IJMMS 2003:43, 2735–2746
PII. S0161171203211121
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON A STOCHASTIC BURGERS EQUATION
WITH DIRICHLET BOUNDARY
CONDITIONS
EKATERINA T. KOLKOVSKA
Received 5 November 2002
We consider the one-dimensional Burgers equation perturbed by a white noise
term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal
approximations for the equation and solving a martingale problem for the weak
limit.
2000 Mathematics Subject Classification: 60H15, 60H40, 35Q35.
1. Introduction. We consider the stochastic partial differential equation
∂2
∂ ∂
u(t, x) =
g t, x, u(t, x)
u(t, x) + f t, x, u(t, x) +
2
∂t
∂x
∂x
∂2
+ σ t, x, u(t, x)
W (t, x)
∂t∂x
(1.1)
with Dirichlet boundary conditions
u(t, 0) = u(t, 1) = 0,
t ≥ 0,
(1.2)
x ∈ [0, 1],
(1.3)
and initial condition
u(0, x) = u0 (x),
where (∂ 2 /∂t∂x)W (t, x) is a spacetime white noise (see [19] for the definition
and properties of the white noise), u0 ∈ L2 ([0, 1]), and f = f (t, x, y), g =
g(t, x, y), and σ = σ (t, x, y) are Borel-measurable functions on R+ ×[0, 1]×R.
A solution to this equation is an L2 ([0, 1])-valued continuous process, adapted
to the filtration generated by the white noise, which solves the equation in a
weak sense (see below).
When f = σ = 0 and g(t, x, y) = y 2 /2, the above equation is called Burgers
equation. It has been proposed as a model for turbulent fluid motion (see
[4, 5, 10]). When g = 0, the equation is a stochastic reaction-diffusion equation
which has been studied intensively (see, e.g., [3, 8, 13, 19] and the references
therein).
2736
EKATERINA T. KOLKOVSKA
When f = 0, g(t, x, y) = y 2 /2, and σ = 0, we obtain the Burgers equation
perturbed by a spacetime white noise. It has been studied by several authors
under Lipschitz coefficients on σ (see, e.g., [1, 2, 7, 9] and the references
therein).
Our aim in this paper is to study a one-dimensional Burgers equation perturbed by a stochastic noise term with a non-Lipschitz coefficient, namely,
∂
u(t, x) = ∆u(t, x) + λ∇u2 (t, x)
∂t
∂2
+ γ u(t, x) 1 − u(t, x)
W (t, x),
∂t∂x
(1.4)
u(t, 0) = u(t, 1) = 0,
u(0, x) = f (x),
x ∈ [0, 1],
where ∆ = ∂ 2 /∂x 2 , ∇ = ∂/∂x, and f (x) : [0, 1] → [0, 1] is a continuous function.
The stochastic term in this equation corresponds to continuous-time steppingstone models in population genetics (see [6, 15]), where u(t, x) models the gene
frequency in colonies.
Equation (1.4) is interpreted in the weak sense, which means that, for each
ϕ ∈ C 2 ([0, 1]),
[0,1]
u(t, x)ϕ(x)dx =
[0,1]
u(0, x)ϕ(x)dx +
t
−λ
0
u2 (s, x)ϕ (x)dx ds
[0,1]
t
+γ
0
u(t, x)ϕ (x)dx
[0,1]
[0,1]
(1.5)
u(s, x) 1 − u(s, x) ϕ(x)W (ds, dx).
Since the coefficients of (1.4) are non-Lipschitz, the standard results on existence and uniqueness of solutions cannot be applied.
In this paper, we prove existence of a nonnegative weak solution of (1.4)
(Theorem 4.2). Our method of proof is briefly described as follows. Following
Funaki [8], in Section 2 we define a discrete version of (1.4), which is a finitedimensional system (2.4) of stochastic differential equations. Next, we prove
existence of a weak solution for this system and use the method of Le Gall
[14] to obtain pathwise uniqueness of weak solutions. This yields existence
of a unique strong solution xkN (t) of (2.4). In Section 3, we define a system
of polygonal approximations uN (t, x) of xkN (t) and use the multidimensional
Kolmogorov-Totoki criterion to obtain tightness of {uN (t, x), N = 1, 2, . . .}. In
Section 4, we use a martingale problem to conclude the proof of existence of a
weak solution of (1.4). Without loss of generality, we will assume that λ = γ = 1.
ON A STOCHASTIC BURGERS EQUATION . . .
2737
2. Existence of a solution of the discretized version. We fix an integer N ≥
1 and consider the discretized version of (1.4) on the set {k/N, 1 ≤ k ≤ N}:
∂
k
k
k
2
XN t,
= ∆N XN t,
+ ∇N XN
t,
∂t
N
N
N
k
k
+ NXN t,
1 − XN t,
dBk (t),
N
N
k
k
XN 0,
=f
, 1 ≤ k ≤ N, t ≥ 0.
N
N
(2.1)
Here, N is the set of the nonnegative integer numbers, {Bk (t)}1≤k≤N is an infinite system of independent one-dimensional Brownian motions, and ∇N and
∆N are, respectively, the discrete approximations of the first and second derivative with respect to the variable x:
XN t, (k + 1)/N − 2XN (t, k/N) + XN t, (k − 1)/N
k
=
,
∆N XN t,
N
1/N 2
h s, (k + 1)/N − h s, k/N
k
=
∇N h s,
, 1 ≤ k ≤ N.
N
1/N
(2.2)
We write xkN (t) = XN (t, k/N). Substituting the above expressions in (2.1), we
obtain the finite-dimensional system of stochastic differential equations
N
N
N
(t) − 2xiN (t) + xi−1
(t) + Nxi+1
(t)2 − NxiN (t)2
dxiN (t) = N 2 xi+1
+ NxiN (t) 1 − xiN (t) dBi (t), i = 1, . . . , N,
(2.3)
which can be written in the more compact form

dxiN (t)
=
N

N
N N
2
aN
dt +
ij xj (t) + bij xj (t)
NxiN (t) 1 − xiN (t) dBi (t),
j=1
i
xiN (0) = f
,
N
(2.4)
1 ≤ i, j ≤ N,
where


N2



N
aij = −2N 2



0


N



N
bij = −N



0
if j = i + 1, i − 1,
if j = i,
otherwise,
(2.5)
if j = i + 1,
if j = i,
otherwise.
2738
EKATERINA T. KOLKOVSKA
Note that for this system we cannot apply the standard results on existence
and uniqueness of solution because Lipschitz assumptions on the drift and
diffusion coefficients fail. We prove the following result.
N
) ∈
Theorem 2.1. For any initial random condition X N (0) = (x1N , . . . , xN
N
[0, 1] , the system

dxiN (t)
=
N
aN
ij xj (t) +
j

N N
bij
xj (t)2 dt
j
+ NxiN (t) 1 − xiN (t) dBi (t),
xiN (0) = xi ,
(2.6)
i = 1, . . . , N,
N
(t)) ∈ C([0, ∞), [0, 1]N ).
admits a unique strong solution X N (t) = (x1N (t), . . . , xN
Proof. We consider the rescaled system

dxiN (t)
=
j
N
aN
ij xj (t) +
N N
bij
xj (t)2 dt
j
+ g xiN (t) dBi (t),
xiN (0) = xi ,
(2.7)
i = 1, . . . , N,
where g : R → R is defined by g(x) = Nx(1−x) for 0 ≤ x ≤ 1, and g(x) = 0 otherwise. Since the coefficients of (2.7) are continuous, by the Skorokhod’s existence theorem (see [11, 17]), we conclude that there exists on some probability
space a weak solution X N (t) of (2.7). We will prove that for each weak solution
N
(t)) of this system, xiN (t) ∈ [0, 1] for all i = 1, . . . , N and
X N (t) = (x1N (t), . . . , xN
t ≥ 0, thus showing that X N (t) is a solution of (2.6).
First, we show that xiN (t) ≥ 0 for each i = 1, . . . , N. Since the coefficients
of the system are non-Lipschitz, the solution may explode in a finite time.
Let τ1 ≤ ∞ denote the explosion time of the solution. If some of the solution
coordinates are negative, then there exists a random time 0 < τ2 ≤ ∞ such that
for 0 < t ≤ τ2 , all such coordinates are between −1 and 0. This is so because
(2.7) is finite-dimensional, and its solution is continuous.
We will use the following lemma (see [14]).
Lemma 2.2. Let Z ≡ {Z(t), t ≥ 0} be a real-valued semimartingale. Suppose
ε
that there exists a function ρ : [0, ∞) → [0, ∞) such that 0 du/ρ(u) = +∞ for
t
all ε > 0, and 0 (1{Zs >0} /ρ(Zs ))d
Zs < ∞ for all t > 0 a.s. Then the local time
at zero of Z, L0t (Z), is identically zero for all t a.s.
2739
ON A STOCHASTIC BURGERS EQUATION . . .
Applying Lemma 2.2 to xiN (t) with ρ(u) = u and using the Tanaka formula
(see [16]), we obtain for xiN (t)− := max[0, −xiN (t)],
N
xiN (t)− = −
t N
0
i=1
≤
≤
=N
N
N
N
xj (s)2 ds
aij xj (s) + bij
j=1
1x N (s)<0 aN
ij xj (s)− ds + N
i
i,j=1
t N
0
i
i=1
t N
0
1x N (s)<0
aN
ij xj (s)− ds + N
i,j=1
t N
0
t N
0
t N
0
1x N (s)<0 xi (s)2 ds
i
i=1
(2.8)
xiN (s)− ds
i=1
xiN (s)− ds,
i=1
N
i aij = 0 to obtain the last equality. Then by Gronwall’s
N
lemma, we obtain that i=1 xiN (t)− = 0, and hence that the solution is nonnegative for each t ≥ 0. By a similar argument applied to (1 − xiN (t))− , it follows
that xiN (t) ≤ 1 for each 1 ≤ i ≤ N.
where we used that
Using Lemma 2.2, we will prove pathwise uniqueness of weak solutions of
1,N
1,N
2,N
2,N
(2.6). Let X 1,N = (x1 , . . . , xN ) and X 2,N = (x1 , . . . , xN ) be two solutions
of (2.6) with the same initial conditions and the same Brownian motions. We
define
l,N
N l,N
2
di X l,N (t) = aN
ij xj (t) + bij xj (t) ,
t ≥ 0, l = 1, 2.
(2.9)
Then
1,N
xi
2,N
(t) − xi
t
(t) =
0
+
1,N
di X (s) − di X 2,N (s) ds
t 1,N
1,N
Nxi (s) 1 − xi (s)
0
2,N
2,N − Nxi (s) 1 − xi
(s) dBi (s),
i = 1, . . . , N.
(2.10)
Since
Xt =
t 0
1,N
Nxi
2
1,N
2,N
2,N (s) ds,
(s) 1 − xi (s) − Nxi (s) 1 − xi
2
1,N
1,N
2,N
2,N t
Nxi (s) 1 − xi (s) − Nxi (s) 1 − xi
(s)
0
1,N
xi
2,N
(s) − xi
(s)
1x 1,N (s)−x 2,N (s)>0 ds
i
i
t
≤
0
2N1x 1,N (s)−x 2,N (s)>0 ds < 2Nt,
i
i
(2.11)
2740
EKATERINA T. KOLKOVSKA
(where we used that ( x(1 − x) − y(1 − y))/(x − y) < 2 for x, y ∈ [0, 1],
x > y, which follows from L’Hospital rule), we can apply Lemma 2.2 to X(t) =
1,N
2,N
1,N
2,N
xi (t) − xi (t) with ρ(x) = x. Therefore, L0t (xi (s) − xi (s)) = 0 for all
i ∈ {1, . . . , N}.
By Tanaka’s formula,
1,N
x (t) − x 2,N (t) =
i
i
t
0
1,N
2,N
sgn xi (s) − xi (s) di X 1,N (s) − di X 2,N (s) ds
t
+
0
1,N
2,N
sgn xi (s) − xi (s)
1,N
1,N
× Nxi (s) 1 − xi (s)
2,N
2,N
− Nxi (s) 1 − xi (s) dBi (s),
i = 1, . . . , N.
(2.12)
N
Since aN
ij and bij are bounded, it follows that
E
t N
N
1,N
1,N
x (t) − x 2,N (t) ≤ E
di X (s) − di X 2,N (s) ds
i
i
0
i=1
t
≤
0
i=1
N
1,N
x (s) − x 2,N (s)ds,
K(N)E
i
(2.13)
i
i=1
where K(N) is a constant depending on N. From Gronwall’s inequality, we
conclude that
E
d
1,N
x (t) − x 2,N (t) = 0
i
i
(2.14)
i=1
for all t ≥ 0, thus proving pathwise uniqueness. By a classical theorem of
Yamada and Watanabe [20], this is sufficient for existence of a unique strong
solution of (2.6).
3. Tightness of the approximating processes. From Theorem 2.1, there exists a strong solution of the system approximating (1.4)
dxiN (t) =
1≤j≤N
N
aN
ij xj (t) +
i
xiN (0) = f
,
N
N N
bij
xj (t)2 + NxiN (t) 1 − xiN (t) dBi (t),
1≤j≤N
i = 1, 2, . . . , N,
(3.1)
2741
ON A STOCHASTIC BURGERS EQUATION . . .
where N ∈ N is fixed. We denote by uN (t, x) the polygonal approximation of
xiN (t)
[Nx] + 1 Nx − [Nx]
uN (t, x) = XN t,
N
[Nx] [Nx] + 1 − Nx ,
+ XN t,
N
(3.2)
t ≥ 0, x ∈ [0, 1],
where by definition, [y] = k/N for k/N ≤ y < (k + 1)/N. Therefore, we have
XN (t, k/N) = xkN (t) = uN (t, k/N), 1 ≤ k ≤ N, and 0 ≤ uN (t, x) ≤ 1 for all t ≥ 0,
x ∈ [0, 1].
Let pN (t, i/N, j/N), t ≥ 0, 0 ≤ i, j ≤ N + 1, be the fundamental solution of
∆N , that is,
i j
i j
∂
pN t, ,
= ∆N pN t, ,
,
∂t
N N
N N
i j
pN 0, ,
= Nδij ,
N N
t > 0, 1 ≤ i, j ≤ N,
(3.3)
with the boundary conditions
j
N +1 j
= pN t,
,
= 0,
pN t, 0,
N
N
N
t > 0, 1 ≤ j ≤ N.
(3.4)
Then (3.1) is equivalent to the system (see [12])
xiN (t) =
N
1
i j
pN t, ,
xjN (0)
N
N
N
j=1

N
1
i
j
N
2
+ 
pN t − s, ,
b(i, j)xj (s) ds
N
N N
0
j=1
t
+

t N 0
j=1
i j NxjN (s) 1 − xjN (s) dBj (s),
pN t − s, ,
N N
1 ≤ i ≤ N,
(3.5)
where in the last integral we used that {(1/N)Bj (s), 1 ≤ j ≤ N} is an independent system of Brownian motions which we also denote by {Bj (s)}.
We define the rescaled polygonal interpolation GN of pN in [0, 1] by
j
[Nx] + 1 j = pN t,
,
GN t, x,
Nx − [Nx]
N
N
N
[Nx] j + pN t,
,
[Nx] + 1 − Nx .
N
N
(3.6)
2742
EKATERINA T. KOLKOVSKA
Using (3.2) and (3.8), we obtain the following representation for the approximate solution. For x ∈ [i/N, (i + 1)/N),
uN (t, x) =
N
1
j
j
GN t, x,
uN 0,
N
N
N
j=1
t N 1
i+1 j
pN t − s,
,
b(i + 1, j)xjN (s)2 Nx − [Nx]
N
N
N
0
j=1
1
i j
b(i, j)xjN (s)2 [Nx] + 1 − Nx ds
+ pN t − s, ,
N
N N
t N
j
j
j
+
GN t − s, x,
NuN s,
1 − uN s,
dBj (s)ds
N
N
N
0
j=1
+
(1)
(2)
(3)
:= uN (t, x) + uN (t, x) + uN (t, x).
(3.7)
Then uN (t, x) satisfies the boundary conditions in (1.4).
Theorem 3.1. For each T > 0, the sequence {uN (t, x), N ≥ 1} is tight in the
space C([0, T ], A), where A = C([0, 1], [0, 1]).
Proof. Using the fact that uN (t, x) ∈ [0, 1], we obtain, as in the proofs of
[8, Lemma 2.2] and [8, Proposition 2.1], that for each T < ∞ and p ∈ N, there
exists C = C(T , p) > 0 such that
p/2
(3) 2p
(3) + |x − y|p/2
E uN t1 , x − uN t2 , y ≤ C t1 − t2 (3.8)
for every t1 , t2 ∈ [0, T ], x, y ∈ [0, 1], and N ∈ N, and that
lim
sup
N→∞ (t,y)∈[0,T ]×[0,1]
(1)
u (t, y) − u(t, y) = 0,
N
(3.9)
where u(t, y) is the fundamental solution of ∆. Since
(2)
uN
t,
k
N
=
t k k+1
k k
N
xk+1
xkN (s)2 ds
pN t − s, ,
(s)2 − pN t − s, ,
N N
N N
0
(3.10)
and pN is a fundamental solution of ∆N , it follows that
k
k
k+1 2
k 2
∂ (2)
(2)
uN t,
= ∆N uN t,
+ uN t,
− uN t,
.
∂t
N
N
N
N
(3.11)
2743
ON A STOCHASTIC BURGERS EQUATION . . .
(2)
(2)
Also, uN (t, 0) = uN (t, 1) = 0 by the boundary condition (3.8). Integration by
parts and an application of Gronwall’s inequality give, as in [12, Theorem 4.2],
t
(2)
(1)
k k
k (3)
≤ e2t
ds.
u
+
u
max
t,
s,
s,
max u
N
N
N
N N
N 1≤k≤N
0 1≤k≤N
(3.12)
(2)
Hence, from the polygonal form of uN and (3.8), (3.9), we obtain that for every
T < ∞ and p ∈ N, there exists C1 = C1 (T , p) > 0 such that
!
p/2
(2)
2p
(2)
E uN (t1 , x) − uN (t2 , y) ≤ C1 t1 − t2 + |x − y|p/2
(3.13)
for every t1 , t2 ∈ [0, T ], x, y ∈ [0, 1], and N ∈ N. It follows from the multidimensional Kolmogorov-Totoki criterion [18] that uN (t, x) ∈ C([0, T ], A) and
that the family {uN (t, x), N ∈ N} is tight in C([0, T ], A) for each positive T .
4. The martingale problem for the stochastic partial differential
equation. Since the sequence {uN (t, x), N ≥ 1} is tight by Theorem 3.1, there
exists a subsequence, which we denote again by {uNk (t, x)}, that converges
weakly in C([0, T ], A) to a limit v(t, x). By the Skorokhod’s representation
theorem, we can construct processes {vN (t, x)}, u(t, x) on some probability
Ᏸ
Ᏸ
space (Ω, Ᏺ, {Ᏺt }, P ) such that {uN } = {vN }, u = v, and {vN (t, x)} converges
to u(t, x) uniformly on compact subsets of [0, T ] × R for any T > 0 as N → ∞.
Obviously, u(t, x) satisfies the boundary conditions in (1.4). We will show that
u(t, x) is a weak solution of (1.4) by solving the corresponding martingale
problem.
Theorem 4.1. For any ϕ ∈ Cc2 ,
ᏹϕ (t) :=
[0,1]
u(t, x)ϕ(x)dx −
−
t
[0,1]
u(0, x)ϕ(x)dx
[0,1]
u(t, x)ϕ (x)dx −
is an {Ᏺt }-martingale with ᏹϕ t =
(4.1)
2
u (s, x)ϕ (x)dx
0
[0,1]
t
0 [0,1] u(s, x)(1 − u(s, x))ϕ
2
(x)dx ds.
Proof. Using that
n∈Z
an bn+1 − bn +
bn+1 an+1 − an = 0
n∈Z
(4.2)
2744
EKATERINA T. KOLKOVSKA
and multiplying both sides of (2.1) by ϕ(k/N)(1/N), we obtain, for fixed N ≥ 1,
N
ᏹϕ
(t) :=
N
k 1 k 1
k
k
ϕ
−
ϕ
uN t,
uN 0,
N
N
N
N
N
N
k=1
k=1
N
t # " N
k 1
k 1
k
k
ϕ
−
ϕ
∆N uN s,
∇N u2N s,
−
N
N
N
N
N
N
0
0
k=1
k=1
t N
=
k
k 1 k 1
k
k
− vN 0,
ϕ
ϕ
vN t,
N
N N
N
N
N
k
t k 1
k 1
k
k
∆N ϕ
−
∇N ϕ
vN s,
vN2 s,
N
N
N
N
N
N
0
0
k
k
k 1 t
k
k
NvN s,
= ϕ
1 − vN s,
dBk (s).
N N 0
N
N
k
−
t
(4.3)
N
N
Hence, ᏹϕ
(t) is a martingale because by (4.3), ᏹϕ
(t) is the sum of a finite
N
number of martingales. Moreover, {ᏹϕ (t)} is uniformly integrable because
N
supN∈N E(ᏹϕ
(t))2 < ∞ uniformly in t ∈ [0, T ]. Indeed, since ϕ2 is integrable,
t 2 N
k 1
k
k
(t) = ϕ2
vN s,
E ᏹϕ
1 − vN s,
ds
N N 0
N
N
k
1
k
≤T
ϕ2
N
N
k
(4.4)
< C(ϕ, T ),
where C(ϕ, T ) is a finite constant depending only on ϕ and T , but not on N.
N
(t) → ᏹϕ (t) as N → ∞, where
Therefore, ᏹϕ
ᏹϕ (t) =
[0,1]
v(t, x)ϕ(x)dx −
t1
−
0
0
v(t, 0)ϕ(x)dx
[0,1]
v(s, x)ϕ (x)dx ds −
t1
0
(4.5)
v 2 (s, x)ϕ (x)dx ds
0
is a martingale. From (4.3), we obtain the quadratic variation of ᏹϕ (t), which
is given by
$
%
ᏹN (ϕ) t =
&
k
=
k
ϕ
N
t
0
k
1
N
'
t
k
k
1 − vN s,
dBk (s)
NvN s,
N
N
0
t
k
k
k 1
1 − vN s,
ϕ2
NvN s,
ds.
N
N
N N2
(4.6)
ON A STOCHASTIC BURGERS EQUATION . . .
Hence, limN→∞ ᏹN (ϕ)t =
and the theorem is proved.
t
0 [0,1] v(s, x)(1 − v(s, x))ϕ
2
2745
(x)dx ds = ᏹ(ϕ)t ,
Now, we proceed to the proof of the main result.
Theorem 4.2. The process u(t, x) is a weak solution of the stochastic partial
differential equation (1.4).
Proof. To the quadratic variation ᏹ(ϕ)t , there corresponds a martingale
measure ᏹ(dt, dx) with quadratic measure
ν(dx dt) = u(t, x) 1 − u(t, x) dx dt
(4.7)
( be a white noise independent of ᏹ (defined possibly on an
(see [19]). Let W
extended probability space). We define
t
Wt (ϕ) =
[0,1]
+
0
1
1{u(s,x)∈{0,1}} ϕ(x)ᏹ(ds, dx)
u(s, x) 1 − u(s, x)
t
[0,1]
0
(4.8)
((ds, dx).
1{u(s,x)∈{0,1}} ϕ(x)W
Then Wt corresponds to a spacetime white noise W (ds, dx) such that
ᏹt (ϕ) =
t
[0,1]
0
u(s, x) 1 − u(s, x) ϕ(x)W (ds, dx).
(4.9)
From (4.5), we conclude that u(t, x) satisfies (1.5), and hence that u(t, x) is a
weak solution of (1.4). The theorem is proved.
Acknowledgments. The author wishes to acknowledge the referees for
their careful reading of the manuscript and their useful comments. This research was partially supported by CONACYT Grant 32401E.
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Ekaterina T. Kolkovska: Centro de Investigacion en Matematicas (CIMAT), Guanajuato, CP 36000, Mexico