SPDE

STOCHASTIC DIFFERENTIAL EQUATIONS WITH PARTIAL
DERIVATIVES.
Gikhman, Ilya
PLLS
49 East Fourth Street
Cincinnati, OH 45202
ph: (513) 763-8308
e-mail: iljogik@yahoo.com
INTRODUCTION
Stochastic partial differential equations are the part of the general theory of random fields
which is actively being developed and applied. There exist different approaches to setting and
solving these problems. We single out the principle ones. In monograph [39] the linear theory
based on the interpretation of parabolic operators as infinitesimal operators of diffusion
processes is given. Generalizations of that approach were established in [33]. In this
monograph using the stochastic characteristic systems like the case of the first order
deterministic partial differential equations it was shown that a first and the second orders
stochastic partial differential systems can be solved. Using A number of application, where the
necessity of solving stochastic parabolic problems appears, have been shown. In [32,36] the
generalization of direct methods of mathematical physics which allows to prove solvability of
non-linear stochastic systems is shown. For these methods it is necessary to obtain apriori
estimates assuming coefficients to be coercive, and also to prove the possibility of limit
transition in finite-dimensional approximations of input equation to be monotone. The main
role in these methods is assigned to a drift coefficient. When it is equal to zero, a diffusion
coefficient can be only a bounded operator, that satisfies a Lipschitz condition. In [41]
martingale statement is given and solvability of a number of evolution problems is proved. It is
well known that the existence of a finite second moment is essential for the existence of
solutions of ordinary stochastic differential equations. Under certain assumptions on
coefficients growth, as it is shown in [21], a solution of an ordinary stochastic differential
equation does not possess a finite second moment. In [22, 28] a solution of parabolic equations
with coefficients of ”white noise” type is constructed. One more approach to the solution of
stochastic parabolic problems consists in the substitution of integral equation of Ito-Volterra
type in corresponding functional spaces for initial problem [3, 6, 34]. This method has been
initial in stating and solving stochastic equations with unbounded operators.
We have mentioned only the works in which the principal approaches are described. At
present the list of papers and books where solutions of stochastic evolution problems are
studied is quite long and covers also physical, chemical and biological literature.
In this paper a direct probabilistic method of a solution of the Cauchy problem for
semi-linear parabolic equation is suggested, and its physical interpretation is considered. Some
of the results presenting here are given in [15]. We note that the results obtained below
correspond to those obtained by probabilistic methods used for a study of deterministic
quasilinear parabolic equations [4, 11, 40]. It is also worthy to note the difference between the
Current address: PLLS,
49 East Fourth Street, Cincinnati, OH 45202

solutions of semilinear and quasilinear systems, although the semilinear case of determined
systems has not been taken separately. The latter allows to prove the existence and uniqueness
of the ”classical” solution of the Cauchy problem when coefficients are sufficiently smooth.
The former makes this possible only for a continuous solution [11, 40]. Moreover, the solution
is local in character when there is no dissipation.
At the last part of the paper we introduce mathematically correct stochastic interpretation
of the Schrodinger equation solution [16-18,23-26] and some model examples that use this
representation. The same complex-valued representation for the Shrödinger equation solution
first was given in [9]. Some applications of this approach to the particular problems were
studied in [1,8,31].
Now, we shall remind the basic aspects of the Lagrange and Euler formalisms used to
describe the dynamics of solid medial. Further, physical interpretation will be generalized for
the stochastic case. We can describe dynamic processes in physical media in two ways. The
first consists in the treatment of medium parameters, for each moment t ≥ 0, as functions of
some fixed system of coordinates  x1, x2, ..., xn . Such a method is called Euler method,
and coordinates are Euler coordinates. The second is called the Lagrange method. It consists in
the interpretation of solid medium as an aggregation of particles. Each particle differs from the
other ones in its initial position. Both of these alternative approaches play an important role for
statistical description of motion in nonhomogeneous media, in the turbulence theory and other
applications [22]. We shall now analyze these methods in the main. It is convenient to interpret
particle motion trajectory via the inverse of time.
Let, for the initial moment t ≥ 0 , a particle occupy the position of y and suppose
V  s, x  to be a velocity of this particle at the point x and moment s ≥ 0. Then the Euler
coordinate x = x  s ; t , y  at the moment s ∈  0, t is calculated by the formula
xi  s ; t, y  = yi −
t
s
∫ v i  r, x  r ; t, y  d r , i = 1, 2, ... n #
Formula ( 0. 1 ) transforms the Lagrange coordinates into the Euler ones. Assume that
F  t, x  is a smooth function of the Euler coordinates and a medium parameter. It is easy to
represent it as a function of the Lagrange coordinates
Fl  s, y  = F  s, y −
t
s
∫ v i  r, x  r ; t, y   d r  #
By differentiating the relation ( 0. 2 ) with respect to s we find
d F l
 s, y 
d s
=
∂ F  s, x 
∂ s
+
n
i = 1
∑ vi  s, x 
∂ F  s, x 
∂ x i
x = x  s, t, y 
#

The expression in square brackets on the right hand side of ( 0. 3 ) is total derivative with
respect to time along particle path. It is called a substantive derivative. In the case, when the
solution of equation ( 0. 1 ) is unique and the function V  t, x  is sufficiently smooth,
formula ( 0.1 ) effects diffiomorfism of the Lagrange coordinates into Euler ones. In order to
show the equivalence of the Lagrange and Euler formalisms we construct the inverse mapping
of the Euler coordinates into Lagrange ones. It means, by definitions to find a sufficiently
smooth function G  s, x  for which the equation
G  s, x  s ; t , x   = y #
has a unique solution. Taking into account of the formula for a substantive derivative, one
can write equation ( 0. 4 ) in the form
G  0, y  +
t
s
∫  ∂
∂ s
+  v  s, x  , ∇  G  s, x  |
x = x  s ; t , y 
d s = y
#
where ∇ =  ∂
∂ x 1
, ∂
∂ x 2
,..., ∂
∂ x n
 . From this it follows that the
function G  t, x  , which defines the transformation is a solution of the Cauchy problem
∂ G  s, x 
∂ s
+
n
i = 1
∑ vi  s, x 
∂ G  s, x 
∂ x i
= 0 , G  0, y  = y #
We next consider the case when particle paths are described by solutions of the ordinary
stochastic differential equations. To do this an accurate theory analogous to the formalism
described above can be establish. It will turn out, that the definition of the function
G  ∗ , ∗  along the trajectory of the solution of stochastic differential equation ( 0. 4 )
can be represented by solution of ( 0. 4 ) in the Euler coordinates with respect to
macroparameter G  t , x  which characterizes solid medium. We note that the equations
used here, with inverse time for the Lagrange trajectories presuppose to consider the Euler
dynamics via a straight direction of time.
Semilinear SPDE.
Before representing our results, let us introduce another approach to the Cauchy problem
study of the SPDE of the parabolic type. After the presentation of the Backward Stochastic
Differential Equations ( BSDE ) by Peng and Pardoux [37] in 1990, this field has received
increasing interest and activity [10,37,38]. Following [38] let

d X s
t,x
= b  s , X s
t,x
 ds + σ  s , X s
t,x
 d w  s 
where s ∈  t , ∞  be a classical Ito equation solution such that X t
t,x
= x. The
associated Peng-Pardoux BSDE is
Y s
t,x
= h  X T
t,x
 +
T
s
∫ f  r , X r
t,x
, Y r
t,x
, Z r
t,x
 dr +
+
T
s
∫ g  r , X r
t,x
, Y r
t,x
, Z r
t,x
 d B  r  −
T
s
∫ Z r
t,x
d w  r 
where s ∈  t , T , and w  t , B  t  , t ≥ 0 are two independent Wiener
processes. They proved that under certain assumptions on coefficients of BSDE , X s
t,x
has a
version whose trajectories are continuous in t, s and twice continuously differentiable in x.
They showed that the random field Z s
t,x
has a modification such that
Z s
t,x
= ∇ Y s
t,x
 ∇ X s
t,x
 − 1
σ  X s
t,x

Thus, substituting this value into BSDE and letting
Y r
t,x
= u  r, X r
t,x
 and then Y t
t,x
= u  t, x .
we can see that BSDE is the type of the stochastic evolution equation that is studied in this
section. An unessential difference between BSDE and the stochastic evolution systems studied
earlier is direction of time. Peng-Pardoux chose forward time for the characteristics X ∗ and
thus implied inverse time for BSDE. In papers [15,19,19,22] were used backward time for the
characteristics and forward time for the evolution which gives us the possibility of study
Cauchy problem for SPDE in the common direction of time. It seems also important to
emphasize that the application BSDE to the study of the deterministic PDE [10] in their
explicit evolutionary form will lead us to the probabilistic methods of the study Cauchy
problem for the nonlinear parabolic deterministic systems [4,7,11,27,40].
We will introduce the notations. Let E
m
be -dimensional Euclid space and
E +
= 0, + ∞ . Assume that the nonrandom Borel functions b  t, x , C  t, x ,
defined for  t , x  ∈ E +
× E
m
and taking values in E
m
and E
d
× E
m
satisfy the
following conditions
| b  t, x  | + | C  t, x  | ≤ K  1 + | x |  #
| b  t, x  − b  t, y  | + | C  t, x  − C  t, y  | ≤ | x − y |
where K > 0, | C |
2
= Tr CC
∗
. Consider the Ito SDE with inverse time

ξ  s ; t, x  = x +
t
s
∫ b  r, ξ  r ; t, x   d r −
t
s
∫ C  r, ξ  r ; t, x   d
←
w  r 
#
Here, the stochastic integral is the inverse stochastic Ito integral, and w  t  is d-dimensional
Wiener process on a some complete probability space  Ω , Ϝ , P  . By definition the
solution of equation ( 1. 2 ) is a random function ξ  s ; t, x  measurable in its variables
s, t, x, ω respectively, and for fixed t, x with respect to the σ - fields
Ϝ
s
= σ  w  t
′
 − w  t
′′
 ; t
′
, t
′′
∈  s, t 
and satisfying the equality ( 1. 2 ) with probability 1 for all s, t, x at once. Taking into
account the fact that SDE theory with inverse time is identical to the one with the standard
time, we are going briefly to state the properties for backward Ito integrals. Let us introduce
two two-parameters σ- fields
→
Ϝ s
t
,
←
Ϝ t
T
are mutually independent of each other, right and
left hand sides continuous respectively and
→
Ϝ s
t 1
⊂
→
Ϝ s
t 2
⊂ Ϝ ;
←
Ϝ t 2
T
⊂
←
Ϝ t 1
T
⊂ Ϝ ; s ≤ t 1
≤ t 2
≤ T
The upper arrow points on time direction for Ϝ. Put
Ϝ t =
→
Ϝ 0
t
∪
←
Ϝ 0
t
and Ϝ
t
=
→
Ϝ t
T
∪
←
Ϝ t
T
.
Let w  t  , Ϝ t
T
, t ∈  0 , T be a Wiener process with the values in E d
, g  t 
be a random Ϝ
t
− measurable d × n matrix-function . Define backward stochastic Ito
integral
←
I =
T
0
∫ g  t  d
←
w  t 
We will show that integral
←
I possesses standard Ito integral properties. First suppose
that g  t  is a step-function
g  t  =
n − 1
i = 0
∑ g  t i + 1  χ  t i , t i + 1  t 
where 0 = t 0 < t 1 < ... < t N = T is a partition of the interval
 0, T . Setting by definition

←
I =
n − 1
i = 0
∑ g  t i + 1  
←
w  t i  −
←
w  t i + 1 
1) If
t ≤ T
sup E | g  t  | < ∞ , then E
←
I = 0. In fact
E
←
I =
n − 1
i = 0
∑ E E  g  t i + 1
 
←
w  t i
 −
←
w  t i + 1
 / Ϝ
t i + 1
 =
=
n − 1
i = 0
∑ E g  t i + 1
 E  
←
w  t i
 −
←
w  t i + 1
 / Ϝ
t i + 1

So as an increment
←
w  t i
 −
←
w  t i + 1
 is
←
Ϝ t i
t i + 1
− measurable and independent
on Ϝ
t i + 1
then conditional expectation coincide with expectation and hence equal to 0.
2) If ∫ 0
T
E | g  t  |
2
d t < ∞ , then E |
←
I |
2
≤
T
0
∫ E | g  t  |
2
d t.
Really, for i < j
E  g  t i + 1
 
←
w  t i
 −
←
w  t i + 1
 
∗
g  t j + 1
 ×
× 
←
w  t j
 −
←
w  t j + 1
 = E E  
←
w  t i
 −
←
w  t i + 1

∗
/ Ϝ
t i + 1

g
∗
 t i + 1
 g  t j + 1
 
←
w  t j
 −
←
w  t j + 1
 = 0
Summing over all i , j note that
E |
←
I |
2
=
n − 1
i = 0
∑ E tr g
∗
 t i + 1
 g  t i + 1
  t i + 1
− t i
 =
= ∫ 0
T
E | g  t  |
2
d t

3) For any positive numbers N and C
P  |
T
0
∫ g  t  d
←
w  t  | > C  ≤ N
C
2
+ P 
T
0
∫ | g  t  |
2
d t ≥ N 
Set g N
 t  = g  t  χ 
t
0
∫ | g  s  |
2
d s ≤ N  . Then
P 
t ≤ T
sup | g N  t  − g  t  | > 0  = P 
T
0
∫ | g  t  |
2
d t ≥ N  .
Hence,
P  |
T
0
∫ g  t  d
←
w  t  | > C  ≤ P  |
T
0
∫ g N
 t  d
←
w  t  | > C  +
+ P  |
T
0
∫  g  t  − g N
 t  d
←
w  t  | > 0  ≤ C
2
E |
T
0
∫ g N
 t  d
←
w  t  |
2
+
+ P 
T
0
∫ | g  t  |
2
d t ≥ N  ≤ N
C
2
+ P 
T
0
∫ | g  t  |
2
d t ≥ N 
Following standard ideas it is easy to generalize these properties for any Ϝ
t
−
measurable random functions such that ∫ 0
T
| g  t  |
2
d t < ∞ with probability 1.
Taking into account that SDE theory with the inverse time is identical to that one with the
standard time , we shall state the result.
Theorem 1.1. Assume that coefficients of equation (1. 2) satisfy the conditions (1.1).
Then there exists a unique solution of equation (1.2), such that for any q ≥ 2 the following
estimates are valid
s , t
sup E | ξ  s ; t, x  |
q
≤ L 1  L 2 + | x |
q
 ,

s , t
sup E | ξ  s ; t, x  − ξ  s ; t, y  |
q
≤ L 3
| x − y |
q
, #
s , t
sup E | ξ  s ; t, x  − x |
q
≤ L 4
 L 5
+ | x |
q
 ,
where L j , j = 1, 2 ... are positive constants dependent on T, k, q . Along the
paths of the solutions of the system (1.2) we consider equation
u  t, x  = ϕ  ξ  s ; t, x   +
t
s
∫ f  r, ξ  r ; t, x , u  r, ξ  r ; t, x    d r −
−
t
s
∫ g  r, ξ  r ; t , x   d
←
w  r  #
The stochastic integral on the right-hand side of equation (1.4) is interpreted as the inverse
stochastic Ito integral. Here we just turn to the case, when the right-hand side of the equality
(1.4) depends on ξ ∗. The equation (1.4) is a generalization of the (0.2), where initial data
for Euler coordinates of the particle ξ  ∗ ; t, x  are the Lagrange variables of the
functions u  t, x  . Note that the process ξ  s ; t, x  in equality (1.4) plays the same
role as characteristics in the deterministic hyperbolic systems of the first order. As a solution of
equation (1.4), we understand the separable random function u  t, x  measurable in its
variables and for any x adapted to a filtration
Ϝ t
= σ  w  t
′
 − w  t
′′
 ; t
′
, t
′′
∈  0, t 
and satisfying the equality (1.4) with probability 1 for all t, x at once.
Theorem 1.2. Assume that non-random continuous in t ∈  0, T coefficients of the
system ( 1. 2 ), ( 1, 4 ) satisfy the conditions (1.1) and the functions
ϕ  x , f  t, x, u , g  t, x  defined for  t, x, u  ∈  0, T × E
m
× E
n
and
taking values in E
n
, E
n
, E
d
× E
n
respectively, satisfy the conditions
| ϕ  x  | + | f  t, x, u  | + | g  t, x  | ≤ k  1 + | u | 

u ≠ v
t
x ≠ y
sup 
| ϕ  x  − ϕ  y  |
| x − y |
γ +
| f  t , x , u  − f  t , y , v  |
| x − y |
γ
+ | u − v |
+
#
+
| g  t , x  − g  t , y  |
| x − y |
γ  < ∞
where γ ∈  0 , 1 . Then equation (1.4) has a unique solution for which
|| u || 2,γ
=
t, x
sup E | u  t, x  |
2
1
2
+
x ≠ y
t ≤ T
sup E
| u  t, x  − u  t, y  |
2
| x − y |
γ
1
2
+
+
t ≠ s , x
sup E
| u  t , x  − u  s , x  |
2
| t − s |
γ
2
 1 + | x | 
1
2
< ∞ .
Proof. We denote by M a set of bounded infinitely differentiable in t , x with
probability 1, random functions u  t, x, ω  measurable with respect to Ϝ t , t ≥ 0 for
each x. Then there exists a separable modification for each function from M that may take
infinite values and an arbitrary set in  0, T can serves as its separability set. Identifying the
class of stochastically equivalent functions with the modification, we fix one separability set for
all functions from M . Identifying the class of stochastically equivalent functions with the
separable modification, we fix one and the same separability set for all functions from M .
Denote B 2 , γ
the Banach space obtained as completion of the set M in norm || u || 2 , γ
. It is
obvious that Ϝ t
-measurable random functions continuous in t , x serve as elements of
space B 2 , γ . Let ( Z u   t, x, ω  be the right-hand side of equality (1.4) and
v  t, x, ω  be an arbitrary function from B 2 , γ . It is easy to verify that
|| Z v || 2 , γ < ∞ . It implies that the operator Z is defined as an operator acting from
space B 2 , γ
into itself. Then the series
∞
i = 1
∑
t , x
sup E | Z
i + 1
v  t, x  − Z
i
v  t, x  |
2
converges and, consequently, there exists a limit of the functions  Z
i
v  t, x, ω  as
i → ∞ for fixed  t , x  with probability 1. Denote this limit by u  t, x, ω  and it is
easy to check that

 Z u   t , x , ω  = u  t , x , ω
and || u || 2 , γ
< ∞ . Thus the function u  t, x, ω  belongs to B 2 , γ
there exists
modification for which equation  Z u   t, x, ω  = u  t, x, ω  holds for all t, x at
once with probability 1. The uniqueness of the solution of equation (1.4) follows from the fact
that a certain power of the operator Z is contraction operator in Banach space B 2
which
completes set M in the norm
|| u || 2 =
t , x
sup E | u  t, x  |
2
1
2
In proving Theorem 2 the same separability set for all elements of the B 2 , γ is used.
Further it is convenient to fix this set.
Theorem 1.3. Assume that the functions b i , b , C i , C satisfy the conditions of
theorem 1.1, and the functions ϕi , ϕ, f i , f , g i , g satisfy the conditions of theorem 1.2
with γ = 1 and
i → ∞
lim
T
0
∫
x
sup  | b i  t, x  − b  t, x  | + | C i  t, x  − C  t, x  | +
+ | g i
 t, x  − g  t, x  | +
u
sup | f i
 t, x, u  − f  t, x, u  | +
+ | ϕ i
 x  − ϕ  x  |  d t = 0
Then, for any q ≥ 2
i → ∞
lim
t , x
sup E | u i
 t, x  − u  t, x  |
q
= 0 .
Here u i  t, x  is a solution of the systems (1. 2) and (1.4) with coefficients
b i , c i , ϕ i , f i , g i .
The proofs of the theorems 1.3,1.4 can be obtained by standard methods using the results
[12, 19] .
Denote by C
q + γ
a set of non-random functions, whose q -order derivatives satisfy
Holder condition with γ ∈  0, 1 .
Theorem 1.4. Assume that coefficients of the system ( 1. 2 ) and ( 1. 4 ) belong to the
space C
1 + γ
∩ C
2 + γ
in variables x and u. Then

x ≠ y, t
sup E |
∇ 2
u  t , x  − ∇
2
u  t , y 
| x − y |
γ |
2
< ∞.
Consider now the solution of the problem ( 1. 4 ) as a functional of the parameters. Denote
by u  t , x ; s , ϕ  , t ≥ s the solution of the problem (1.4) for which
u  s , x ; s , ϕ  = ϕ  x  .
Lemma 1.1. Suppose the conditions of theorem 2 hold and u  t, x ; s, ϕ, ω  is the
solution of equation ( 1. 4 ). Then for arbitrary r ∈  s , t  for all
s ∈  0 , t  , t ∈  0 , ∞  , x ∈ E
n
with probability 1
u  t , x ; s , ϕ , ω  = u  t , x ; r , u  r , ∗ ; s , ϕ , ω  , ω 
Proof. The random function u  t , x ; s , ∗ , ω  is separable and continuous in its
variables t , x , s and the random functions u  t , x ; r , ψ , ω  and
u  r , x ; s , ϕ , ω  are independent for any r ∈  s , t  and any for nonrandom
functions ϕ , ψ , since the function u  t , x ; r , ψ , ω  , for nonrandom ψ, depends
only on the increments of the Wiener process on the interval  r , t , and function
u  r , y ; s , ϕ , ω  depends only the increments of this process over the interval
 s ,r . It immediately follows from equality (1.4) that
u  t, x ; s, ϕ, ω  =  ϕ  ξ  s ; r , y   +
+
s
r
∫ f  l, ξ  l ; r, y , u  l, ξ  l ; r, y  ; s, ϕ   d l
−
r
s
∫ g  l, ξ  l ; r, y   d
←
w  l  y = ξ  r ; t, x 
+
+
t
r
∫ f  l, ξ  l ; t, x , u  l, ξ  l ; t, x  ; s, ϕ, ω   dl −
−
t
r
∫ g  l , ξ  l ; t, x   d
←
w  l  = u  r, ξ  r ; t, x ; s, ϕ, ω  +
+
t
r
∫ f  l, ξ  l ; t, x  u  l, ξ  l ; t, x  ; s, ϕ    d l −

−
t
r
∫ g  l , ξ  l ; t , x   d
←
w  l 
Theorem 3 implies that the function u  t, x ; s , ϕ , ω  is continuous in ϕ . It is
easy to observe that the function satisfies the equation
u  t , x ; r , u  r , ∗ ; s , ϕ   = u  r, ξ  r ; t , x  ; s , ϕ  +
+
t
r
∫ f  l, ξ  l ; t, x , u  l, ξ  l ; t, x ; r, ϕ   dl −
−
t
r
∫ g  l, ξ  l ; t, x   d
←
w  l 
By subtracting this equality from the preceding one, we find
x
sup | Δ u  t , x  | < K
t
s
∫
x
sup | Δ u  r , x  | d r
where Δ u  t , x  = u  t, x ; s, ϕ  − u  t, x ; r, u  r, ∗ ; s, ϕ   and K is a
Lipschitz constant of a function f with respect to u. Using Gronwall’s lemma we complete the
proof.
Theorem 1.5. Assume that the coefficients of equation ( 1 .2 ) are continuous in t and
for each t , second derivatives of the coefficients of the system ( 1. 2 ), ( 1. 4 ) satisfy
Holder’s condition with γ ∈  0 , 1 in variable x and u are uniformly bounded. Then
the solution of the problem (1. 4) is also a solution of the following Cauchy problem
u  t , x  = ϕ  x  +
t
0
∫ 
m
i = 1
∑ ∂ u  s, x 
∂ x i
b i
 s, x  +
+ 1
2
m
i, j = 1
∑
d
k = 1
∑ ∂
2
u  s, x 
∂ x i ∂ x j
ci k  s, x  cj k  s, x  + f  s, x, u    d s +
#

+
m
i = 1
∑
d
k = 1
∑
t
0
∫ ci k  s , x 
∂ u  s , x 
∂ x i
d w k
 s  +
t
0
∫ g  s, x  d w  s 
for all t, x with probability 1. Here, the stochastic integrals are interpreted in a ”classical”
sense, i. e. as Ito’s integrals with the standard time.
Before proving the theorem, we illustrate its applications. First let us remark that equation
( 1. 6 ) is the equation in Lagrange coordinates. Using this equation, it is easy to construct the
function G  t , x  , which transform the Euler coordinates into Lagrange coordinates.
Indeed, let G  t , x  = G  t , x, ω  be a smooth random function. Let
0 = t 0 < t 1 < t 2 < .... < t N = t be a partition of the interval  0 , t and
λ =
i
max  t i + 1 − t i . Then
G  t , y  − G  0 , ξ  0 ; t , y   =
=
i = 0
N − 1
∑ G  t i + 1 , ξ  t i + 1 ; t , y   − G  t i , ξ  t i ; t , y   =
=
i = 0
N − 1
∑
∂ G  t i , ξ  t i ; t , y  
∂ t
Δ t i +  ξ  t i + 1 ; t , y  − ξ  t i ; t , y  ⋅
∇ G  t i , ξ  t i ; t , y   +
1
2
tr  ξ  t i + 1 ; t , y  − ξ  t i ; t , y  ∗
⋅
ΔG  t i , ξ  t i ; t , y    ξ  t i + 1 ; t , y  − ξ  t i ; t , y  + α λ  ω 
where α λ  ω  = 0. Taking the limits of all summations as λ → 0 in the formula we
get
G  t, y  − G  0, ξ  0 ; t , y   =
t
0
∫ 
∂ G  s, ξ  s 
∂ t
−
− b  s, ξ  s ; t, y  ∇ G  s, ξ  s ; t, y  +
+ 1
2
Tr C
∗
 s, ξ  s ; t, y  Δ G  s, ξ  s ; t, y  C  s, ξ  s ; t, y   d s

−
t
0
∫ C
∗
 s, ξ  s ; t, y  ∇ G  s, ξ  s ; t , y   d
←
w  s 
In order to find a function that transforms the Euler coordinates into Lagrange coordinates,
we put G  0, ξ  0 ; t , y   = y. Then
y = G  t, y  −
t
0
∫ 
∂ G  s, ξ  s; t, y  
∂ s
−
− b  s, ξ  s ; t, y  ∇ G  s, ξ  s ; t, y  +
+ 1
2
Tr C
∗
 s, ξ  s ; t, y  Δ G  s, ξ  s ; t, y  C  s, ξ  s ; t, y   d s +
+
t
0
∫ C
∗
 s, ξ  s ; t, y  ∇ G  s, ξ  s ; t , y   d
←
w  s 
Thus, let G  s , x, ω  be a formal solution of the inverse Cauchy problem
∂ G  s, y 
∂ s
− b  s, y  ∇ G  s, y  + #
+ 1
2
Tr C
∗
 s, y   ΔG  s, y C  s, y   + C
∗
 s, y  ∇G  s, y 
d
←
w  s 
d s
= 0 ,
s ∈  0, t  with boundary condition on the end of the span
G  t , y  = y
Then G  0, ξ  0 ; t , y   = y and therefore G  t , y, ω  transforms Euler
coordinates into Lagrange’s. The question concerning the construction of the first integrals of
the solution of equation ( 1. 2 ) can serve as another application of Theorem 1.5. Recall that the
function V  t , x  is a first integral of the solution of the equation (1.2), if for all
s ∈  0 , t , V  s ,  s , ξ  s ; t , y   = const. Analogously to what has been
said above, it is easy to observe that the first integrals satisfy equation (1.7) with initial
condition. V  t , x  = const. The first integrals were considered in [7, 39].
Proof of theorem 1.5. Let s = t 0
< t 1
< t 2
< .... < t N
= t be a partition
of the interval  s , t . Then
u  s, ξ  s ; t, y   − u  s, y  =

=
i = 1
N
∑ u  s, ξ  t i − 1 ; t, y   − u  s, ξ  t i; t, y  
In order to simplify our notations, we set ξ  s  = ξ  s ; t , y  . Since the function
u  t, x  is twice continuously differentiable with probability 1, and using Taylor’s formula
we have
N
i = 1
∑ u  s, ξ  t i − 1  − u  s, ξ  t i  =
=
i
∑ u
 1 
 s, ξ  t i
   ξ  t i − 1
 − ξ  t i
 +
+ 1
2
Tr  ξ  t i − 1
 − ξ  t i

∗
u
 2 
 s, ξ  t i
   ξ  t i − 1
 − ξ  t i
 +
+ 1
2
1
0
∫  1 − l  Tr  ξ  t i − 1
 − ξ  t i

∗
 u
 2 
 s, ξ l
 t i
  −
− u
 2 
 s , ξ  t i
   ξ  t i − 1
 − ξ  t i
 d l = I 1
+ I 2
+ I 3
Here, u
 1 
 t , x = ∇ u  t , x , u
 2
 t , x = ∇ 2
u  t , x, and
ξ l  t i  = l ξ  t i − 1  +  1 − l  ξ  t i  . We shall now show that I 3 tends to
0 in probability for λ = max  t i
− t i − 1
 → 0 . By Chebyshev’s inequality for
arbitrary ε > 0
P  | I 3
| > ε  ≤  2 ε 
− 1
i = 1
N
∑ E
1
0
∫  ξ  t i − 1
 − ξ  t i

2
×
×  u
 2 
 s, ξ l  t i   − u
 2 
 s, ξ  t i   d l ≤  2 ε − 1
| u
 2 
| 1,γ ×

×
N
i = 1
∑ E
1
0
∫  ξ  t i − 1
 − ξ  t i

2 + γ
≤ R ε
− 1
 t − s  λ
γ
2
,
where R is some constant, dependent on T and on coefficients of the equation. We now
estimate the terms I 1
and I 2
. We have
I = ∑ i
u
 1 
 s , ξ  t i
   ξ  t i − 1
 − ξ  t i
 =
= ∑ i u
 1 
 s , ξ  t i    b  t i , ξ  t i    t i − t i − 1  +
+ C  t i , ξ  t i    w  t i − 1  − w  t i  + o 1  λ  
where
o 1
 λ  =
N
i = 1
∑ u
 1 
 s , ξ  t i
  
t i
t i − 1
∫  b  r, ξ  r   −
− b  t i , ξ  t i   d r +
t i
t i − 1
∫  C  r, ξ  r   − C  t i , ξ  t i   d
←
w  r  | 
Then
E | o 1
 λ  |
2
≤ 2 λ || u
 1 
||
2
| s 1 − s 2 | ≤ λ
sup E | b  s 1
, ξ  s 1
  −
− b  s 2 , ξ  s 2   |
2
 t − s  + 2 E | ∑ i u
 1 
 s , ξ  t i   ×
×
t i
t i − 1
∫  C  r, ξ  r   − C  t i
, ξ  t i
  d
←
w  r  |
2
.
Estimating the second term we set i < j . Since random variables

t j
t j − 1
∫  C  r , ξ  r   − C  t i
, ξ  t i
  d
←
w  r  and ξ  t j

are Ϝ
t i
− measurable, then
E | u
 1 
 s, ξ  t i  
t i
t i − 1
∫  C  r, ξ  r   − C  t i , ξ  t i   d
←
w  r  ×
× u
 1 
 s, ξ  t j
 
t j
t j − 1
∫  C  r, ξ  r   − C  t j
, ξ  t j
  d
←
w  r  | =
= E 
t i
t i − 1
∫  C  r, ξ  r   − C  t i , ξ  t i   d
←
w  r 
∗
×
× E   u
 1 
 s, z j

∗
u
 1 
 s, z i
 ×
×
t j
t j − 1
∫  C  r, ξ  r   − C  t i
, ξ  t i
  d
←
w  r  / Ϝ
t i

z j = ξ  t j

z i = ξ  t i 
The expression under the sign for the conditional mathematical expectation, doesn’t
depend on σ − algebra Ϝ
t i
. Hence the conditional mathematical expectation considers
with the unconditional one. Then
E  u 1 
 s, z j

∗
u 1 
 s, z i

t i
t i − 1
∫ C  r, ξ  r   − C  ti , ξ  t i
  d
←
w r  =
= E E   u
 1 
 s, z j 
∗
u
 1 
 s, z i 
t i
t i − 1
∫  C  r, ξ  r   −

− C  t i , ξ  t i   d
←
w  r  / Ϝ s = E  u
 1 
 s, z j 
∗
u
 1 
 s, z i  ×
× E
t i − 1
t i
∫  C  r, ξ  r   − C  t i
, ξ  t i
  d
←
w  r  = 0
Thus the second term of the estimate of the variable doesn’t exceed
E
i
∑ | u
 1 
 s, ξ  t i
 |
2
|
t j
t j − 1
∫  C  r, ξ  r   − C  t i
, ξ  t i
  d
←
w  r  |
2
≤
≤ || u
 1 
||
2
sup E | C  s 1
, ξ  s 1
  − C  t 2
, ξ  t 2
  |
2
 t − s 
where sup is taken over | s 1 − s 2 | < λ . Hence
E | o 1
 λ  |
2
≤ 2 λ || u
 1 
||
2
 t − s 
| s 1 − s 2 | < λ
sup E  λ | b  s 1
, ξ  s 1
  −
− b  s 2
, ξ  s 2
  |
2
+ | C  s 1
, ξ  s 1
  − C  t 2
, ξ  t 2
  |
2
 .
By the conditions of the theorem
λ → 0
lim E | o 1  λ  |
2
= 0 . Further
I 2 = 1
2
i
∑ Tr C
∗
 s, ξ  t i  u
 2 
 s, ξ  t i  C  s, ξ  t i    t i − t i − 1 +
+ o 2
 λ 
where
| o 2  λ  | ≤ 1
2
i
∑ | u
 2 
 s, ξ  t i  |
2

t i
t i − 1
∫ | C  r, ξ  r   C
∗
 r, ξ  r   −

− C  t i
, ξ  t i
  C
∗
 t i
, ξ  t i
  | d r + |
t i
t i − 1
∫ b  r , ξ  r   d r |
2

Analogously to the estimate I 1 it is easy to observe that
λ → 0
lim E | o 2  λ  |
2
= 0.
Thus
I 1 + I 2 =
=
N
i = 1
∑ u
 1 
 s, x  b  s, x  ti − ti − 1 + C  s, x   w  t i − 1
 − w  t i
  +
+ 1
2
N
i = 1
∑ Tr C
∗
 s, x   u
 2 
 s , x  C  s, x   t i
− t i − 1
 +
+ o 1
 λ  + o 2
 λ  + o 3
 t − s 
where
o 3
 t − s  =
N
i = 1
∑  u
 1 
 s, ξ  t i
  b  t i
, ξ  t i
  −
− u
 1 
 s, x  b  s, x   t i
− t i − 1
 +  u
 1 
 s, ξ  t i
  C  t i
, ξ  t i
  −
− u
 1 
 s, x  C  s, x   w  t i − 1  − w  t i  +
+ 1
2
N
i = 1
∑  Tr C
∗
 s, ξ  t i
  u
 2 
 s, ξ  t i
  C  s, ξ  t i
  −

− Tr C
∗
 s , x   u
 2 
 s , x   C  s , x   t i − t i − 1  .
We now estimate the sum in which the increments of the Wiener process w  t  are
contained. Then
P  |
i
∑  u
 1 
 s, ξ  t i
  C  t i
, ξ  t i
  − u
 1 
 s, x  C  s, x  ×
×  w  t i − 1  − w  t i  | > 2 ε  ≤ P  |
i
∑  u
 1 
 s , ξ  t i   −
− u
 1 
 s , x  C  t i
, ξ  t i
   w  t i − 1
 − w  t i
 | > ε  +
+ P  |
i
∑  u 1 
 s, x   C  t i, ξ  t i   − C  s, x  ×
×  w  t i − 1
 − w  t i
 | ≥ ε 
The second summand on the right can be estimated by Chebyshev’s inequality. Since the
random functions ξ  t i
 = ξ  t i
; t , x  , w  t i − 1
 − w  t i

are Ϝ
s
− measurable for any i and u  s , x  is independent of the σ −algebra, the
second term doesn’t exceed
ε
− 2
x
sup E | u
 1 
 s , x  |
2
r ∈  s , t
sup | E | C  r, ξ  r   C
∗
 r, ξ  r   −
− C  t i , ξ  t i   C
∗
 t i , ξ  t i   |
2
 t − s 
Then
P  | ∑ i
 u
 1 
 s, ξ  t i
  − u
 1 
 s , x  C  t i
, ξ  t i
  ×
×  w  t i − 1 − w  t i  | > ε  ≤ ε
− 2
|| u
 2 
||
2
K
2
×
×
r ∈  s , t
sup | E | C  r , ξ  r   − x |
2
 t − s  ≤ R 1
 t − s 
2

where the positive constant R 1
is easily defined by formula (1.3). Estimate the remained
terms of the o 3  t − s  . We have
P 
i
∑ |  u
 1 
 s, ξ  t i
  b  t i
, ξ  t i
  − u
 1 
 s, x  b  s, x  +
+ 1
2
∑ i  Tr C
∗
 s, ξ  t i   u
 2 
 s, ξ  t i   C  s , ξ  t i   −
− Tr C
∗
 s , x   u
 2 
 s , x   C  s , x  |  t i
− t i − 1
 > ε  ≤
≤ ε
− 1
∑ E  | u
 1 
 s, ξ  t i   b  t i , ξ  t i   − u
 1 
 s, x  b  s, x  | +
+ 1
2
Tr C
∗
 t i
, ξ  t i
  u
 2 
 s, ξ  t i
  C  t i
, ξ  t i
  −
− Tr C
∗
 s, x   u
 2 
 s, x   C  s, x  |  t i − t i − 1  ≤
≤ ε
− 1
R 2
  t − s 
1
2
+
r ∈  s , t
sup E | b  r , ξ  r   − b  s , x  | +
+
r ∈  s , t
sup E | C
∗
 r, ξ  r   C  r, ξ  r   − C
∗
 s, x  C  s, x  |   t − s 
where R 2
≥ 0 is some constant. Following the assumption of the theorem, we observe
that
λ → 0
lim I 1 + I 2 + I 3 =
= u
 1 
 s, x   b  s, x   t − s  + C  s, x   w  s  − w  t  
+ 1
2
∑ i Tr C
∗
 s, x   u
 2 
 s, x   C  s, x   t − s  + o  t − s 
where

P.
t → s
lim  t − s 
− 1
o  t − s  = 0 .
Let 0 = s 0
< s 1
< ... s N
= t . Applying the estimates obtained above to each
interval  s i , s i + 1 we find
N − 1
i = 0
∑ u  si + 1, x  − u  si, x  =
N − 1
i = 0
∑ u  si, ξ  si ; si + 1, x   − u  si, x  +
+
s i + 1
s i
∫ f  s, ξ  s ; s i + 1 , x  , u  s, ξ  s ; s i + 1, x    d s −
−
s i + 1
s i
∫ g  s , ξ  s ; s i + 1
, x  d
←
w  s  =
N − 1
i = 0
∑  u
 1 
 s i
, x  b  s i
, x  +
+ 1
2
Tr C
∗
 s i
, x   u
 2 
 s i
, x   C  s i
, x   s i + 1
− s i
 +
+
N − 1
i = 0
∑  u
 1 
 s i
, x  C  s i
, x   w  s i + 1
 − w  s i
 +
+ f  s i
, x, u  s i
, x    s i + 1
− s i
 +
+ g  s i , x   w  s i + 1  − w  s i   + o  λ 
Here λ = max  s i + 1
− s i
 and P.
λ → 0
lim | o  λ  | = 0 . Passing to the limit
in this equality as λ → 0 , we obtain the function u  t , x  to be a solution of equation
(1.6) and Theorem1.5 is proved.
Next we consider a generalization of theorem 1.5. Let w i
 t  be independent Wiener
processes taking values in E
d i
, i = 1 , 2 , ... l . Denote by ϜΔ
 i 
the σ −algebra

generated by the increments of the w i  t  , t ∈ Δ and Ϝ Δ =
I
i = 1
∪ ϜΔ
 i 
. Consider
the system
ξ  s ; t, x  = x +
t
s
∫ b  r, ξ  r ; t, x  d r −
i = 1
I
∑
t
s
∫ Ci  r, ξ  r ; t, x  d
←
wi  r 
u  t, x  = ϕ  ξ  s ; t, x   + #
+
t
s
∫ f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r −
t
s
∫ g  r, ξ  r ; t, x   d
←
w1  r 
where v  t , x  = E  u  t , x  | ϜΔ
 1 
 and C i  t , x  are nonrandom
matrices with elements c k j
 i 
 t , x  ; k = 1, 2, ... d i
, j = 1, 2 , .. m . It is easy
to show, that the results proved in theorem 1.1-4 hold for the system (1.8 ).
Theorem 1.6. Assume that the coefficients of the system (1.8 ) satisfy the conditions of
theorem 1.5. Then the function v  t , x , ω  is a ”classical” solution of the Cauchy
problem
v  t , x  = ϕ  x  +
t
0
∫   b  s , x  , ∇  v  s , x  +
+ 1
2
 C  s, x  C
∗
 s, x  ∇ , ∇  v  s, x  + f  s, x, v   d s + #
+
t
0
∫  C 1  s, x  d w 1  s  , ∇  v  s, x  +
t
0
∫ g  s, x  d w 1  s 
where ,
C  s , x  C
∗
 s , x  =
I
i = 1
∑ C i  s , x  C i
∗
 s , x 
and stochastic integrals on the right in (1.9) are interpreted in the classical Ito sense. The
the proof of the theorem 1.6 follows that of theorem 1.5 in the main. Therefore we underline
only the moments, which differ. We have

u  t , x  − ϕ  x  =
N − 1
j = 0
∑ u  s j + 1 , x  − u  s j , x  =
=
N − 1
j = 0
∑
I
i = 1
∑  C i  s j , x  Δ w i  s j  ,∇  u  s j , x  +
N − 1
j = 0
∑   b  s j
, x  , ∇  u  s j
, x  +
+ 1
2
 C
∗
 s , x  C  s , x  ∇ , ∇  u  s , x   s j + 1
− s j
 +
+
N − 1
j = 0
∑ f  s j
, x, v  s j
, x    s j + 1
− s j
 +
+
N − 1
j = 0
∑ g  s j , x   Δ w 1  s j  + o  λ  .
Here 0 = s 0
< s 1
< ... s N
= t is a partition of the interval  0 , t ,
Δ w i  s j  = w i  s j + 1  − w i  s j  , λ = max  s j + 1 − s j  ,
and random variable o  λ  is such, that lim λ → 0
| o  λ  | = 0 . Calculating the
conditional expectation with respect to σ − algebra Ϝ  0 , t 
 1
and replacing the operator ∇
and the conditional expectation, we have
E   C i
 s j
, x  Δ w i
 s j
 , ∇  u  s j
, x  / Ϝ  0 , t 
 1
 =

= C i  s j , x  ∇ E  u  s j , x  Δ w i  s j  / Ϝ  0 , t 
 1

We now consider the conditional mathematical expectation on the right-hand side of this
equality. Set
Ϝ 1
= Ϝ  s j , t 
 1
, Ϝ 2
= Ϝ  0 , s J 
 1
, Ϝ 3
=
I
i = 2
∪ Ϝ  s j , s j + 1 
 i 
∪ Ϝ  0 , s j 
It is easy to show that σ − algebras Ϝ 1
and Ϝ 3
are conditionally independent with
respect to σ −algebra Ϝ 2
. Let Y k
are arbitrary nonnegative random variables , measurable
with respect to σ −algebra Ϝ k , k = 1, 3 . Taking into account that Ϝ 2 ⊂ Ϝ 3 and
that σ −algebra Ϝ 1
does not depend on σ −algebras Ϝ 2
and Ϝ 3
, we have
E  Y 1
Y 3
/ Ϝ 2
 = E  Y 3
E  Y 1
/ Ϝ 3
 / Ϝ 2
 =
= E  Y 3 / Ϝ 2  E Y 1 = E  Y 3 / Ϝ 2  E  Y 1 / Ϝ 2  #
The equality obtained coincides with the definition of conditional independence of
σ −algebras Ϝ 1
and Ϝ 3
. Hence, (see[32]), with probability 1
E  Y 3
/ Ϝ 1
∪ Ϝ 2
 = E  Y 3
/ Ϝ 2
 #
Assuming Y 3
= u  s j
, x  Δ w i
 s j
 , for i = 2 , 3 , ... ,l we have
E  u  s j , x  Δ w i  s j  / Ϝ  0 , t 
 1 
 =
= E  u  s j
, x  Δ w i
 s j
 | Ϝ  0 , s j 
 1 
∪ Ϝ  s j , t 
 1 
 =
= E  u  s j , x  Δ w i  s j  / Ϝ  0 , s j 
 1 
 =
= E  u  s j
, x  E  Δ w i
 s j
 / Ϝ  0 , s j 
 / Ϝ  0 , s j 
 1 
 = 0
In this case i = 1 we assume that

Ϝ 1 = Ϝ  s j + 1 , t 
 1
, Ϝ 2 = Ϝ  0 , s J + 1 
 1
, Ϝ 3 = Ϝ  s j , s j + 1 
 1 
∪ Ϝ  0 , s j  .
Analogously to the preceding the equality (1.10) is valid and taking account of (1.11) we
have
E  u  s j
, x  Δ w 1
 s j
 / Ϝ  0 , t 
 1 
 =
= E  u  s j , x  Δ w 1  s j  / Ϝ  0 , s j + 1 
 1 
∪ Ϝ  s j + 1 , t 
 1 
 =
= E  u  s j
, x  / Ϝ  0 , s j + 1 
 1 
 Δ w 1
 s j
 = v  s j
, x  Δ w 1
 s j

Hence, assuming s j + 1
= t we obtain
E  u  s j
, x  / Ϝ  0 , t 
 1 
 = E  u  s j
, x  / Ϝ  0 , s j 
 1 
 = v  s j
, x 
Therefore
E 
i
∑ ∑
j
 C i
 s j
, x  Δ w i
 s j
 , ∇  u  s j
, x  / Ϝ  0 , t 
 1
 =
= ∑ j
 C 1
 s j
, x  Δ w 1
 s j
 , ∇  v  s j
, x  ,
E 
N − 1
j = 0
∑   b  s j , x  , ∇  + 1
2
 C
∗
 s j , x  C  s j , x  ∇ , ∇  ×
× u  s j , x  Δ s j / Ϝ  0 , t 
 1
 =
N − 1
j = 0
∑   b  s j , x  , ∇  +
+ 1
2
 C
∗
 s j , x  C  s j , x  ∇ , ∇  v  s j , x  Δ s j
Passing to the limit as λ → 0 , we can see that theorem 1.6 is proved.
Now we shall study the inverse problem. Using equation (1.6), we shall find the equation
of particle trajectories. Moreover the uniqueness of the solution of the Cauchy problem (1.6) is

not supposed, but it follows from the uniqueness of the solution of the corresponding
trajectory- problems.
Let B 2
2 + γ
denote the Banach space of Ϝ t − measurable separable random
functions V  t , x , ω  with the norm
|| V || 2 , 2 + γ
= max || ∇
i
V || 2 , γ
.
Theorem 1.7. Assume that V  t , x , ω  is an ’classical ’ solution of equation
(1.6) belonging to the space B 2
2 + γ
with coefficients satisfy the conditions of the theorem 1.5.
Then the function V  t , x , ω  is the solution of equation (1.4).
Proof. Let the function V  t , x , ω  be and arbitrary ’ classical’ solution of equation
(1.6) and  s , t be an arbitrary subinterval of the interval  0 , T and ξ  s ; t , x 
be a unique solution of equation (1.2) with the Wiener process w  t  that used in equation
(1.6). Then for an arbitrary function Q  t , x , ω  ∈ B 2
2 + γ
we have
Q  s, ξ  s ; t, x   − Q  t, x  =
=
N
i = 1
∑ Q  s, ξ  t i − 1 ; t, x   − Q  s, ξ  t i ; t, x   .
For the sake of simplicity of notations we assume further that ξ  s  = ξ  s ; t , x .
By Taylor’s formula with remainder term in the integral form:
N
i = 1
∑ Q  s , ξ  t i − 1   − Q  s , ξ  t i   =
=
N
i = 1
∑  ξ  t i − 1
 − ξ  t i
 ∇ Q  s , ξ  t i
  +
+ 1
2
Tr  ξ  t i − 1
 − ξ  t i

∗
Δ Q  s , ξ  t i
   ξ  t i − 1
 − ξ  t i
 +
+
1
0
∫  1 − θ  1
2
Tr  ξ  t i − 1  − ξ  t i 
∗
 Δ Q  s, ξ θ  t i   −

− Δ Q  s, ξ  t i    ξ  t i − 1  − ξ  t i  d θ = I 1 + I 2 + I 3 ,
where ξ θ
 t i
 =  1 − θ  ξ  t i
 + θ ξ  t i + 1
 . For the function
Q  t , x  ∈ B 2
2 + γ
, analogously to theorem 5 we obtain
P  | I 3 | > ε  ≤ R ε
− 1
 t − s  λ
γ
2
,
where R is a constant independent of x , ε , λ . For the expressions I 1 and I 2 using
the methods applied above, we find
I 1
= ∑ i
 b  t i
, ξ  t i
   t i
− t i − 1
 +
+ C  t i
, ξ  t i
   w  t i − 1
 − w  t i
  ∇ Q  s, ξ  t i
  + o 1
 λ  ,
I 2 = 1
2
i
∑ Tr C
∗
 t i , ξ  t i    Δ Q  s, t i   ×
× C  t i
, ξ  t i
   t i
− t i − 1
 + o 2
 λ 
where
λ → 0
lim E | o 1
 λ  | + | o 2
 λ  | = 0 .
Hence, we can note that
I 1
+ I 2
+ I 3
+
+
t
r
∫ f  l, ξ  l ; t, x , Q  l, ξ  l ; t, x    d l −
t
r
∫ g  l, ξ  l ; t, x   d
←
w  l  =
=
i
∑   b  s , x   Δ t i + C  s , x   Δ w  t i  ∇ Q  s , x  +

+ 1
2
i
∑ Tr C
∗
 s, x  ΔQ  s, x  C  s, x   Δ t i
+
+ f  s, x,Q  s, x   Δ t i
+ g  s , x  Δ w  t i
  + o  t − s 
Here P.
t → s
lim | o  t − s  | = 0 . Consequently,
Q  s, x  +  b  s, x    t − s  + C  s, x    w  t  − w  s   ∇ Q  s, x  +
+  1
2
Tr C∗
 s, x    ΔQ  s, x  C  s, x   + f  s, x,Q  s, x    t − s 
+ g  s, x   w  t  − w  s  = Q  s, ξ  s   + #
t
+
s
∫ f  l, ξ  l ; t, x , Q  l, ξ  l ; t, x    d l −
−
t
s
∫ g  l, ξ  l ; t, x   d
←
w  l  + o  t − s 
In the equality (1.12) which is valid for any function Q from the space B 2
2 + γ
, we choose
a given ’classical’ solution V  t , x , ω  of the Cauchy problem (1.6) with initial condition
V  0 , x , ω  = ϕ  x  . Let 0 = t 0
< t 1
< t 2
< .... < t N
= t be a
partition of the interval  0 , t , λ = max Δ t i
. Then applying the equality (1.12) to
 t 0 , t 1 , we find
V  t 1
, x  = ϕ  ξ  t 0
; t 1
, x   +
+
t 1
t 0
∫ f  l, ξ  l ; t 1 , x , V  l, ξ  l ; t 1 , x    d l −
−
t 1
t 0
∫ g  l, ξ  l ; t 1
, x   d
←
w  r  + o  λ  + γ  t 1
− t 0
, x  ,
where

γ  t 1
− t 0
, x  =
t 1
t 0
∫   b  s , x   ∇ V  s , x  +
+ 1
2
Tr C
∗
 s, x    Δ V  s, x   C  s, x   + f  s, x, V  s, x   −
−  b  t0 , x   ∇V  t0 , x  + 1
2
Tr C
∗
 t0 , x    ΔV  t0, x   C  t0 , x   +
+ f  t 0 , x, V  t 0 , x    d s +
t 1
t 0
∫   C  s, x  ∇ V  s, x  + g  s, x  −
−  C  t 0
, x  ∇ V  t 0
, x  + g  t 0
, x   d w  s 
Suppose, that for any k the following representation holds
V  t k − 1
, x  = ϕ  ξ  t 0
; t k − 1
, x   +
+
t k − 1
t 0
∫ f  l, ξ  l ; t k − 1 , x  , V  l, ξ  l ; t k − 1 , x    d l −
−
t k − 1
t 0
∫ g  l, ξ  l ; t k − 1
, x   d
←
w  l  +
+  k − 1  o  λ  +
k − 1
i = 1
∑ γ  t i
− t i − 1
, x 
We show that the similar representation also holds for the moment t k
, and then estimate
the sum of variables γ. By (1.12) we have
V  t k , x  = V  t k − 1 , y  +

+
t k
t k − 1
∫ f  l , ξ  l ; t k , x  , V  l , ξ  l ; t k , x    d l −
−
t k
t k − 1
∫ g  l, ξ  l ; t k , x   d
←
w  l  + o  λ  + γ  t k − t k − 1 , x  =
=  ϕ  ξ  t 0 ; t k − 1 , y   +
+
t k − 1
t 0
∫ f  l, ξ  l ; t k − 1
, y , V  l, ξ  l ; t k − 1
, y    d l −
−
t 1
t 0
∫ g  l, ξ  l ; t k − 1
, x   d
←
w  l  +
+
k − 1
i = 1
∑ γ  t i
− t i − 1
, y 
y = ξ  t
k − 1
; t
k
, x 
+ #
+
t k − 1
t 0
∫ f  l , ξ  l ; t k
, x  , V  l , ξ  l ; t k
, x    d l −
−
t k
t k − 1
∫ g  l, ξ  l ; t k
, x   d
←
w  l  + k o  λ  + γ  t k
− t k − 1
, x 
Since,
k − 1
i = 1
∑ γ  t i
− t i − 1
, y  |
y = ξ  t
k − 1
; t
k
, x 
+ γ  t k
− t k − 1
, x  = 0
then, it is sufficient to show, that

λ → 0
lim
N
i = 1
∑ γ  t i − t i − 1 , x  = 0
Both the ordinary integrals and the stochastic Ito integrals are contained in the sum under
consideration. Ordinary integrals are estimated by standard methods. So, as an example, we
shall estimate one of the sums of stochastic integrals
E |
N
i = 1
∑
t i
ti − 1
∫  C  r, x  ∇V  r, x  − C  ti − 1, x  ∇V  ti − 1, x  d w  r  |
2
≤
≤ t 
| t  − t  | < λ
max | C  t

, x  − C  t

, x  |
2
max
x
E | ∇ V  t , x  |
2
+
+
t
max | C  t , x  |
2
| t  − t  | < λ
max E | V  t

, x  − V  t

 |
2

From the condition of theorem 1.7 it follows that the expression in brackets tends to 0 as
λ → 0 . Passing to the limit in equality ( 1. 12 ) as and supposing k = N we can verify
that the function v  t, x  = v  t , x ; 0, ϕ  being a solution of equation (1.6) is also a
solution of equation (1.4). Thus by conditions of theorem 1.7 each ”classical” solution of the
Cauchy problem (1.6) is also a solution of the equation (1.4) and vice versa. By virtue of
theorem 1.2 the solution of the problem (1.4) is unique and consequently the solution of the
problem (1.6) is also unique. That theorem is now proved.
We return now to the problem ( 1.9 ). Let us recall that function v  t , x  belonging to
the space B 2
2 + γ
measurable with the flow of σ −algebras Ϝ  0 , t 
 1 
independent of the
Wiener processes w j
 t  , j = 2, 3, ..., l contained in the system (1.8) and satisfying
the equality (1.9) for all t and x at once with probability is called the ”classical” solution of
the problem (1.9). It is rather difficult to apply the methods analogous to those in the proof of
Theorem 1.7 directly. Therefore another technique of proving the uniqueness of the solution of
equation (1.9) will be used.
Theorem 1.8. Let the conditions of the theorem 1.6 hold and assume that the functions
b  t , x  , C i
 t , x  i = 1, 2, ..l for any t ∈  0 , T are equal 0 outside
some compact. Then the ’classical ’ solution of the problem (1.9) is unique.
Proof. Suppose equation ( 1.9 ) has two ”classical” solutions v i
 t , x  i = 1, 2.
Denote their difference by h  t , x  =  h 1  t , x  , ..., h n  t , x  . Then

h q
 t , x  =
t
0
∫ 
m
i = 1
∑ b i
 s , x 
∂ h q
 s , x 
∂ x i
+
+
m
i , j = 1
∑
I
k = 1
∑
d k
ν = 1
∑ c i , ν
 k 
 s, x  c j , ν
 k 
 s, x 
∂
2
h  s, x 
∂ x i ∂ x j
+ f q
 s, x, v 1
 s, x   −
− f q s, x, v2  s, x   d s +
m
i = 1
∑
d 1
ν = 1
∑
t
0
∫ c i , ν
 1 
 s, x 
∂ hq s, x 
∂ xi
d w ν
 1
 s 
q = 1, 2, ..., m . Treating the variable x in the preceding equality as a parameter, we
apply Ito’s formula to the function h q
2
 s , x . Then
h q
2
 s, x  =
t
0
∫ 2 h q
 s, x  
m
i = 1
∑ b i
 s, x 
∂ h q  s, x 
∂ x i
+
+
m
i , j = 1
∑
I
k = 1
∑
d k
ν = 1
∑ c i , ν
 k 
 s, x  c j , ν
 k 
 s, x 
∂
2
h  s, x 
∂ x i ∂ x j
+
+  f q
 s, x, v 1
 s , x   − f q
 s, x, v 2
 s, x    d s +
+
t
0
∫
m
i , j = 1
∑
d 1
ν = 1
∑ c i , ν
 1 
 s, x  c j , ν
 1 
 s, x 
∂ h q
 s, x 
∂ x i
∂ h q
 s, x 
∂ x j
 d s +
+
m
i = 1
∑
d 1
ν = 1
∑
t
0
∫ c i , ν
 1 
 s , x 
∂ h q
2
 s , x 
∂ x i
d w ν
 1 
 s 
We shall now take the mathematical expectation from the equality , and integrate the
equality obtained with respect to d x . Note, that according to the assumption of the theorem,
the order of integration can be changed. Indeed, for the Riemann integral it is sufficient, that
integrand function be absolutely integrable with respect to measure dx × dt with the

probability 1. For the order of integration to be reversed in the stochastic integral it is sufficient
[39] , that with probability1
T
0
∫
E m
∫ | C
 1 
 t , x  ∇ h
2
 t , x  d x d t < ∞ #
Since the function h  ∗  belongs to the space B 2
2 + γ
, and the diffusion coefficient
C
 1 
 ∗  is equal to 0 outside some compact , then condition (1.14) holds. Integrating
by parts in the inner integral in the first two terms on the right-hands side, we obtain
E m
∫ E h q
2
 s , x  d x =
t
0
∫ d s
E m
∫  
m
i = 1
∑
∂ b i  s , x 
∂ x i
+
+
m
i , j = 1
∑
I
k = 1
∑
d k
ν = 1
∑
∂
2
c i , ν
 k 
 s , x  c j , ν
 k 
 s , x 
∂ x i
∂ x j
 E h q
2
 s, x  + #
+
m
i , j = 1
∑
I
k = 1
∑
d k
ν = 1
∑ c i , ν
 k 
 s, x  c  s, x 
∂ h q
2
 s, x 
∂ x i
∂ h q
2
 s, x 
∂ x j
+
+
m
i , j = 1
∑
d 1
ν = 1
∑ c i , ν
 1 
 s, x  c j , ν
 1 
 s, x 
∂ h q
2
 s, x 
∂ x i
∂ h q
 s, x 
∂ x ij
+
+ k f E h q
2
 s , x   d x .
Here, k f
= sup t , x , v
| ∇ v
f  t , x , v  | . Summing over all q we establish
estimates for the function h  t , x  = v 1
 t , x  − v 2
 t , x  :
E m
∫ E h q
2
 t , x  d x ≤
t
0
∫ d s
E m
∫  div b  s , x  +
+ 1
2
 ∇ , ∇ C  s, x  C
∗
 s, x   + k f
n  E | h  s , x  | d s
By Granules’s lemma,

E m
∫ E | v 1
 t , x  − v 2
 t , x  |
2
d x = 0.
Since subintegral function is continuous, the solutions coincide for all t , x at once with
probability 1, which is what had to be proved.
Now we are going to study other types of stochastic dynamic system. Note, that one of the
possible applications of the theory introduced above is design of the methods of resolving the
problem Cauchy for nonlinear stochastic parabolic equation. Turn back to the Theorem 1.5 we
would to emphasize that there are two types of ’white’ noise terms involved into the equations.
They are the terms which we can conditionally call ’external’ and ’internal’. Internal noise is
generated by the diffusion of the process ξ  ∗ . External noises in the equation ( 1.9 ) are
’white’ noise terms that perturb the second equation of the system (1.8) . Analyzing equations
(1.4), (1.8 ) we can see that the macro-parameters u  t , x  or v  t , x  change along
the trajectories of the characteristics ξ  ∗ . Now we represent a scheme where these
parameters will changed along the distribution of the characteristics. This is the direct
extension of Kolmogorov’s equations and can be considered as a generalization of probabilistic
methods of studying deterministic nonlinear parabolic equations [ 4 ]. Given the conditions of
the theorem (1.5) we introduce the problem
v  t, x  = E ϕ  ξ  0 ; t, x   +
+
t
0
∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r
 d r +
+
t
0
∫ E  g  r, ξ  r ; t, x  , v  r, ξ  r ; t, x    / Ϝ r
 d w  r  #
The major deference between equations (1.16) and (1.4) is that the function v  ∗  in
(1.16) depends upon the averaged characteristics and the function u  ∗  in the (1.4)
depends upon the characteristics itself. The proof of the existence theorem is almost the same
as before. Some insignificant modifications involved based on the new term in (1.16). To use
contraction mapping principle we need to substitute random functions
v  ,  , ξ  ∗ ; t, x  into conditional expectation.
We will show that after substitution correspondent stochastic integral remain measurable
function. First, let B be a set of Borel’s bounded random functions depending on t , x and
v  t , x, ω  be an arbitrary function belonging to B . Function
G  s, t, x, ω  = g  r , ξ  r ; t, x  , v  r , ξ  r ; t, x   is measurable over
s, t, x, ω as a composition of measurable functions and

s , t , x
sup E | G  s, t, x, ω  | < ∞
Now , we can exhibit that there exist measurable modification of the conditional
expectation E  G  s, t, x, ω  | Ϝ s  .
Lemma 1.2. Let G  s, t, x, ω  be a measurable random field ,and
sup s , t , x E | G  s, t, x, ω  | < ∞. Then there exist s , t , x ω − measurable
modification of the conditional expectation E  G  s, t, x, ω  / Ϝ s
 .
Proof. Denote U a set of random functions for which Lemma is true. Let A be a
s, t, x, ω − measurable set of a type A = A 1
× A 2
× A 3
× A 4
, where
A 1 , A 2 , A 3 are Borel sets from  0 , t ,  0 ,T , E
m
respectively, and
A 4 ∈ Ϝ . Then
E χ A
 s, t, x, ω  / Ϝs = E χ A 1
 s  χ A 2
 t  χ A 3
 x  χ A 4
 ω  / Ϝs =
= χ A 1
 s  χ A 2
 t  χ A 3
 x  E  χ A 4
 ω  / Ϝ s

From general martingale theory [35] it follows that there exist measurable modification of
right hand side of the equality. Set U is an algebra and monotone class and contains indicators
of the measurable sets of chosen type. Hence, U contains indicators of all
s, t, x, ω − measurable sets. Evidently, that U is linear and closed with respect to
monotone limit transition. So that U contains all measurable nonnegative functions. Since, any
measurable function can be represented as a difference of two nonnegative functions one gets a
proof of the lemma.
Turn back to the problem (1.16). The solution of the problem exist, unique and satisfies
relationship
v  t, x  = E v  ξ  s ; t, x   / Ϝ s
 +
+
t
s
 d r + #
+
t
s
∫ E  g  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r
 d w  r 
We will establish a derivation of the first term in right hand side of (1.17). Two others
terms can be establish without any difficulties. Since, the random variable ξ  s ; t, x  is
independent on Ϝ s
then

E  E ϕ  ξ  0 ; t, y   | y = ξ  s ; t, x   =
= E  E ϕ  ξ  0 ; t, x   | y = ξ  s ; t, x 
/ Ϝ s
Taking into account that the right hand side of (1.16) is t, x, ω − measurable note
E ϕ ξ 0 ; t, x   +
0
s
 d r +
+
s
0
∫ E  g  r, ξ  r ; t, x  , v  r, ξ  r ; t, x    / Ϝ r  d w  r  =
= E  E ϕ  ξ  0 ; s, y   +
+
s
0
∫ E  f  r, ξ  r ; s, y , v  r, ξ  r ; s, y    / Ϝr  d r +
+
s
0
∫ Eg r, ξ r; s, y , v r, ξ r ; s, y  / Ϝr dw r  y = ξ  s; t, x / Ϝs
= E  v  ξ  s ; t, x   / Ϝ s

It is not difficult to check that the statements of the theorems1.2 - 1.4 remain correct for the
system (1.16). Granting these we are able to formulate
Theorem 1.9. Under conditions of the theorem 1.5 the solution of the equation (1.16) is
a solution of the Cauchy problem
v  t , x  = ϕ  x  +
t
0
∫ 
m
i = 1
∑ ∂ v  s , x 
∂ x i
b i
 s , x  +
+ 1
2
i , j = 1
m
∑ ∂
2
v  s, x 
∂ x i
∂ x j
k = 1
d
∑ ci k  s, x  cj k  s, x  + f  s, x, v  s, x    d s +

+
t
0
∫ g  s, x, v  s, x   d w  s 
The proof of the theorem is almost the same as the theorem 1.5 . We just have to start from
equality
v  t, x  − v  s, x  = E  v  ξ  s ; t, x   − v  s, x  / Ϝ s  +
+
t
s
∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r  d r +
+
t
s
∫ E  g  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r  d w  r 
and then to repeat the way of the proof of the theorem 1.5.
Quasilinear evolutionary stochastic systems
In this section we will consider quasilinear stochastic parabolic systems. Two significant
differences between a quasilinear and a semilinear systems should be noted. The first
difference is that the solution of quasilinear systems is local in time and the solution of
semilinear systems exists on an arbitrary finite interval of time. The second one is that the
solution of quasilinear systems with smooth coefficients is continuous in x and therefore the
solution must be interpreted as generalized. For semilinear systems differentiability of the
solution in x corresponds to the differentiability of the coefficients in x . Some types of
stochastic partial differential equations were studied in [12,36,41].
We introduce the needed notation. Let the w j  t  be mutually independent Weiner
processes with values in d j
-dimensional Euclidean space E
d j
, d j
≥ 1 , j = 1, 2 ,
and let w  t  = w 1
 t  , w 2
 t  . For an arbitrary interval
Δ ⊂  0 , T let
Ϝ Δ
 j 
= σ  w j  t 1  − w j  t 2  ; t 1, t 2 ∈ Δ  and Ϝ Δ =
2
j = 1
∪ Ϝ Δ
 j 
.
Consider the quasilinear system

ξ  s ; t, x  = x +
t
s
∫ b  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r −
−
t
s
∫ E  c  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ  r , t 
 d
←
w  r  #
u  t, x  = ϕ  ξ  0 ; t, x   +
t
0
∫ f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r +
+
t
0
∫ g  r , ξ  r ; t, x  d
←
w 1  r 
where v  t , x  = E  u  t , x  / Ϝ  0 , t 
 1 
 and 0 ≤ s ≤ t ≤ T . The arrow
here over the Wiener process w  t  means that the stochastic integral is to be interpreted as
Ito’s inverse integral . A solution of the system (2.1) on the interval  0 , T  is understood
to be a pair of random functions ξ  s ; t, x  , u  t , x  that are defined for
s ≤ t < T and satisfy the estimate (2.2), are measurable in all their arguments, are
measurable with the respective flows of σ −algebras Ϝ  s , t and Ϝ  0 , t  , and satisfy (2.1)
for all s, t, and x with probability 1. In what follows to simplify the notation the exact same
symbol | ∗ | will be used for the norm of a vector in Euclidean space and for the norm of the
trace of a matrix.
THEOREM 2.1 Suppose that the nonrandom functions ϕ  x  , f  t , x , u  with
values in E
n
and Borel measurable in all their arguments, the function b  t , x , u  with
values in E
m
, the matrix-function g  t , x  of dimension d 1
× n and matrix- function
c  t , x , u  =  c 1
 t , x , u  , c 2
 t , x , u   , where c j
 t , x , u  is of
dimension d j × m, are uniformly bounded by a positive constant K and satisfy a Lipschitz
condition in x and u with a positive constant L. Then there exist a time interval  0 , T  in
which the system (2.1) has unique solution u  t , x  , ξ  s ; t , x  in the class of the
functions
x , t 1 ≠ t 2
sup E |
ξ  s ; t 1 , x  − ξ  s ; t 2 , x 
| t 1
− t 2
|
0 . 5
| p
+
+
x , s 1 ≠ s 2
sup E |
ξ  s 1
; t , x  − ξ  s 2
; t , x 
| s 1
− s 2
|
0 . 5
|
p
+

+
x
sup E | u  t, x  |
p
+
x ≠ y
sup E |
u  t , x  − u  t , y 
x − y
|
p
+ #
s , x
sup E |
ξ  s ; t, x 
1 + | x |
|
p
+
s , x ≠ y
sup E |
ξ  s ; t, x  − ξ  s ; t, y 
x − y
|
p
+
+
t ≠ s , x
sup E |
u  t , x  − u  s , x 
| t − s |
0 . 5
|
p
< ∞
for any p ≥ 2 , and 0 ≤ s ≤ t < T . In addition, a solution to system (2.1) is unique
in the class of function that satisfy ( 2.2 ).
Proof. Before proceeding to prove the theorem , we point out that the conditional
expectation in the integrand of stochastic integral has a predictable modification. It can be
proved the same way as Lemma 2 . Rewriting the system (2.1) in operator form
ξ  s ; t , x  = L  s ; t , x   ξ , u 
u  t , x  = L  t , x   ξ , u 
we form successive approximations
ξ
 0 
 s ; t , x  = x ,
ξ
 n 
 s ; t , x  = L  s ; t , x   ξ
 n − 1 
, u
 n 
 ,
u
 n 
 t , x  = L  t , x   ξ
 n − 1 
, u
 n 
.
This choice of the successive approximations give us possibility to use the results of the
seilinear case that considered above. The first approximation ξ
 0 
 s ; t, x  = x,
determines a function u
 1 
 t , x  that satisfies (2.2) in an arbitrary interval of  0 , ∞ .
Suppose now that the functions ξ
 n 
 s ; t , x  , u
 n 
 t , x  were found and let
L n
denote the constant on right -hand of (2.2). Show that there is a constant L n + 1
that
majorized the left-hand side of (2.2), where
ξ  s ; t, x  = ξ
 n 
 s ; t, x  , u  t, x  = u
 n 
 t, x 
This shows that the procedure of construction successive approximations does not

terminate. We now indicate the time-interval in which the sequence of numbers L n
is
uniformly bounded. Let Δ be an arbitrary interval of time p ≥ 2 , and R  t , ξ , u  be
the expression of the right hand side of ( 2.2 ). By definition we put
R n
=
t ∈ Δ
sup R  t , ξ
 n 
, u
 n 

We will establish the constant R n + 1 for which
R n + 1 =
t ∈ Δ
sup R  t , ξ
 n + 1 
, u
 n + 1 

We have
 1 + | x |
p

− 1
E | ξ
 n 
 s ; t , x  |
p
≤ 3
p − 1
 1 + K
p
α p
 t 
E | u
 n + 1 
 t , x  |
p
≤ 3
p − 1
K
p
 1 + α p  t 
where α p  t  = p
2
 p − 1  t
p
2
− 1
+ t
p − 1
. Denote
u t
 n 
p =
x ≠ y
sup E |
u  t , x  − u  t , y 
x − y
|
p
,
ξ s , t
 n 
p
=
x ≠ y
sup E |
ξ  s ; t , x  − ξ  s ; t , y 
x − y
|
p
Then
u t
 n + 1 
p ≤ 3
p − 1
L
p
 ξ 0 , t
 n 
p +
+ α p  t 
t
0
∫  1 + u s
 n 
p ξ 0 , t
 n 
p d s 
With Gronwall’s lemma, we obtain

t ≤ T
sup u t
 n + 1 
p ≤ U n , p
where
U n , p
= 3
p − 1
 L R n

p
 1 + T α p
 T   exp 3
p − 1
α  T   L R n

p
T
Similarly ,
ξ s , t
 n 
p
≤ V n , p
,
t ≠ s , x
sup E |
u
 n + 1 
 t, x  − u
 n + 1 
 s, x 
| t − s |
0 . 5
|
p
≤ 3
p − 1
 L R n

p
×
×  1 + 2
p − 1
T α p
 T   1 + U n , p
 + 2
p − 1
T α p
 T  ,
x , t 1 ≠ t 2
sup E |
ξ
 n + 1 
 s ; t 1
, x  − ξ
 n + 1 
 s ; t 2
, x 
| t 1
− t 2
|
0 . 5
|
p
≤
≤  4
3

p − 1
 V n , p
+ K
p
T
p
+ α p
 T  ,
x , s 1 ≠ s 2
sup E |
ξ
 n + 1 
 s 1; t, x  − ξ
 n + 1 
 s 2 ; t, x 
| s 1 − s 2 |
0 . 5
|
p
≤
≤ 2
p − 1
K
p
1 + α p
 T 
where
V n , p
= 3
p − 1
 1 + T α p
 T   L R n

p
 1 + U n , p

One can point out an interval of time where numbers V n , p , U n , p are uniformly

bounded. We have
E | u
 n + 1 
 t , x  − u
 n + 1 
 t , y  |
p
≤
≤ β p
 t 
s
sup E | ξ
 n 
 s ; t, x  − ξ
 n 
 s ; t, y  |
p
;
s
sup E | ξ
 n 
 s ; t, x  − ξ
 n 
 s ; t, y  |
p
≤ 3
p − 1
 | x − y |
p
+
+ μ p  t 
s
sup E | ξ
 n − 1 
 s ; t, x  − ξ
 n − 1 
 s ; t, y  |
p

where
β p  t  = 3
p − 1
L
p
 1 + t   1 + α p  t   exp 3
p − 1
L
p
t α p  t  ,
μ p  t  = t  1 + α p  t    1 + β p  t   L
p
The function μ p
 t  is continuous and monotone increasing and μ p
 0  = 0. Let
t 1
denote the root of equation μ p
 t  = 3
1 − p
. Then, for any t < t 1
s
sup E | ξ
 n 
 s ; t, x  − ξ
 n 
 s ; t, y  |
p
≤
≤
3
p − 1
1 − 3
p − 1
μ p
 t 
| x − y |
p
#
E | u
 n 
 t, x  − u
 n 
 t, y  |
p
≤
3
p − 1
β p
 t 
1 − 3
p − 1
μ p  t 
| x − y |
p
Using these estimates, we can prove that the successive approximations converge. Let
B p
t
, N p
t
be Banach spaces of the random functions
Ϝ t , Ϝ
t
, t ∈  0 , T with norms

|| u || t , p
p
=
x , s ≤ t
sup E | u  s , x  |
p
,
||| ξ ||| t , p
p
=
x , s ≤ t
sup E |
ξ  s ; t , x 
1 + | x |
|
p
,
respectively. Putting for simplicity p = 2 we have
|| u
 n + 1 
− u
 n 
|| t , 2
2
≤ 3 L
2
  1 + t α 2  t   +
+ 2α 2  t 
t
0
∫ || u
 n + 1 
− u
 n 
|| s , 2
2
d s + 2α 2  t  ×
×
t
0
∫
x
sup EE | u
 n + 1 
 s, z 1
 − u
 n + 1 
 s, z 2
 |
2

z 1 = ξ
 n 
 s ; t , x 
z 2 = ξ
 n − 1 
 s ; t , x 
d s
Applying the formulas (2.3) we find that
t
0
∫
x
sup E E | u
 n + 1 
 s, z 1
 − u
 n + 1 
 s, z 2
 |2

z 1 = ξ
 n 
 s ; t , x 
z 2 = ξ
 n − 1 
 s ; t , x 
d s ≤
≤
3 β 2
 t 
1 − 3 μ 2
 t 
||| ξ
 n 
− ξ
 n − 1 
||| t , 2
2
Applying Gronwall’s lemma, we obtain
|| u
 n + 1 
− u
 n 
|| t , 2
2
≤ γ  t  ||| ξ
 n 
− ξ
 n − 1 
||| t , 2
2
where γ  t  = 3 L
2

 1 + t α 2  t   + 6 t β 2  t 
1 − 3 μ 2  t 
 exp 6 L
2
t α 2
 t  .
Similarly to the preceding
||| ξ
 n 
− ξ
 n − 1 
||| t , 2
2
≤ 2 L
2
α 2
 t   t ||| ξ
 n − 1 
− ξ
 n − 2 
||| t , 2
2
+

+ 2t || u
 n 
− u
 n − 1 
|| t , 2
2
+ 2β 2
 t 
t
0
∫
x
sup E  E | ξ
 n − 2 
 s ; t, z 1
 −
− ξ
 n − 2 
 s ; t, z 2  |
2
/ z 1 = ξ
 n − 1 
 s ; t , x  , z 2 = ξ
 n − 2 
 s ; t , x 
d s  ≤
≤ λ  t  ||| ξ
 n − 1 
− ξ
 n − 2 
||| t , 2
2
where λ  t  = 2 L
2
α 2  t 
 1 + 2 γ  t   + 6 β 2  t 
1 − 3 μ 2  t 
. We point out that
λ  t  is continuous and monotone increasing, and
λ  0  = 0 . Thus, λ  t 2
 = 1 for some positive t 2
and therefore
||| ξ
 n 
− ξ
 n − 1 
||| t , 2
2
≤ λ
n
 t  ||| ξ
 1 
− ξ
 0 
||| t , 2
2
|| u
 n 
− u
 n − 1 
|| t , 2
2
≤ γ  t  λ
n
 t  ||| ξ
 1 
− ξ
 0 
||| t , 2
2
Let T = min  t 1
, t 2
 . Then, for t < T , the preceding inequalities lead to
n , m → ∞
lim  || u
 n 
− u
 m 
||| t , 2
2
+ ||| ξ
 n 
− ξ
 m 
||| t , 2
2
 = 0
This means that there exist processes ξ  s ; t , x  , u  t , x  for which
m → ∞
lim  || u − u
 m 
||| t , 2
2
+ ||| ξ − ξ
 m 
||| t , 2
2
 = 0
when t < T .It easy to verify that
E | ξ
 n 
 s 1 ; t , x  − ξ
 n 
 s 2 ; t , x  |
p
≤
≤ 2
p
K
p
 1 + α p  t   | s 1 − s 2 |
p
2
,

E | ξ
 n 
 s ; t 1 , x  − ξ
 n 
 s ; t 2 , x  |
p
≤ k
 1 
 t
∗
 | t 1 − t 2 |
p
2
,
E | u
 n 
 t 1
, x  − u
 n 
 t , x  |
p
≤ k
 2 
 t
∗
 | t 1
− t 2
|
p
2
,
where t
∗
= max  t 1 , t 2  and
k
 1 
 t  = 4
p − 1
K
p
 1 + α p
 t   exp 4
p − 1
L
p
 1 + α p
 t   t ;
k
 2 
 t  = 6
p − 1
K
p
 1 + α p
 t   + 3
p − 1
L
p
 1 +
+ 2
p − 1
 1 + α p  t    1 +
3 β p
 t 
1 − 3 μ p  t 
 k
 1 
 t  .
Letting n → ∞ in formula (2.3) we can show that the processes
ξ  s ; t , x  , u  t , x  belong to the spaces B p
t
, N p
t
, satisfy (2.2) and therefore
have measurable separable modification. Retain the same notation for them as before. Now,
we are able to show that this modifications are solutions of the system(2.1). To this end it
suffices to show possibility to pass to the limit in each of the terms in the system of successive
approximations. Justifying the passage to the limit in each of the terms occurring in the system
is basically the same and so we shall explain it just for one of the stochastic integrals. Applying
the properties of conditional expectations and using (2.2) for t < T we note
E |
t
s
∫ E  c  r, ξ
 n − 1 
 r ; t, x  , v  r , ξ
 n − 1 
 r ; t, x    −
− c  r , ξ  r ; t , x  , v  r , ξ  r ; t , x    / Ϝ
r
 d w←
 r  | 2
≤
≤ 2 L
2
t
s
∫ E  | ξ
 n − 1 
 r ; t, x  − ξ  r ; t, x  |
2
 1 + || v
 n 
|| r
2
 +
+ || u
 n 
− u || r
2
 d r ≤ N 1  ||| ξ
 n − 1 
− ξ ||| t
2
+ || u
 n 
− u || t
2


In what follows, N i = N i
 t  , i = 1 , 2 , ... will denote continuous increasing
functions on  0 , T  . By means of analogous reasoning , by passing to the limit in the
system of successive approximations as n → ∞ we can show that the functions
ξ  s ; t , x  , u  t , x  give a solution to the system (2.1) on  0 , T  for which the
inequality (2.2) holds. To prove uniqueness of the solution let suppose that
ξ j
 s ; t , x  and u j
 t , x  , j = 1, 2 be two solutions of the system (2.1)
satisfying (2.2) and specified on the same time interval. Then
E | ξ 1  s ; t, x  − ξ 2  s ; t, x  |
2
≤ 2  1 + t  L
2
×
×
t
s
∫ E  | ξ 1
 r ; t, x  − ξ 2
 r ; t, x  |
2
+
+ | u 1
 r, ξ 1
 r ; t, x   − u 2
 r, ξ 2
 r ; t, x   |
2
 d r ≤
≤ N 2
t
s
∫ E  | ξ 1
 r ; t, x  − ξ 2
 r ; t, x  |
2
+
+
x
sup E | u 1  r, x  − u 2  r, x  |
2
 d r .
By Grownwall’s lemma , we find that
E | ξ 1
 s ; t, x  − ξ 2
 s ; t, x  |
2
≤
≤ N 2
t
s
∫
x
sup E | u 1
 r, x  − u 2
 r, x  |
2
d r .
In exactly the same way one can establish
x
sup E | u 1
 t, x  − u 2
 t, x  |
2
≤ N 4
||| ξ 1
− ξ 2
||| t
2
.
Taking these inequalities into account, we have
x
sup E | u 1
 t, x  − u 2
 t, x  |
2
≤ .

≤ N 5
t
s
∫
x
sup E | u 1
 r, x  − u 2
 r, x  |
2
d r
From this, we easily deduce that
x
sup E | u 1
 t , x  − u 2
 t , x  |
2
= 0 ,
and hence
s , x
sup E | ξ 1
 s ; t , x  − ξ 2
 s ; t , x  |
2
= 0 .
Theorem is proved.
In what follows, it will be shown that the function v  t , x  determined by the system
(1) form generalized solution of a stochastic Cauchy problem for a class of quasilinear
systems. First we find estimates for the derivatives of u  t , x  The system determining
ξ
 1 
 s ; t , x  = ∇ ξ  s ; t , x  ,
u
 1 
 t , x  = ∇ u  t , x  ,
where ∇ = 
∂
∂ x 1
,....
∂
∂ x m
 , which is founded by formally differentiating the
original one, has more complicated form and we have not succeeded in proving its solvability
by the same method as Theorem 2 1. More precisely, the right-hand side of the system
contains an integral of the product of u
 1 
and ξ
 1 
which is not allowed by the
hypotheses of the theorem 2.1. Therefore when speaking of estimates of derivatives of the
solution of the system (2.1) in what follows, we shall always bear in mind their a priori nature.
Nevertheless, it should be noted that if the first derivatives ξ
 1 
 s ; t , x  or
u
 1 
 t , x  have been shown to exist , then proving the existence of the high order
derivatives is of no difficulty, since the equations for them satisfy the hypotheses of Theorem
2.1 and the proof is similar to the theorem on the continuous dependence of the solution to the
system (2.1) on the coefficients. Formula (2.2) may be used to obtain a priori estimates for
ξ
 1 
 s ; t , x  and u
 1 
 t , x  . However, it is inadequate for obtaining estimates for
the higher derivatives. Assume for arbitrary p ≥ 2 that there exists a function
u
 1 
 t , x  such that
h → 0
lim
x
sup E |
u  t , x + h  − u  t , x 
h
− u
 1 
 t , x  |
p
= 0
for any t in some subinterval of  0 , T  introduced in Theorem 2.1. All of the

subsequence discussion will be carried out on this subinterval.
THEOREM 2.2 Assume that the functions
ϕ n  x  , b n  t , x , u  , c n  t , x , u  , f n  t , x, u  , g n  t , x 
satisfy the conditions of theorem 2.1 and
n → ∞
lim
x
sup | ϕ n  x  − ϕ  x  | +
T
0
∫
x , u
sup  | b n  t, x, u  − b  t, x, u  | +
+ | c n  t, x, u  − c  t, x, u  | +
+ | g n
 t, x  − g  t, x  | + | f n
 t, x, u  − f  t, x, u  |  d t = 0
Then , solutions u n
 t, x  , ξ n
 s ; t, x  converge to u  t, x  , ξ  s ; t, x 
n → ∞
lim
t , x
sup E | u n
 t, x  − u  t, x  |
2
+ | ξ n
 s ; t, x  − ξ  s ; t, x  |
2
 = 0
Here u  t, x  , ξ  s ; t , x  is the solution of the system ( 2.1 ).
Proof. The proof of this theorem is similar to correspondent proof of the theorem 1.2 and
so we just briefly remind the main stand points. One has
s , x
sup E | ξn  s ; t, x  − ξ  s ; t, x  |
2
≤
≤ 8
t
0
∫
u , x
sup  t | b n
 r, x, u  − b  r, x, u  |
2
+
+ | c n
 t, x, u  − c  t, x, u  |
2
 d s + 4 L
2
 1 + t   1 + 2 || u || t
2
 ×
×
t
0
∫
x
sup  E  | u n  r, x  − u  r, x  |
2
+ | ξ n  r ; t, x  − ξ  r ; t, x  |
2
 d r ;

t , x
sup E  | u n
 t, x  − u  t, x  |
2
≤ 12
t
0
∫
u , x
sup  | g n
 s, x  − g  s, x  |
2
+
+ | f n
 s, x, u  − f  s, x, u  |
2
 d s + 3
x
sup | ϕ n
 x  − ϕ  x  |
2
+
+ 12 L
2
 1 + t  1 + 2 || u || t
2

t
0
∫
x
sup  E  | u n  r, x  − u  r, x  |
2
+
+ | ξ n
 r ; t, x  − ξ  r ; t, x  |
2
 d r .
Adding together these two inequalities and applying Grownwall’s lemma , we get the
proof. Consider the equation
η  s ; t, x  = I +
t
s
∫ b
 1 
 r ; t, x, η  r ; t, x   d r − #
−
t
s
∫ E  c
 1 
 r ; t, x, η  r ; t, x   / Ϝ  d
←
w  r 
where
b
 1 
 r ; t, x, η  r ; t, x   = 
∂ b
∂ x
 r, ξ  r ; t, x  , v  r, ξ  r ; t, x    +
+
∂ b
∂ u
 r, ξ  r ; t, x , v  r, ξ  r ; t, x    v 1 
 r, ξ  r ; t, x   η  r ; t, x ;
c
 1 
 r ; t, x, η  r ; t, x   = 
∂ c
∂ x
 r, ξ  r ; t, x , v  r, ξ  r ; t, x    +
+
∂ c
∂ u
 r, ξ  r ; t, x , v  r, ξ  r ; t, x    v 1 
 r, ξ  r ; t, x   η  r ; t, x ;

v
 1 
 t , x  = E  u  t , x  | Ϝ  0 , t 
 1 

The system (2.4) is derived from the equation for characteristic occurring in (2.1) by
differentiating with respect to parameter x .
Lemma 2.1. Suppose that the hypotheses of Theorem 2.1 hold and that the functions
b  t, x, u  and c  t, x, u  have bounded continuous first derivatives in x and u . Then
the system (2.4) has a unique solution with η  r ; t, x  = = ξ
 1 
 r ; t, x  , and
the derivative is understood in the sense of mean square.
Proof. Since the proof of the lemma is similar to the corresponding one for ordinary
stochastic differential equations, we shall state only main steps. Denote the right hand side of
(2.4) by
L η  s ; t, x  . Then
E | Lη  s ; t, x  |
2
≤ 3  1 +  1 + T  K
2
×
× || 1 + u
 1 
|| t
2
t
s
∫ E | η  r ; t, x  |
2
d r  ;
E | L η  s ; t, x  − L ζ  s ; t, x  |
2
≤
≤ 2  1 + T  K
2
|| 1 + u
 1 
|| t
2
t
s
∫ E | η  r ; t, x  − ζ  r ; t, x  |
2
d r
From this inequalities, it is easy to see that some power of operator L is a contracting
operator in the Banach space of random functions with finite second order moments. This
implies the first assertion of the lemma. We then have
E | η  s ; t, x  −
ξ  s ; t, x + Δ x  − ξ  s ; t, x 
Δ x
|
2
≤
≤ 2
t
s
∫ E  t | B
 1 
 r ; t, x  − B Δ
 1 
 r ; t, x  |
2
+

+ | C
 1 
 r ; t, x  − C Δ
 1 
 r ; t, x  |
2
 d r
To simplify notation we put B
 1 
 r ; t, x  = b
 1 
 r ; t, x, η  r ; t, x   , and
we have taken for B Δ
 1 
 r ; t, x  a similar expression corresponding to the drift
coefficient of the process
ξ  s ; t, x + Δ x  − ξ  s ; t, x 
Δ x
and C
 1 
 r ; t, x  , C Δ
 1 
 r ; t, x  are given similarly. More precisely,
B Δ
 1 
 r ; t, x  = Δx
− 1
b  r, ξ  r ; t, x + Δx , v  r, ξ  r ; t, x + Δx    −
− b  r , ξ  r ; t, x  , v  r , ξ  r ; t, x   
C Δ
 1 
 r ; t, x  = Δx
− 1
 c  r, ξ  r ; t, x + Δx , v  r, ξ  r ; t, x + Δx    −
− c  r, ξ  r ; t, x , v  r, ξ  r ; t, x   
Since the estimation of both terms in the integrand is identical, we shall do one of them.
We find that
E | B
 1 
 r ; t, x  − B Δ
 1 
 r ; t, x  |
2
≤
≤ 4 E | ∇ x
b  r , ξ  r ; t, x , v  r, ξ  r ; t, x    −
−
1
0
∫  1 − θ  ∇xb  r, ξθ  r ; t, x , v  r, ξ  r ; t, x + Δ x    d θ |
2
||| η ||| t
2
+
+ 4 K
2
E | η  s ; t, x  −
ξ  s ; t, x + Δ x  − ξ  s ; t, x 
Δ x
|
2
+
+ 6 E | ∇ u b  r, ξ  r ; t, x , v  r, ξ  r ; t, x    −

−
1
0
∫  1 − θ  ∇ u
b  r , ξ  r ; t, x  , v  r , ξ θ
 r ; t, x    d θ |
2
×
× | η  r ; t, x  |
2
| v θ  r , ξ  r ; t, x   |
2
+
+ 6 K
2
E || u 1 
|| t
2
E | η  s ; t, x  −
ξ  s ; t, x + Δ x  − ξ  s ; t, x 
Δ x
|
2
+
+ 6 K
2
E | ξ
 1 
 r ; t, x  |
2
| v
 1 
 r , ξ  r ; t, x   −
−
t
0
∫  1 − θ  v
 1 
 r , ξ θ  r ; t, x    d θ |
2
where ξ θ  r ; t, x  = θ ξ  r ; t, x  +  1 − θ  ξ  r ; t, x + Δ x . Applying
Holder’s inequality and using the conditions of the Lemma and finiteness of the moments of
the random functions we easily find that
x
sup E | η  s ; t, x  −
ξ  s ; t, x + Δ x  − ξ  s ; t, x 
Δ x
|
2
≤   Δ x  +
+ R
0
t
∫
x
sup E | η  r ; t, x  −
ξ  r ; t, x + Δ x  − ξ  r ; t, x 
Δ x
|
2
d r
where
Δ x → 0
lim   Δ x  = 0 and R = R  || v
 1 
|| , K  . The proof of Lemma 2.1
is easily completed by resorting to Gronwall’s lemma.
Remark. Taking lemma 2.1 as our starting assumption and applying the theorem on
continuous dependence of a solution on the coefficients with some insignificant complications,
we can prove that the process u
 1 
 t , x  satisfies the equation obtained from (2.1) by
differentiating formally with respect to x . Therefore the system
u
 1 
 t, x  = Φ
 1 
 0 ; t, x  +
t
0
∫ F
 1
 r ; t, x  d r +

+
t
0
∫ G
 1
 r ; t, x  d
←
w 1  r  #
ξ
 1 
 s ; t, x  = I +
t
s
∫ B
 1
 r ; t, x  d r +
+
t
s
∫ E  C
 1
 r ; t, x  / Ϝ  0 , t
 d
←
w  r 
will be considered as we investigate further the a priori smoothness of the function
u  t , x  . Here Φ
 1 
 0 ; t , x  = ∇ ϕ  ξ  0 ; t, x   ξ
 1 
 0 ; t, x  and
F
 1
 r ; t , x  and G
 1
 r ; t , x  are given by expressions similar to
B
 1
 r ; t , x  in Lemma 2.1.
LEMMA 2.2. Under hypotheses of Lemma2.1, suppose that the partial derivatives of the
coefficients of the system (2.1) satisfy a Holder condition in x and u with exponent
γ ∈  0 , 1 . Then, for p ≥ 2
x ≠ y
sup E  |
u
 1 
 t, x  − u
 1 
 t, y 
| x − y |
γ |
p
+
+ |
ξ
 1 
 s ; t, x  − ξ
 1 
 s ; t, y 
| x − y |
γ |
p
 < ∞
Proof. We have
E | ξ
 1 
 s ; t , x  − ξ
 1 
 s ; t , y  |
p
≤
≤ 2
p − 1
s
t
∫ E  t
p − 1
| B
 1 
 r ; t, x  − B Δ
 1 
 r ; t, x  |
p
+
+
p
2
 p − 1  t
1
2
 p − 1 
| C
 1 
 r ; t, x  − C Δ
 1 
 r ; t, x  |
p
 d r ;
E | u
 1 
 t, x  − u
 1 
 t, y  |
p
≤

≤ 3
p − 1
E  | Φ
 1 
 0 ; t, x  − Φ
 1 
 0 ; t, y  |
p
+
+ ∫ s
t
E  t
p − 1
| F
 1 
 r ; t, x  − F Δ
 1 
 r ; t, x  |
p
+
+
p
2
 p − 1  t
1
2
 p − 1 
| G
 1 
 r ; t, x  − G Δ
 1 
 r ; t, x  |
p
 d r .
As an example, we shall estimate one of the terms on the right-hand side:
t
0
∫ E | F
 1 
 r ; t, x  − F Δ
 1 
 r ; t, x  |
p
≤
≤ 2
p − 1
E
t
0
∫  | ∇ x
f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    ξ
 1 
 r ; t, x  −
− ∇ x
f  r, ξ  r ; t, y , v  r, ξ  r ; t, y    ξ
 1 
 r ; t, y  |
p
 d r +
+ | ∇u f r, ξ r ; t, x , v  r, ξ r ; t, x  v 1 
 r, ξ r ; t, x  ξ 1 
 r ; t, x  −
− ∇ u
f  r, ξ  r ; t, y , v  r, ξ  r ; t, y    v
 1 
 r, ξ  r ; t, y   ×
× ξ
 1 
 r ; t, y  |
p
 d r = 2
p − 1
E
t
0
∫  I 1  r  + I 2  r  d r .
In the subsequent computations, N i
, i = 1,2... , denote positive constants not
depending on x or y . Applying Hölder inequality, we easily obtain the estimate
I 2
 r  ≤ N 1
E | ξ
 1 
 r ; t , x  − ξ
 1 
 r ; t , y  |
p
+
+ N 2 E | u
 1 
 r , x  − u
 1 
 r , y  |
p
+ N 3 | x − y |
p γ
.
A similar estimate also holds for I 1  r 
I 1  r  ≤ N 5 E | ξ
 1 
 r ; t, x  − ξ
 1 
 r ; t, y  |
p
+ N 4 | x − y |
p γ
.

Since
E | Φ
 1 
 0 ; t, x  − Φ
 1 
 0 ; t, y  |
p
≤
≤ 2
p − 1
E  G γ
p
| ξ
 1 
 r ; t, x  − ξ
 1 
 r ; t, y  |
p γ
+
+ K
p
| ξ
 1 
 r ; t , x  − ξ
 1 
 r ; t , y  |
p
,
it is not hard to see that
+ |
ξ
 1 
 s ; t , x  − ξ
 1 
 s ; t , y 
| x − y |
γ |
p
 ≤ N 6
+
+ N 7
t
0
∫
x ≠ y
sup E  |
u
 1 
 r , x  − u
 1 
 r , y 
| x − y |
γ |
p
+
+ |
ξ
 1 
 r ; t , x  − ξ
 1 
 r ; t , y 
| x − y |
γ |
p
 d r
Applying Gronwall’s lemma, we see that Lemma 2.2 is true.
Define expressions B
 2 
 r ; t, x  , C
 2 
 r ; t, x  , Φ
 2 
 r ; t, x  ,
F
 2 
 r ; t, x  , G
 2 
 r ; t, x  by formal differentiation with respect to x of the
correspondent coefficients of the system ( 2.1 ). For instance
Φ
 2 
 0 ; t, x  =
= Tr   ξ
 1 
 0 ; t, x 
∗ ∂
2
ϕ  ξ  0 ; t, x  
∂ x
2
ξ
 1 
 0 ; t, x  +
+
∂ ϕ  ξ  0 ; t , x  
∂ x
ξ
 2 
 0 ; t , x   ,

B
 2 
 r ; t, x  = Tr ξ
 1 
 r ; t, x 
∗ ∂
2
b  ξ  r ; t, x  
∂ x
2
ξ
 1 
 r ; t, x  +
+  v
 1 
 r , ξ  r ; t, x   ξ
 1 
 r ; t, x 
∗
×
×
∂
2
b  ξ  r ; t , x  
∂ x ∂ u
ξ
 1 
 r ; t, x  +  ξ
 1 
 r ; t, x 
∗
×
×
∂
2
b  ξ  r ; t, x  
∂ x ∂ u
v
 1 
 r, ξ  r ; t, x   ξ
 1 
 r ; t, x  +
 v
 1 
 r, ξ  r ; t, x   ξ
 1 
 r ; t, x 
∗ ∂
2
b  ξ  r ; t, x  
∂ u
2
×
× v
 1 
 r, ξ  r ; t, x   ξ
 1 
 r ; t, x  +
+
∂ b  ξ  r ; t, x  
∂ x
ξ
 2 
 r ; t, x  +
∂ b  ξ  r ; t, x  
∂ u
×
×  ξ
 1 
 r ; t, x 
∗
v
 2 
 r , ξ  r ; t, x   ξ
 1 
 r ; t, x  +
+
∂ b  ξ  r ; t, x  
∂ u
v
 1 
 r , ξ  r ; t, x   ξ
 2 
 r ; t, x   .
Lemma 2.3. Let the functions ξ  s ; t , x  and u  t , x  exist and be in the
mean in all their arguments and satisfy, for all q ≥ 2
x
sup E  | ξ  s ; t, x  |
q
+ | u  t, x  |
q
 < ∞
Assume that coefficients of system (2.1) have continuous bounded second-order
derivatives in x and u . Then the second-order derivatives ξ
 2 
 r ; t , x  =
∂
2
ξ
∂ x
2
,

u
 2 
 t , x  =
∂
2
u
∂ x
2 exist in the sense of convergence in the mean, and they satisfy
the system
u
 2 
 t , x  = Φ
 2 
 0 ; t , x  +
t
0
∫ F
 2 
 r ; t , x  d r +
+
t
0
∫ G
 2 
 r ; t , x  d
←
w 1  r  #
ξ
 2 
 s ; t, x  =
t
s
∫ B
 2 
 r ; t, x  d r +
+
s
t
∫ E  C
 2 
 r ; t , x  / Ϝ  0 , t
 d
←
w  r 
and
x
sup E  | u
 2 
 t , x  |
p
+ | ξ
 2 
 s ; t , x  |
p
 < ∞
If the second order partial derivatives of the coefficients of the system (2.1) satisfy Hölder
condition with exponent γ ∈  0 ,1 , then
x ≠ y
sup E  |
u
 2 
 t , x  − u
 2 
 t , y 
| x − y |
γ |
p
+
+ |
ξ
 2 
 s ; t , x  − ξ
 2 
 s ; t , y 
| x − y |
γ |
p
 < ∞ #
The coefficients in system (2.6) are obtained from those of (2.1) by means of repeated
formal differentiation with respect to parameter x . In contrast to the corresponding assertion
for the first order derivatives, the a priori existence of u
 2 
 t , x  is not assumed. The
other statements in the lemma are proved similarly to the preceding one and so it will not be
done. To show that u  2 
 t , x  , ξ
 2 
 s ; t , x  exist it is necessary to repeat the
proof of Theorem 2.1 with some minor additions.
We pause now to consider the question of relationship between a solution to the system
(2.1) and quasilinear parabolic equations.
Lemma 2.4. Let u  t , x  , ξ  s ; t , x  be a solution to system (2.1). Then

ξ  s ; t , x  = ξ  s ; r , ξ  r ; t , x  
#
for any r ∈  s , t with probability 1.
Proof. From the system (2.1), we obtain
ξ  s ; t, x  = ξ  r ; t, x  +
r
s
∫ b  l, ξ  l ; t, x , v  l , ξ  l ; t, x    d r −
−
r
s
∫ E  c  l, ξ  l ; t, x  , v  l, ξ  l ; t, x    / Ϝ  l , t
 d
←
w  r 
Note, that the random function ξ  s ; r , ξ  r ; t , x   is measurable in all the
arguments s , t , x being the composition of measurable mappings, and is consistent with the
flow of σ-algebras Ϝ  s , t
and
ξ  s ; r , ξ  r ; t , x   = L  s ; t , ξ  r ; t , x    ξ , u ,
where the operator L  s ; t , x  was defined in the proof of Theorem 2.1.
Therefore
E | ξ  s ; t, x  − ξ  s ; r , ξ  r ; t , x   |
2
≤
≤ 2 L
2
 1 + T 
r
s
∫  1 +
x ≠ y
sup E  |
u  l , x  − u  l , y 
x − y
|
2
 ×
× E | ξ  l ; t, x  − ξ  l ; r , ξ  r ; t , x   |
2
d l
Taking into account estimate (2.2) and applying Gronwall’s lemma, we can easily
complete the proof of Lemma2.4.
Random function ξ  s ; t, x  is continuous over all its arguments, and so has a
separable modification such that equality ( 2.8) holds with probability 1 over the introduced
time interval. In what follows this modification we will mean considering ξ  s ; t, x . From
lemma 2.4 ensue that

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