−s, x−y)dBs,y (17)

Use Riemann-Stieljes-type sums, and choose between Ito andStratonovich integral

We use Ito calculus from now on


SPDE Existence and Uniqueness

A general stochastic heat equation is written as

U(t, x) =

∫RG (t, x−y)φ(y)dy+

∫ t

0

∫RG (t−s, x−y)f (U(s, y))dBs,y

(18)Or, by the shorthand:

∂U

∂t=∂2U

∂x2+ f (U(t, x))W (t, x) (19)

We impose an inital condition U(0, x) = φ(x).

Theorem

If f is Lipschitz continuous and the initial condition φ(x) is compactlysupported, then almost surely there exists a unique continuous function on[0,T ]× R which satisfies the SPDE.



Theorem

If f is Lipschitz continuous and the initial condition φ(x) is compactlysupported, then almost surely there exists a unique continuous function on[0,T ]× R which satisfies the SPDE.

Proof Sketch.

(Existence) By Picard’s iteration method: Set

U0(t, x) =φ(x) (20)

Un+1(t, x) =

∫RG (t, x − y)φ(y)dy (21)

+

∫ t

0

∫RG (t − s, x − y)f (Un(s, y))dBs,y

Each Un(t, x) is continuous in t and x .



Proof Sketch (continued).

Define the supremum of the L2(Ω) norm of the adjacent differences

zn(t) = supx∈R

sup0≤s≤t

E[|Un+1(s, x)− Un(s, x)|2

]. (22)

These are bounded as

zn(t) ≤ C1

∫ t

0zn−1(s)ds. (23)

By induction:

zn(t) ≤ C2(C1t)n+1

(n + 1)!≤ C2

(C1T )n+1

(n + 1)!(24)

Since this is summable, we get that Un(t, x) has an L2(Ω)-limit functionU(t, x). Since the convergence is uniform in t, the limit function U(t, x) iscontinuous. Now check that U satisfies that SPDE and is unique.


SPDE Example 1

Consider the stochastic heat equation with f (u) = 1, anddeterministic initial condition φ(x) = 1[1,3](x).


SPDE Example 2

Consider the stochastic heat equation with f (u) = |1− u|, anddeterministic initial condition φ(x) = 1[1,3](x).


SPDE Example 3

Consider the stochastic heat equation with f (u) = 1[0,∞)(u), anddeterministic initial condition φ(x) = 1[1,3](x).


Research Questions

“Smoothness” of solutions to S(P)DE

Existence and Uniqueness of SPDE solutions other than thestochastic heat equation

Relationship between Ito and Stratonovich calculus in the SPDE case

How to simulate S(P)DE efficiently and precisely with a computer?

Long-time behavior of certain SPDE models of interest


Thank you!


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