The variational approach to Hamilton–Jacobi equationsdriven by a Gaussian noise
Viorel Barbu
Al.I. Cuza University and Octav Mayer Institute of Mathematics, Romanian Academy, Iasi, Romania
a r t i c l e i n f o a b s t r a c t
Article history:Received 18 June 2013Revised 18 July 2013Available online 29 August 2013
MSC:60H1560H0570H20
Keywords:Hamilton–Jacobi equationBrownian motionViscosity solutionStochastic process
The global existence and uniqueness of viscosity solutions to theCauchy problem for the Hamilton–Jacobi equations in R
N drivenby additive and multiplicative Wiener processes are studied forconvex Hamiltonians via variational techniques. The finite speed ofpropagation is also established in the multiplicative noise case forequations with Lipschitzian Hamiltonians.
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3833
and H : [0, T ] × RN × R
N → R is a Hamiltonian function to be made precise later on. In (2),{μ j}m
j=1 are smooth real-valued functions on RN and {β j}m
j=1 is a system of linear independent realBrownian motions in a probability space {Ω,F ,P} with the natural filtration (Ft)t�0. The solutionX = X(t, x,ω) : [0, T ] × R
n × Ω → R to (1) is taken in a certain weak sense which will be madeprecise later on in Section 2. Besides (1), we shall study also the stochastic equation (1) with themultiplicative white noise X W , that is,
dX(t, x) + H(t, x, Xx(t, x)
)dt = X(t, x)dW (t),
X(0, x) = X0(x), x ∈Rn, t ∈ [0, T ], (3)
where X dW is considered in the Itô sense. Eqs. (1), (3) arise as models for dynamics of mechanicalparticles in a field perturbed by white noise (a few examples are given below in Section 2) as wellas in the study of optimal control problems with systems driven by Gaussian processes. In fact, thesolution X to (1) may be viewed as the action functions corresponding to the stochastic Hamiltoniansystem
was studied in the context of stochastic homogenization (see, e.g., [12,13]) but there are a few resultsfor Eq. (1) or (3) under general conditions on H . In this context, we mention the works [11,10], whereexistence in (1) is studied for the smooth Hamiltonian function H by the method of the stochasticcharacteristic defined by the Hamiltonian system (4). The method we use here is different and leadsto existence and uniqueness results of solutions under more general assumptions on H , which coversin particular the case of the eikonal stochastic equation.
By the substitution X = Y + W , Eq. (1) reduces to a random Hamilton–Jacobi equation of the form(5), but the existence of an adapted stochastic process t → Y (t, x,ω), which is a viscosity solutionto (5), is not a direct consequence of the existence theory of viscosity solutions to Hamilton–Jacobiequations and requires a special analysis based on the variational theory. As a matter of fact, theprincipal effort is to show that the viscosity solution to corresponding random equations in Y definesan (Ft)t�0-adapted stochastic process.
As regards Eq. (3), it reduces via the rescaling procedure (see [2,3]) to a random Hamilton–Jacobiequation of the form ϕt + H0(t, x,ϕ,ϕx,ω) = 0. Both problems are treated in Sections 2–4 below.Eqs. (1) and (3) with Dirichlet stochastic boundary conditions on bounded domains can be similarlytreated, but we do not give details. Note also that the Stratonovich equation
dX + H(t, x, Xx)dt = X ◦ dW , X(t) = X0, (6)
can be treated by the same technique.
Notation. Everywhere in the following, RN is the Euclidean space with the norm denoted | · |N andscalar product denoted x · y. By B R we denote the ball {x ∈ R
N ; |x|N � R}.Denote by BUC(RN ), BUC([0, T ] ×R
N ) and BUC([0, T ] ×RN × B R) the space of bounded and uni-
formly continuous functions on RN (respectively, [0, T ] ×R
N , [0, T ] ×RN × B R ).
3834 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
We denote by L1(0, T ;RN ) and C([0, T ];RN ) the space of Lebesgue-summable functions u :(0, T ) → R
N , respectively continuous functions u : [0, T ] → RN . By C1([0, T ] × R
N ) we denote thespace of continuously differentiable functions on [0, T ] ×R
N .Finally, Ck
b(RN ) is the space of bounded and continuously k-differentiable functions on R
N withbounded derivatives up to order k.
We also use the notations:
ϕx(t, x) = ∇xϕ(t, x), x ∈ RN , y′(s) = dy
ds(s), s ∈R, ϕt = ∂ϕ
∂t.
Given the Hamiltonian function H : [0, T ] ×RN ×R
N → R , we denote by L : [0, T ] ×RN ×R
N → Rthe corresponding Lagrangian function
L(t, x, u) = supp
{p · u − H(t, x, p); p ∈R
N}. (7)
We recall that u → L(t, x, u) is convex and lower semicontinuous and, if p → H(t, x, p) is convex andlower semicontinuous, then
H(t, x, p) = supu
{p · u − L(t, x, u); u ∈ R
N}, ∀p ∈R
N . (8)
A continuous monotonically increasing function m : [0,∞) → [0,∞) such that m(0) = 0 is called mod-ulus.
2. The main results
The following hypotheses will be considered in the sequel.
(i) H = BUC([0, T ] ×RN × B R) for each R > 0.
(ii) For each (t, x) ∈ [0, T ] ×RN , the function p → H(t, x, p) is convex.
(iii) There is a modulus m such that∣∣H(t, x, p) − H(t, x, p)∣∣ � m
(|x − y|N(1 + |p|N
)),
for all t ∈ [0, T ] and all x, x, p ∈ RN .
(iv) H(t, x, p) � ν|p|N , ∀t ∈ [0, T ], x, p ∈ RN .
(v) μ j ∈ C2b (RN ), j = 1,2, . . . ,m.
By the substitution Y = X − W , we reduce Eq. (1) to the random Hamilton–Jacobi equation
Yt(t, x) + H(t, x, Yx(t, x) + W x(t, x)
) = 0, x ∈RN ,
Y (0, x) = X0(x), x ∈RN , (9)
which will be studied below in the framework of viscosity solutions.We recall (see [7,6]) that the continuous function (t, x) → Y (t, x) is said to be a viscosity solution
to (9) (for a fixed ω ∈ Ω) if Y (0, x) ≡ X0(x) and, for each ϕ ∈ C1([0, T ] × RN ), we have ϕt(t0, x0) +
H(t0, x0,ϕx(t0, x0)) � 0 (respectively, � 0) at each local maxima (respectively, minima) of Y − ϕ .
Definition 2.1. The stochastic process X = X(t, x) is said to be a viscosity solution to (1) if t → X(t, x)is, for each x ∈ R
N , (Ft)t�0–adapted and Y = X − W is a viscosity solution to (9), P-a.s., i.e., forP-almost all ω ∈ Ω .
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3835
Theorem 2.2. Let X0 ∈ BUC(RN ) and 0 < T < ∞. Then, under hypotheses (i), (ii), (iii), (v), Eq. (1) has aunique viscosity solution
X ∈ BUC([0, T ] ×R
N), P-a.s. (10)
Moreover, the solution X depends P-a.s. continuously on X0 from BUC(RN ) to C([0, T ];RN ).
The viscosity solution to Eq. (3) is defined via the rescaling transformation
X = eW Y . (11)
If X is a strong solution to (3) in Itô’s integral sense, that is, P-a.s.,
X(t, x) +t∫
0
H(s, x, Xx(s, x)
)ds = X0(x) +
t∫0
X(s, x)dW (s), t ∈ [0, T ], x ∈RN , (12)
then, by Itô’s formula,
dX = eW Y dW + eW dY + μeW Y dt,
and so Y is the solution to the random Hamilton–Jacobi equation
Yt(t, x) + e−W (t,x)H(t, x, eW (t,x)(Yx(t, x) + W x(t, x)
)) + μ(x)Y (t, x) = 0, t ∈ [0, T ], x ∈RN ,
Y (0, x) = X0(x), x ∈ RN , (13)
where
μ(x) = 1
2
m∑j=1
μ2j (x), x ∈R
N .
Conversely, every strong P-a.s. solution Y to (13) (if any) which is adapted to the filtration (Ft)t�0is a solution to (12). The equivalence of the stochastic equation and the corresponding deterministicequation via the rescaling transformation (11) can be proved by the arguments used in [2,3]. Sinceunder our assumptions the random Hamilton–Jacobi equation has not, in general, a strong solution,the equivalence between (12) and (13) is only conceptual, but we are entitled, however, to say that Xis a viscosity solution to (12) if the process t → X(t, x) is (Ft)t�0-adapted for each x ∈ R
N and Y = e−W X isP-a.s. a viscosity solution to (13).
We have
Theorem 2.3. Let X0 ∈ BUC(RN ) and 0 < T < ∞. Then, under hypotheses (i), (ii), (iii), (iv), (v), Eq. (3)has a unique viscosity solution X ∈ BUC([0, T ] ×R
N ), P-a.s. The solution depends continuously on the initialdata X0 .
We note that, by substitution (11), the Stratonovich stochastic Eq. (6) reduces to
Yt + e−W H(t, x, eW (Yx + W x)
) = 0, Y (0) = X0, (14)
and the definition of the viscosity solution remains the same.
3836 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
Let us now briefly recall a few examples for which Theorems 2.2 and 2.3 apply.1. The eikonal equation
dX + ρ|Xx|N dt + a(t, x) · Xx dt = X dW in (0, T ) ×RN ,
X(0) = X0 in RN , (15)
where ρ > 0 and a ∈ BUC([0, T ] ×RN ).
For a ≡ 0 this is the classical geometrical optic equation perturbed by a linearly multiplicativeGaussian process. The same equation models the flame propagation driven by the multiplicative Gaus-sian noise X W and the linear transport term a · Xx .
2. The inviscid stochastic Burger equation
dX + 1
2
∣∣X(t)∣∣2x dt = dW (t), t ∈ (0, T ), x ∈R,
X(0) = X0, (16)
reduces via the transformation X = Xx to the Hamilton–Jacobi equation
dX + 1
2| Xx|2N dt = dW (t), X(0) =
0∫−∞
X0(ξ)dξ, (17)
where W = ∑Nj=1 β j
∫ x−∞ μ j(ξ)dξ .
In a similar way, one reduces the stochastic conservation law equation dX + (F (X))x dt = dW in1 − D to a Hamilton–Jacobi equation of the form (1).
3. The stochastic erosion equation
dh + H(hx)dt = h dW , t � 0, x ∈ RN (18)
can be taken as a model for the temporary change of the surface height h = h(t, x) at time t inpresence of a stochastic perturbation hW proportional with the height. In the special case of (18),with h(p) = 1
2 |p|2N and additive noise dW , this is a limit case of the Kardar–Parisi–Zhang equation forthe stochastic interface growth.
3. Proof of Theorem 2.2
It should be said that, under assumptions (i)–(iii) and (v), for fixed ω ∈ Ω , the Hamilton–Jacobiequation (9) has a unique viscosity solution Y (see [5,7,6,8]). The main effort here is to show that theprocess t → Y (t, x) is (Ft)t -adapted and, to this purpose, we shall invoke a variational technique tothe existence theory for (9) which though is not new (see, e.g., [1,4]), it has, however, some specificfeatures in the present case. Namely, consider for a fixed ω ∈ Ω the optimal value function
Y (t, x) = infu
{ t∫0
(L(s, y(s), u(s)
) − u(s) · W y(s, y(s)
))ds + X0
(y(0)
);y′(s) = u(s), a.e. s ∈ (0, t), y(t) = x, u ∈ L1(0, t;RN)}
(19)
for (t, x) ∈ [0, T ] ×RN .
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3837
Lemma 3.1. The function Y : [0, T ] ×RN → R is continuous and is a viscosity solution to (9).
Proof. We note first that, for each (t, x) ∈ [0, T ] × RN , the infimum in (19) is attained. Indeed, we
have by (7)
L(s, y, u) � ρ|u|N − H
(s, y,ρ
u
|u|N
), ∀u ∈ R
N , ρ > 0, (20)
and so, by hypothesis (i), it follows that
L(s, y, u) � ρ|u|N − C(ρ), ∀u ∈RN , ∀ρ > 0. (21)
Now, if {(yn, un)} is a minimizing sequence in (19), it follows by (21) via the Dunford–Pettis theoremthat {un} is weakly compact in L1(0, t;RN ) and so, by the Arzela theorem, {yn} is strongly compactin C([0, t];RN ). Hence, on a subsequence, again denoted {n}, we have, for n → ∞,
un −→ u∗ weakly in L1(0, t;RN),
yn −→ y∗ in C([0, t];RN)
,
un · W (·, yn) −→ u∗ · W(·, y∗) weakly in L1(0, t;RN)
(22)
where y∗(s) = x − ∫ ts u∗(τ )dτ , s ∈ [0, t].
By (7), we have
t∫0
L(s, yn(s), un(s)
)ds �
t∫0
un(s) · p(s)ds −t∫
0
H(s, yn(s), p(s)
)ds, ∀p ∈ L∞(
0, t;RN),
and so, it follows by virtue of assumption (i) that
lim infn→∞
t∫0
L(s, yn(s), un(s)
)ds �
t∫0
(u∗(s) · p(s) − H
(s, y∗(s), p(s)
))ds,
∀p ∈ L∞(0, t;RN)
. (23)
Taking into account that, by (ii), the function p → H(s, y∗; p) is convex and continuous, we have (see,e.g., Theorem 1 in [14])
supp
{ t∫0
(u∗(s) · p(s) − H
(s, y∗(s), p(s)
))ds; p ∈ L∞(
0, t;RN)} =t∫
0
L(s, y∗(s), u∗(s)
)ds.
Hence, it follows that
Y (t, x) =t∫
0
(L(s, y∗(s), u∗(s)
) − u∗(s) · W y(s, y∗(s)
))ds + X0
(y∗(0)
),
as claimed.
3838 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
It is easily seen that (t, x) → Y (t, x) is continuous. Indeed, if (tn, xn) → (t0, x0), for n → ∞, wehave
Y (tn, xn) =tn∫
0
(L(s, yn(s), un(s)
) − W y(s, yn(s)
) · un(s))
ds + X0(
yn(0))
�tn∫
0
(L(s, y(s), u(s)
) − W y(s, y(s), u(s)
) · u(s))
ds + X0(
y(0)), (24)
for all admissible pairs (y, u) in the optimal control problem (19). Arguing as above, it follows by (24)that
Y (tn, xn)�tn∫
0
(p · un − H(s, yn, p)
)ds −
tn∫0
W y(s, yn(s)
) · un(s)ds + X0(
yn(0))
for all p ∈ L∞(0,∞,RN ). Letting n → ∞, we obtain that
lim supn→∞
Y (tn, xn) � Y (t0, x0).
On the other hand, by the lower semicontinuity of the integral functional arising in the right handside of (24), it is easily seen that
lim infn→∞ Y (tn, xn) � Y (t0, x0).
Let us show now that Y is a viscosity solution to (9). We set Y (t, x) = Y (T − t, x) and note that
Y (t, x) = infv
{ T∫t
(L(T − s, z(s), v(s)
) − v(s) · W z(T − s, z(s)
))ds + X0
(z(T )
);z′ = −v on (t, T ); z(t) = x
}. (25)
By the dynamic programming principle, we have, for all (t0, x0) ∈ [0, T ] ×RN ,
Y (t0, x0) = infv
{ s∫t0
(L
(T − τ , x0 −
τ∫t0
v(θ)dθ
), v(τ )
)dτ + Y
(s, x0 −
s∫t0
v(θ)dθ
),
v ∈ L1(t0, s;RN)}, (26)
where L(s, y, v) ≡ L(s, y, v) − v · W y(s, y).
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3839
Let (t0, x0) ∈ [0, T ] ×RN and ϕ ∈ C1([0, T ] ×R
N ) be such that
Y (t0, x0) − ϕ(t0, x0) � Y (t, x) − ϕ(t, x),
for all (t, x) in a neighborhood of (t0, x0).By (26), this yields, for s = t0, x = x0 − (t − t0)v and v ∈ R
N ,
ϕ(t0, x0) − ϕ(t, x) � Y (t0, x0) − Y (t, x)
�t∫
t0
L(T − τ , x0 − (τ − t0)v, v
)dτ ,
and, therefore,
−ϕt(t0, x0) + ϕx(t0, x0) · v � L(T − t0, x0, v) − v · W x(T − t0, x0), ∀v ∈RN .
This yields
−ϕt(t0, x0) + H(T − t0, x0, Y x(t0) + W x(T − t0, x0)
)� 0.
Similarly, if (t0, x0) is a local minima for Y − ϕ , we get that
and this clearly implies that Y is a viscosity solution to (9), as claimed. �Lemma 3.2. We have P-a.s.,
Y ∈ BUC([0, T ] ×R
N).
Proof. The argument is well known, but we sketch it for convenience (see, e.g., [5, Theorem 8.1]). Weconsider the function
ψ(t, x, y) = Y (t, x) − Y (t, y) − γ(t, |x − y|N
),
where γ is a smooth function such that ψ < 0 on [0, T ] ×RN ×R
N . Then, if max ψ > 0 and
(x, t, y, s) = arg max
{Y (t, x) − Y (s, y) − γ
(t, |x − y|N
) − (t − s)2
α2− δ
(|x|2N + |y|2N)}
,
where α, δ > 0, it follows by the definition of the viscosity solution that
γt(t, |x − y|N
) + H(t, x, p) − H(s, y, p) � 0,
3840 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
where
p = γx
(t,
x − y
|x − y|N
).
By (iii), this yields
γt(t, |x − y|N
) − m(|x − y|N(1 + p)
) + 0(α) < 0. (27)
If pγ (t) = sup{|γx(t, x)|; |x|N < 1} and, choosing γ such that
γt(t, r) − m(r(1 + pγ (t)
))> δ1 > 0,
γ (0, r) = sup{∣∣X0(x) − X0(y)
∣∣; |x − y|N � r}, γ (t,0) = 0,
we arrive at a contradiction. Hence, we have
Y (t, x) − Y (t, y) � γ(t, |x − y|N
), ∀x, y ∈R
N ,
and this implies that Y ∈ BUC([0, T ] ×RN ), as desired. �
Lemma 3.3. Y is the unique viscosity solution to Eq. (9).
Proof. Nothing remains to be done except to combine Lemma 3.2 with the uniqueness of BUC([0, T ]×R
N ) viscosity solutions of Eq. (9) under hypotheses (i)–(iii) (see, [5,7,6]). �We shall establish now the following key lemma.
Lemma 3.4. For each x ∈RN , the stochastic process t → Y (t, x) is adapted to the filtration (Ft)t�0 .
Proof. The Ft -measurability of the function ω → Y (t, x,ω) does not follow directly from (19), though,as easily seen, the integrand is Fs-measurable. To show this, we shall proceed as follows. Replacingthe function L by the approximating Lagrangian
Lε(t, x, u) =∫RN
L(t, y, v)ρε(x − y, u − v)dy dv + ε|u|2N , ε > 0,
where ρε is a C1-mollifier, we may assume without any loss of generality that L is of class C1([0, T ]×R
N ×RN), and strictly convex in u. Indeed, as easily seen, for ε → 0, the sequence of functions
Yε(t, x) = infu
{ t∫0
(Lε(s, y, u) − W y
(s, y(s)
) · u(s))
ds + X0(
y(0)),
y(s) = x −t∫
s
u(τ )dτ , u ∈ L1(0, t);RN )
}, (28)
is P-a.s. convergent to Y on [0, T ] ×RN . In a similar way, we may assume without any loss of gener-
ality that Y0 ∈ C1b (RN ). Now, with this assumption we define the sequence of functions
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3841
Y n(t, x) = infu
{ t∫0
(L(s, y(s), u(s)
) − W y(s, y(s)
) · u(s))
ds + X0(
y(0));
y′(s) = u(s), a.e. s ∈ (0, t),∣∣u(s)
∣∣N � n, y(t) = x, u ∈ L1(0, t;RN)}
, n ∈N. (29)
It is easily seen that, P-a.s.,
limn→∞ Y n(t, x) = Y (t, x), ∀(t, x) ∈ [0, T ] ×R
N . (30)
Hence, it suffices to show that, for each n and all (t, x) ∈ [0, T ] × RN , ω → Y n(t, x) is
Ft -measurable. To this end, we note that, by the Pontriaghin maximum principle (see, e.g., [1, p. 45]),any optimal pair (y∗, u∗) in problem (29) satisfies the Euler–Lagrange system
(y∗(s)
)′ = u∗(s),(
p∗(s))′ = L y
(s·y∗(s), u∗(s)
) − W yy(s, y∗(s)
)(u∗(s)
),
p∗(s) ∈ Lu(s, y∗(s), u∗(s)
) − W y(s, y∗(s)
) + NUn
(u∗)(s), a.e. s ∈ (0, t),
y∗(t) = x, p∗(0) = (X0)y(
y∗(0)), (31)
where Lu = ∂u L is the subdifferential of the convex function u → L(t, x, u) and NUn ⊂ L1(0, t;RN ) isthe normal cone to the set
Un = {u ∈ L1(0, t;RN); ∣∣u(s)
∣∣N � n, a.e. s ∈ (0, t)
},
that is, NUn (u∗) = {η ∈ L∞(0, t;RN )}, where
η(s) ∈⎧⎨⎩
[0,∞) if u∗(s) = n,
(−∞,0] if u∗(s) = −n,
0 if u∗(s) ∈ (−n,n),
a.e. s ∈ (0, t).
Since, as mentioned earlier, the function u → L(·, u) is strictly convex, we have, for some λ > 0,
(Lu(·, u) − Lv(·, · v)
) · (u − v)� λ|u − v|2N , ∀u, v ∈RN ,
and so (Lu + NUn )−1 is Lipschitzian for each (t, x) ∈ [0, T ] ×R
N . Then, by (31), we see that
u∗(s) = (Lu(t, x, ·) + NUn
)−1(p∗(s) + W y
(s, y∗(s)
))−1, s ∈ [0, t],
and, therefore, u∗ is continuous on [0, t]. Then, we may rewrite (29) as
Y n(t, x) = inf
{ t∫0
(L
(s, x −
t∫s
u(τ )dτ , u(s)
)− u(s) · W y
(s, x −
t∫s
u(τ )dτ
))ds
+ X0
(x −
t∫u(τ )dτ
); u ∈ Un
}, (32)
0
3842 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
where
Un = {u ∈ C
([0, t];RN); ∣∣u(s)∣∣
N � n, ∀s ∈ [0, T ]}.Since Un is a separable metric space and the functions L, W y are in BUC([0, T ]×R
N × Bn), we seethat the infimum in (32) can be taken on a countable subset V of Un . Taking into account that, foreach u ∈ V , the function
ω →t∫
0
L
(s, x −
t∫s
u(τ )dτ , u(s)
)ds −
t∫0
u(s) · W y
(s, x −
t∫s
u(τ )dτ
)ds
is Ft -measurable (because each β j is (Ft)t�0-adapted), we conclude by (32) that ω → Y n(t, x) isFt -measurable and therefore, by virtue of (29), so is ω → Y (t, x) for all x ∈ R
N and t ∈ [0, T ], asdesired. �Proof of Theorem 2.2. (Continued.) By Lemmas 3.1–3.4, it follows that X is the unique viscosity so-lution to (1) and that X ∈ BUC([0, T ] × R
N × RN ), P-a.s. By (19), it follows also that, for P-almost
all ω ∈ Ω , Y and consequently X depend continuously on X0 as functions from C([0, T ];RN ) toBUC(RN ). This completes the proof. �Remark 3.5. If X0 is Lipschitzian and assumption (iii) is strengthen to
then it follows by (19) that x → Y (t, x) is Lipschitz continuous for each t ∈ [0, T ] and
∥∥Yx(·, t)∥∥
L∞(RN )� C, ∀t ∈ [0, T ].
(This also follows by Theorem 9.1 in [5].) Moreover, Y ∈ Lip([0, T ] ×RN ) and satisfies a.e. on (0, T ) ×
RN Eq. (13). In other words, X is a strong solution to (1) in Itô’s sense (13).
Remark 3.6. By (19) we have
X(t, x) = Y (t, x) + W (t, x)
= infu
{ t∫0
L(s, y(s), u(s)
)ds +
t∫0
m∑j=1
μ j(
y(s))
dβ j(s) + X0(
y(0));
y′(s) = u(s), a.e. s ∈ (0, t), y(t) = x, u ∈ L1(0, t;RN)}. (33)
In other words, X can be viewed as the action function of the stochastic variational problem with theHamiltonian system (4). Of course, under our general assumptions, this system does not have a strongsolution in Itô’s sense, so (4) cannot be used to construct, as in [9,10], a strong solution to Eq. (1), viastochastic characteristics defined by system (4).
In the special case of the stochastic eikonal equation
dX + ρ|Xx|N dt = dW , X(0) = X0,
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3843
we obtain by (33) the Lax–Hopf type formula, for the viscosity solution X ,
X(t, x) = inf
{X0
(y(0)
) +m∑
j=1
t∫0
μ j(
y(s))
dβ j(s), |u|N � ρ
}, t ∈ [0, T ], x ∈R
N .
We have a similar formula in the general case of Eq. (15) with the additive noise W .
4. Proof of Theorem 2.3
It suffices to show that, for each ω ∈ Ω , the Hamilton–Jacobi equation (13) has a unique viscositysolution Y such that t → Y (t, x) is (Ft)t�0-adapted and Y ∈ BUC([0, T ] × R
N ) P-a.s. To this end, wefix ω ∈ Ω and, for each Z ∈ BUC([0, T ];RN ), we consider the viscosity solution Y = F (Z) to theequation
Yt(t, x) + e−W (t,x)H(t, x, eW (t,x)(Yx(t, x) + W x(t, x)Z(t, x)
)) + μ(x)Z(t, x) = 0,
Y (0, x) = X0(x), x ∈RN , t ∈ [0, T ]. (34)
By Lemma 3.1, we know that Eq. (34) has a unique viscosity solution, Y ∈ BUC([0, T ]×RN ), which
is given by (see (19))
Y (t, x) = infu
{ t∫0
L(s, y(s), u(s)
)ds + X0
(y(0)
); y′(s) = u(s),
a.e. s ∈ (0, t), y(t) = x, u ∈ L1(0, t;RN)}, (35)
where L is the Lagrangian function associated with the Hamiltonian
H(t, x, p) = e−W (t,x)H(t, x, eW (t,x)(p + W x(t, x)Z(t, x)
3844 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
− (μ
(y(s)
) + u(s) · W y(s, y(s)
)e−W (s,y(s)))Z
(s, y(s)
))ds + X0
(y(0)
);y′(s) = u(s), a.e. s ∈ (0, t), u ∈ L1(0, t;RN)
, y(t) = x
}. (37)
By (36) and (iv), it follows that
L(t, x, u) = +∞ for |u|N > ν, t ∈ [0, T ], x ∈RN , (38)
and, therefore, the infimum into (37) is constrained to the set {u; |u(s)|N � ν , a.e. s ∈ (0,1)}.Next, by (37) and hypothesis (v), we see that (recall that Y = F (Z))
F (Z)(t, x) � F (Z)(t, x) + supu
{ t∫0
(μ
(y(s)
) + u(s)
· W y(s, y(s)
)e−W (s,y(s)))(Z
(s, y(s)
) − Z(s, y(s)
))ds;
y′ = u; y(t) = x,∣∣u(s)
∣∣N � ν, a.e. s ∈ (0, t)
}
� F (Z)(t, x) + C supu
{ t∫0
∣∣Z(s, y(s)
) − Z(s, y(s)
)∣∣ds;
y(s) = x −t∫
s
u(τ )dτ ,∣∣u(s)
∣∣N � ν, a.e. s ∈ (0, t), u ∈ L1(0, t;RN)}
. (39)
We introduce in the space X = {Z ∈ BUC([0, T ] ×RN )} the norm
‖Z‖α = sup{
eαt∣∣Z(t, x)
∣∣; (t, x) ∈ [0, T ] ×RN}
,
where α > 1. Then, by (39), we see that
∥∥F (Z) − F (Z)∥∥α� 1
α‖Z − Z‖α, ∀Z , Z ∈ BUC
([0, T ] ×RN)
,
and, therefore, by the Banach fixed point theorem, Eq. (37) (i.e., Z = F (Z)) has a unique viscosity so-lution Y ∈ BUC([0, T ]×R
N ). Moreover, since, as seen in Lemma 3.4, t → F (Z(t, x)) is (Ft)t�0-adaptedfor each x and the solution Y to Z = F (Z) is obtained as limit of the sequence {Zn}, whereZn = F (Zn−1), we infer that the process t → Y (t, x) is (Ft)t�0-adapted, too.
The uniqueness of the viscosity solutions Y ∈ BUC([0, T ] ×RN ) to (34) follows by assumption (iv)
(see, e.g., [7,6]). This completes the proof.
Remark 4.1. As easily follows from the previous proofs, Theorems 2.2 and 2.3 remain true for moregeneral Wiener processes
W (t, x) =m∑
j=1
μ j(t, x)β j(t), (t, x) ∈ [0,∞) ×RN ,
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3845
where μ j ∈ BUC([0, T ] ×RN ). In this case, Eq. (1) reduces to a random equation by the change of the
variable X = Y − ∫ t0
∑mj=1 μ j(s, x)dβ j(s), while in (3), instead of (11), we may use the substitution
X(t, x) =(
exp
( t∫0
m∑j=1
μ j(s, x)dβ j(s)
))Y (t, x), (t, x) ∈ [0, T ] ×R
N ,
to obtain for Y a random Hamilton–Jacobi equation of the form (34), for which hypotheses (i)–(iii)hold. The details are omitted.
Remark 4.2. Taking into account (14), we see that the previous proof applies word by word to proveTheorem 2.3 for the viscosity solutions to the Stratonovich equation (6).
5. The finite speed propagation
We establish here the finite speed propagation of the viscosity solutions X to Eq. (3). Namely, wehave
Theorem 5.1. Assume that H satisfies hypotheses (i), (ii), (iii), (iv), (v), H � 0 and that H(t, x,0) ≡ 0. LetX0 ∈ BUC(RN ) be such that support(X0) ⊂ B R . Then P-a.s. support(X(t, ·)) ⊂ B R+νt , for all t > 0.
Proof. It suffices to prove that support(Y (t, ·)) ⊂ B R+νt , P-a.s. for the solution Y to the randomEq. (34) or, equivalently (see (37)),
Y (t, x) = infu
{ t∫0
(e−W (s,y(s))L
(s, y(s), u(s)
) − (μ
(y(s)
)+ u(s)e−W (s,y(s)))Y
(s, y(s)
))ds + X0
(y(0)
),
y′(s) = u(s), a.e. s ∈ (0, t), u ∈ L1(0, t;RN), y(t) = x
}. (40)
As seen earlier in the proof of Theorem 2.3, by the Banach contraction principle, we have, for eachT > 0,
Y = limn→∞ Yn in C
([0, T ] ×RN)
, P-a.s.,
where Yn = F (Yn−1), that is,
Yn(t, x) = infu
{ t∫0
(e−W (s,y(s))L
(s, y(s), u(s)
) − (μ
(y(s)
)+ u(s)e−W (s,y(s)))Yn−1
(s, y(s)
))ds + X0
(y(0)
);y′(s) = u(s), a.e. s ∈ (0, t), u ∈ L1(0, t;RN)
, y(t) = x
}, n ∈N. (41)
3846 V. Barbu / J. Differential Equations 255 (2013) 3832–3847
On the other hand, by (38), it follows that each minimizing arc y in (41) must satisfy the constraint
|x|N � ν|t − s| + ∣∣y(s)∣∣
N � νt + ∣∣y(s)∣∣
N , ∀s ∈ (0, t).
Since support(X0) ⊂ BR , this implies that, for |x|N � νt + R , we have
Yn(t, x) = infu
{ t∫0
(e−W (s,y(s))L
(s, y(s), u(s)
)ds − (
μ(
y(s)) + u(s)e−W (s,y(s)))Yn−1
(s, y(s)
))ds;
y′(s) = u(s),∣∣u(s)
∣∣N � ν, a.e. s ∈ (0, t), u ∈ L1(0, t;RN)
, y(t) = x
}, n � 1, (42)
and, in particular, for n = 1, we obtain that for Y0 = X0
Y1(t, x) = infu
{ t∫0
e−W (s,y(s))L(s, y(s), u(s)
)ds; y′(s) = u(s),
∣∣u(s)∣∣
N � ν, a.e. s ∈ (0, t), u ∈ L1(0, t;RN), y(t) = x
}. (43)
Since H � 0, H(t, x,0) ≡ 0, we have L � 0 on (0, T ) ×RN ×R
N and
L(t, x,0) = infu
{L(t, x, u); u ∈R
N} = 0, ∀(t, x) ∈ (0, T ) ×RN .
Then (43) yields
Y1(t, x) = 0, for |x|N � νt + R, t � 0,
and, iterating by induction (42), we get that, for all n ∈ N,
Yn(t, x) = 0, for |x|N � νt + R, t � 0.
Hence
Y (t, x) = 0, for |x|N � νt + R, t � 0,
which completes the proof. �Remark 5.2. It is interesting that the finite speed of the propagation property is independent of thecoefficients μ j of the Wiener process W (t). Theorem 5.1 applies, in particular, to the eikonal equation(15) and so, if X0(x) = 0 for |x|N � R , then we have
If we view (15) as a model of the flame propagation in R N , N = 1,2,3, then {x ∈ RN ; X(t, x) =
0} = Ot represents the burnt region at time t and (44) shows that this region extends with finitespeed � ν .
V. Barbu / J. Differential Equations 255 (2013) 3832–3847 3847
Acknowledgments
This work was supported by a grant of the Romanian National Authority for Scientific Research,CNCS–UEFISCDI, project PN-II-PCE-2011-3-0027.
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