SPDE

308 views

Published on

Stochastic Partial Differential Equations

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
308
On SlideShare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

SPDE

  1. 1. STOCHASTIC DIFFERENTIAL EQUATIONS WITH PARTIALDERIVATIVES.Gikhman, IlyaPLLS49 East Fourth StreetCincinnati, OH 45202ph: (513) 763-8308e-mail: iljogik@yahoo.comINTRODUCTIONStochastic partial differential equations are the part of the general theory of random fieldswhich is actively being developed and applied. There exist different approaches to setting andsolving these problems. We single out the principle ones. In monograph [39] the linear theorybased on the interpretation of parabolic operators as infinitesimal operators of diffusionprocesses is given. Generalizations of that approach were established in [33]. In thismonograph using the stochastic characteristic systems like the case of the first orderdeterministic partial differential equations it was shown that a first and the second ordersstochastic partial differential systems can be solved. Using A number of application, where thenecessity of solving stochastic parabolic problems appears, have been shown. In [32,36] thegeneralization of direct methods of mathematical physics which allows to prove solvability ofnon-linear stochastic systems is shown. For these methods it is necessary to obtain aprioriestimates assuming coefficients to be coercive, and also to prove the possibility of limittransition in finite-dimensional approximations of input equation to be monotone. The mainrole in these methods is assigned to a drift coefficient. When it is equal to zero, a diffusioncoefficient 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 iswell known that the existence of a finite second moment is essential for the existence ofsolutions of ordinary stochastic differential equations. Under certain assumptions oncoefficients growth, as it is shown in [21], a solution of an ordinary stochastic differentialequation does not possess a finite second moment. In [22, 28] a solution of parabolic equationswith coefficients of ”white noise” type is constructed. One more approach to the solution ofstochastic parabolic problems consists in the substitution of integral equation of Ito-Volterratype in corresponding functional spaces for initial problem [3, 6, 34]. This method has beeninitial in stating and solving stochastic equations with unbounded operators.We have mentioned only the works in which the principal approaches are described. Atpresent the list of papers and books where solutions of stochastic evolution problems arestudied 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 forsemi-linear parabolic equation is suggested, and its physical interpretation is considered. Someof the results presenting here are given in [15]. We note that the results obtained belowcorrespond to those obtained by probabilistic methods used for a study of deterministicquasilinear parabolic equations [4, 11, 40]. It is also worthy to note the difference between theCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  2. 2. solutions of semilinear and quasilinear systems, although the semilinear case of determinedsystems has not been taken separately. The latter allows to prove the existence and uniquenessof 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 solutionis local in character when there is no dissipation.At the last part of the paper we introduce mathematically correct stochastic interpretationof the Schrodinger equation solution [16-18,23-26] and some model examples that use thisrepresentation. The same complex-valued representation for the Shrödinger equation solutionfirst was given in [9]. Some applications of this approach to the particular problems werestudied in [1,8,31].Now, we shall remind the basic aspects of the Lagrange and Euler formalisms used todescribe the dynamics of solid medial. Further, physical interpretation will be generalized forthe stochastic case. We can describe dynamic processes in physical media in two ways. Thefirst consists in the treatment of medium parameters, for each moment t ≥ 0, as functions ofsome 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 inthe interpretation of solid medium as an aggregation of particles. Each particle differs from theother ones in its initial position. Both of these alternative approaches play an important role forstatistical description of motion in nonhomogeneous media, in the turbulence theory and otherapplications [22]. We shall now analyze these methods in the main. It is convenient to interpretparticle motion trajectory via the inverse of time.Let, for the initial moment t ≥ 0 , a particle occupy the position of y and supposeV  s, x  to be a velocity of this particle at the point x and moment s ≥ 0. Then the Eulercoordinate x = x  s ; t , y  at the moment s ∈  0, t  is calculated by the formulaxi  s ; t, y  = yi −ts∫ 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 thatF  t, x  is a smooth function of the Euler coordinates and a medium parameter. It is easy torepresent it as a function of the Lagrange coordinatesFl  s, y  = F  s, y −ts∫ v i  r, x  r ; t, y   d r  #By differentiating the relation ( 0. 2 ) with respect to s we findd F l s, y d s=∂ F  s, x ∂ s+ni = 1∑ vi  s, x ∂ F  s, x ∂ x ix = x  s, t, y #Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  3. 3. The expression in square brackets on the right hand side of ( 0. 3 ) is total derivative withrespect to time along particle path. It is called a substantive derivative. In the case, when thesolution 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 toshow the equivalence of the Lagrange and Euler formalisms we construct the inverse mappingof the Euler coordinates into Lagrange ones. It means, by definitions to find a sufficientlysmooth function G  s, x  for which the equationG  s, x  s ; t , x   = y #has a unique solution. Taking into account of the formula for a substantive derivative, onecan write equation ( 0. 4 ) in the formG  0, y  +ts∫  ∂∂ 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 thefunction G  t, x  , which defines the transformation is a solution of the Cauchy problem∂ G  s, x ∂ s+ni = 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 ordinarystochastic differential equations. To do this an accurate theory analogous to the formalismdescribed above can be establish. It will turn out, that the definition of the functionG  ∗ , ∗  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 tomacroparameter G  t , x  which characterizes solid medium. We note that the equationsused here, with inverse time for the Lagrange trajectories presuppose to consider the Eulerdynamics via a straight direction of time.Semilinear SPDE.Before representing our results, let us introduce another approach to the Cauchy problemstudy of the SPDE of the parabolic type. After the presentation of the Backward StochasticDifferential Equations ( BSDE ) by Peng and Pardoux [37] in 1990, this field has receivedincreasing interest and activity [10,37,38]. Following [38] letCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  4. 4. d X st,x= b  s , X st,x ds + σ  s , X st,x d w  s where s ∈  t , ∞  be a classical Ito equation solution such that X tt,x= x. Theassociated Peng-Pardoux BSDE isY st,x= h  X Tt,x +Ts∫ f  r , X rt,x, Y rt,x, Z rt,x dr ++Ts∫ g  r , X rt,x, Y rt,x, Z rt,x d B  r  −Ts∫ Z rt,xd w  r where s ∈  t , T , and w  t , B  t  , t ≥ 0 are two independent Wienerprocesses. They proved that under certain assumptions on coefficients of BSDE , X st,xhas aversion whose trajectories are continuous in t, s and twice continuously differentiable in x.They showed that the random field Z st,xhas a modification such thatZ st,x= ∇ Y st,x ∇ X st,x − 1σ  X st,xThus, substituting this value into BSDE and lettingY rt,x= u  r, X rt,x and then Y tt,x= u  t, x .we can see that BSDE is the type of the stochastic evolution equation that is studied in thissection. An unessential difference between BSDE and the stochastic evolution systems studiedearlier is direction of time. Peng-Pardoux chose forward time for the characteristics X ∗ andthus implied inverse time for BSDE. In papers [15,19,19,22] were used backward time for thecharacteristics and forward time for the evolution which gives us the possibility of studyCauchy problem for SPDE in the common direction of time. It seems also important toemphasize that the application BSDE to the study of the deterministic PDE [10] in theirexplicit evolutionary form will lead us to the probabilistic methods of the study Cauchyproblem for the nonlinear parabolic deterministic systems [4,7,11,27,40].We will introduce the notations. Let Embe -dimensional Euclid space andE += 0, + ∞ . Assume that the nonrandom Borel functions b  t, x , C  t, x ,defined for  t , x  ∈ E +× Emand taking values in Emand Ed× Emsatisfy thefollowing 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 timeCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  5. 5. ξ  s ; t, x  = x +ts∫ b  r, ξ  r ; t, x   d r −ts∫ C  r, ξ  r ; t, x   d←w  r #Here, the stochastic integral is the inverse stochastic Ito integral, and w  t  is d-dimensionalWiener process on a some complete probability space  Ω , Ϝ , P  . By definition thesolution of equation ( 1. 2 ) is a random function ξ  s ; t, x  measurable in its variabless, t, x, ω respectively, and for fixed t, x with respect to the σ - fieldsϜs= 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 intoaccount the fact that SDE theory with inverse time is identical to the one with the standardtime, we are going briefly to state the properties for backward Ito integrals. Let us introducetwo two-parameters σ- fields→Ϝ st,←Ϝ tTare mutually independent of each other, right andleft hand sides continuous respectively and→Ϝ st 1⊂→Ϝ st 2⊂ Ϝ ;←Ϝ t 2T⊂←Ϝ t 1T⊂ Ϝ ; s ≤ t 1≤ t 2≤ TThe upper arrow points on time direction for Ϝ. PutϜ t =→Ϝ 0t∪←Ϝ 0tand Ϝt=→Ϝ tT∪←Ϝ tT.Let w  t  , Ϝ tT, t ∈  0 , T  be a Wiener process with the values in E d, g  t be a random Ϝt− measurable d × n matrix-function . Define backward stochastic Itointegral←I =T0∫ g  t  d←w  t We will show that integral←I possesses standard Ito integral properties. First supposethat g  t  is a step-functiong  t  =n − 1i = 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 definitionCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  6. 6. ←I =n − 1i = 0∑ g  t i + 1  ←w  t i  −←w  t i + 1  1) Ift ≤ Tsup E | g  t  | < ∞ , then E←I = 0. In factE←I =n − 1i = 0∑ E E  g  t i + 1 ←w  t i −←w  t i + 1  / Ϝt i + 1 ==n − 1i = 0∑ E g  t i + 1 E  ←w  t i −←w  t i + 1  / Ϝt i + 1So as an increment←w  t i −←w  t i + 1 is←Ϝ t it i + 1− measurable and independenton Ϝt i + 1then conditional expectation coincide with expectation and hence equal to 0.2) If ∫ 0TE | g  t  |2d t < ∞ , then E |←I |2≤T0∫ E | g  t  |2d t.Really, for i < jE  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 ∗/ Ϝt i + 1g∗ t i + 1 g  t j + 1 ←w  t j −←w  t j + 1  = 0Summing over all i , j note thatE |←I |2=n − 1i = 0∑ E tr g∗ t i + 1 g  t i + 1  t i + 1− t i == ∫ 0TE | g  t  |2d tCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  7. 7. 3) For any positive numbers N and CP  |T0∫ g  t  d←w  t  | > C  ≤ NC2+ P T0∫ | g  t  |2d t ≥ N Set g N t  = g  t  χ t0∫ | g  s  |2d s ≤ N  . ThenP t ≤ Tsup | g N  t  − g  t  | > 0  = P T0∫ | g  t  |2d t ≥ N  .Hence,P  |T0∫ g  t  d←w  t  | > C  ≤ P  |T0∫ g N t  d←w  t  | > C  ++ P  |T0∫  g  t  − g N t   d←w  t  | > 0  ≤ C2E |T0∫ g N t  d←w  t  |2++ P T0∫ | g  t  |2d t ≥ N  ≤ NC2+ P T0∫ | g  t  |2d t ≥ N Following standard ideas it is easy to generalize these properties for any Ϝt−measurable random functions such that ∫ 0T| g  t  |2d t < ∞ with probability 1.Taking into account that SDE theory with the inverse time is identical to that one with thestandard 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 followingestimates are valids , tsup E | ξ  s ; t, x  |q≤ L 1  L 2 + | x |q ,Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  8. 8. s , tsup E | ξ  s ; t, x  − ξ  s ; t, y  |q≤ L 3| x − y |q, #s , tsup 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 thepaths of the solutions of the system (1.2) we consider equationu  t, x  = ϕ  ξ  s ; t, x   +ts∫ f  r, ξ  r ; t, x , u  r, ξ  r ; t, x    d r −−ts∫ 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 inversestochastic 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 datafor Euler coordinates of the particle ξ  ∗ ; t, x  are the Lagrange variables of thefunctions u  t, x  . Note that the process ξ  s ; t, x  in equality (1.4) plays the samerole as characteristics in the deterministic hyperbolic systems of the first order. As a solution ofequation (1.4), we understand the separable random function u  t, x  measurable in itsvariables and for any x adapted to a filtrationϜ 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 thesystem ( 1. 2 ), ( 1, 4 ) satisfy the conditions (1.1) and the functionsϕ  x , f  t, x, u , g  t, x  defined for  t, x, u  ∈  0, T  × Em× Enandtaking values in En, En, Ed× Enrespectively, satisfy the conditions| ϕ  x  | + | f  t, x, u  | + | g  t, x  | ≤ k  1 + | u | Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  9. 9. u ≠ vtx ≠ ysup | 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, xsup E | u  t, x  |212+x ≠ yt ≤ Tsup E| u  t, x  − u  t, y  |2| x − y |γ12++t ≠ s , xsup E| u  t , x  − u  s , x  |2| t − s |γ2 1 + | x | 12< ∞ .Proof. We denote by M a set of bounded infinitely differentiable in t , x withprobability 1, random functions u  t, x, ω  measurable with respect to Ϝ t , t ≥ 0 foreach x. Then there exists a separable modification for each function from M that may takeinfinite values and an arbitrary set in  0, T  can serves as its separability set. Identifying theclass of stochastically equivalent functions with the modification, we fix one separability set forall functions from M . Identifying the class of stochastically equivalent functions with theseparable modification, we fix one and the same separability set for all functions from M .Denote B 2 , γthe Banach space obtained as completion of the set M in norm || u || 2 , γ. It isobvious that Ϝ t-measurable random functions continuous in t , x serve as elements ofspace B 2 , γ . Let ( Z u   t, x, ω  be the right-hand side of equality (1.4) andv  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 fromspace B 2 , γinto itself. Then the series∞i = 1∑t , xsup E | Zi + 1v  t, x  − Ziv  t, x  |2converges and, consequently, there exists a limit of the functions  Ziv  t, x, ω  asi → ∞ for fixed  t , x  with probability 1. Denote this limit by u  t, x, ω  and it iseasy to check thatCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  10. 10.  Z u   t , x , ω  = u  t , x , ωand || u || 2 , γ< ∞ . Thus the function u  t, x, ω  belongs to B 2 , γthere existsmodification for which equation  Z u   t, x, ω  = u  t, x, ω  holds for all t, x atonce with probability 1. The uniqueness of the solution of equation (1.4) follows from the factthat a certain power of the operator Z is contraction operator in Banach space B 2whichcompletes set M in the norm|| u || 2 =t , xsup E | u  t, x  |212In 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 oftheorem 1.1, and the functions ϕi , ϕ, f i , f , g i , g satisfy the conditions of theorem 1.2with γ = 1 andi → ∞limT0∫xsup  | b i  t, x  − b  t, x  | + | C i  t, x  − C  t, x  | ++ | g i t, x  − g  t, x  | +usup | f i t, x, u  − f  t, x, u  | ++ | ϕ i x  − ϕ  x  |  d t = 0Then, for any q ≥ 2i → ∞limt , xsup 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 coefficientsb 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 Cq + γa set of non-random functions, whose q -order derivatives satisfyHolder condition with γ ∈  0, 1 .Theorem 1.4. Assume that coefficients of the system ( 1. 2 ) and ( 1. 4 ) belong to thespace C1 + γ∩ C2 + γin variables x and u. ThenCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  11. 11. x ≠ y, tsup E |∇ 2u  t , x  − ∇2u  t , y | x − y |γ |2< ∞.Consider now the solution of the problem ( 1. 4 ) as a functional of the parameters. Denoteby u  t , x ; s , ϕ  , t ≥ s the solution of the problem (1.4) for whichu  s , x ; s , ϕ  = ϕ  x  .Lemma 1.1. Suppose the conditions of theorem 2 hold and u  t, x ; s, ϕ, ω  is thesolution of equation ( 1. 4 ). Then for arbitrary r ∈  s , t  for alls ∈  0 , t  , t ∈  0 , ∞  , x ∈ Enwith probability 1u  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 itsvariables t , x , s and the random functions u  t , x ; r , ψ , ω  andu  r , x ; s , ϕ , ω  are independent for any r ∈  s , t  and any for nonrandomfunctions ϕ , ψ , since the function u  t , x ; r , ψ , ω  , for nonrandom ψ, dependsonly on the increments of the Wiener process on the interval  r , t  , and functionu  r , y ; s , ϕ , ω  depends only the increments of this process over the interval s ,r  . It immediately follows from equality (1.4) thatu  t, x ; s, ϕ, ω  =  ϕ  ξ  s ; r , y   ++sr∫ f  l, ξ  l ; r, y , u  l, ξ  l ; r, y  ; s, ϕ   d l−rs∫ g  l, ξ  l ; r, y   d←w  l   y = ξ  r ; t, x ++tr∫ f  l, ξ  l ; t, x , u  l, ξ  l ; t, x  ; s, ϕ, ω   dl −−tr∫ g  l , ξ  l ; t, x   d←w  l  = u  r, ξ  r ; t, x ; s, ϕ, ω  ++tr∫ f  l, ξ  l ; t, x  u  l, ξ  l ; t, x  ; s, ϕ    d l −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  12. 12. −tr∫ 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 iseasy to observe that the function satisfies the equationu  t , x ; r , u  r , ∗ ; s , ϕ   = u  r, ξ  r ; t , x  ; s , ϕ  ++tr∫ f  l, ξ  l ; t, x , u  l, ξ  l ; t, x ; r, ϕ   dl −−tr∫ g  l, ξ  l ; t, x   d←w  l By subtracting this equality from the preceding one, we findxsup | Δ u  t , x  | < Kts∫xsup | Δ u  r , x  | d rwhere Δ u  t , x  = u  t, x ; s, ϕ  − u  t, x ; r, u  r, ∗ ; s, ϕ   and K is aLipschitz constant of a function f with respect to u. Using Gronwall’s lemma we complete theproof.Theorem 1.5. Assume that the coefficients of equation ( 1 .2 ) are continuous in t andfor each t , second derivatives of the coefficients of the system ( 1. 2 ), ( 1. 4 ) satisfyHolder’s condition with γ ∈  0 , 1  in variable x and u are uniformly bounded. Thenthe solution of the problem (1. 4) is also a solution of the following Cauchy problemu  t , x  = ϕ  x  +t0∫ mi = 1∑ ∂ u  s, x ∂ x ib i s, x  ++ 12mi, j = 1∑dk = 1∑ ∂2u  s, x ∂ x i ∂ x jci k  s, x  cj k  s, x  + f  s, x, u    d s +#Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  13. 13. +mi = 1∑dk = 1∑t0∫ ci k  s , x ∂ u  s , x ∂ x id w k s  +t0∫ 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 thefunction 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. Let0 = t 0 < t 1 < t 2 < .... < t N = t be a partition of the interval  0 , t  andλ =imax  t i + 1 − t i . ThenG  t , y  − G  0 , ξ  0 ; t , y   ==i = 0N − 1∑ G  t i + 1 , ξ  t i + 1 ; t , y   − G  t i , ξ  t i ; t , y   ==i = 0N − 1∑∂ G  t i , ξ  t i ; t , y  ∂ tΔ t i +  ξ  t i + 1 ; t , y  − ξ  t i ; t , y   ⋅∇ G  t i , ξ  t i ; t , y   +12tr  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 wegetG  t, y  − G  0, ξ  0 ; t , y   =t0∫ ∂ G  s, ξ  s ∂ t−− b  s, ξ  s ; t, y  ∇ G  s, ξ  s ; t, y  ++ 12Tr C∗ s, ξ  s ; t, y  Δ G  s, ξ  s ; t, y  C  s, ξ  s ; t, y   d sCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  14. 14. −t0∫ 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. Theny = G  t, y  −t0∫ ∂ G  s, ξ  s; t, y  ∂ s−− b  s, ξ  s ; t, y  ∇ G  s, ξ  s ; t, y  ++ 12Tr C∗ s, ξ  s ; t, y  Δ G  s, ξ  s ; t, y  C  s, ξ  s ; t, y   d s ++t0∫ 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  + #+ 12Tr 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 spanG  t , y  = yThen G  0, ξ  0 ; t , y   = y and therefore G  t , y, ω  transforms Eulercoordinates into Lagrange’s. The question concerning the construction of the first integrals ofthe solution of equation ( 1. 2 ) can serve as another application of Theorem 1.5. Recall that thefunction V  t , x  is a first integral of the solution of the equation (1.2), if for alls ∈  0 , t  , V  s ,  s , ξ  s ; t , y   = const. Analogously to what has beensaid above, it is easy to observe that the first integrals satisfy equation (1.7) with initialcondition. 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 partitionof the interval  s , t . Thenu  s, ξ  s ; t, y   − u  s, y  =Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  15. 15. =i = 1N∑ 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 functionu  t, x  is twice continuously differentiable with probability 1, and using Taylor’s formulawe haveNi = 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  ++ 12Tr  ξ  t i − 1 − ξ  t i ∗u 2  s, ξ  t i   ξ  t i − 1 − ξ  t i  ++ 1210∫  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 3Here, u 1  t , x = ∇ u  t , x , u 2 t , x = ∇ 2u  t , x, andξ l  t i  = l ξ  t i − 1  +  1 − l  ξ  t i  . We shall now show that I 3 tends to0 in probability for λ = max  t i− t i − 1 → 0 . By Chebyshev’s inequality forarbitrary ε > 0P  | I 3| > ε  ≤  2 ε − 1i = 1N∑ E10∫  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 ×Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  16. 16. ×Ni = 1∑ E10∫  ξ  t i − 1 − ξ  t i 2 + γ≤ R ε− 1 t − s  λγ2,where R is some constant, dependent on T and on coefficients of the equation. We nowestimate the terms I 1and I 2. We haveI = ∑ iu 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  whereo 1 λ  =Ni = 1∑ u 1  s , ξ  t i  t it i − 1∫  b  r, ξ  r   −− b  t i , ξ  t i    d r +t it i − 1∫  C  r, ξ  r   − C  t i , ξ  t i    d←w  r  | ThenE | o 1 λ  |2≤ 2 λ || u 1 ||2| s 1 − s 2 | ≤ λsup 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 it 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 variablesCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  17. 17. t jt j − 1∫  C  r , ξ  r   − C  t i, ξ  t i   d←w  r  and ξ  t jare Ϝt i− measurable, thenE | u 1  s, ξ  t i  t it i − 1∫  C  r, ξ  r   − C  t i , ξ  t i    d←w  r  ×× u 1  s, ξ  t j t jt j − 1∫  C  r, ξ  r   − C  t j, ξ  t j   d←w  r  | == E t it 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 jt j − 1∫  C  r, ξ  r   − C  t i, ξ  t i   d←w  r  / Ϝt iz j = ξ  t jz i = ξ  t i The expression under the sign for the conditional mathematical expectation, doesn’tdepend on σ − algebra Ϝt i. Hence the conditional mathematical expectation considerswith the unconditional one. ThenE  u 1  s, z j ∗u 1  s, z it it 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 it i − 1∫  C  r, ξ  r   −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  18. 18. − C  t i , ξ  t i    d←w  r  / Ϝ s = E  u 1  s, z j  ∗u 1  s, z i  ×× Et i − 1t i∫  C  r, ξ  r   − C  t i, ξ  t i   d←w  r  = 0Thus the second term of the estimate of the variable doesn’t exceedEi∑ | u 1  s, ξ  t i |2|t jt j − 1∫  C  r, ξ  r   − C  t i, ξ  t i   d←w  r  |2≤≤ || u 1 ||2sup 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 . HenceE | o 1 λ  |2≤ 2 λ || u 1 ||2 t − s | s 1 − s 2 | < λsup 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 theoremλ → 0lim E | o 1  λ  |2= 0 . FurtherI 2 = 12i∑ 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  | ≤ 12i∑ | u 2  s, ξ  t i  |2t it i − 1∫ | C  r, ξ  r   C∗ r, ξ  r   −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  19. 19. − C  t i, ξ  t i  C∗ t i, ξ  t i  | d r + |t it i − 1∫ b  r , ξ  r   d r |2Analogously to the estimate I 1 it is easy to observe thatλ → 0lim E | o 2  λ  |2= 0.ThusI 1 + I 2 ==Ni = 1∑ u 1  s, x  b  s, x  ti − ti − 1 + C  s, x   w  t i − 1 − w  t i  ++ 12Ni = 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 whereo 3 t − s  =Ni = 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   ++ 12Ni = 1∑  Tr C∗ s, ξ  t i  u 2  s, ξ  t i  C  s, ξ  t i  −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  20. 20. − 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  arecontained. ThenP  |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 therandom functions ξ  t i = ξ  t i; t , x  , w  t i − 1 − w  t iare Ϝs− measurable for any i and u  s , x  is independent of the σ −algebra, thesecond term doesn’t exceedε− 2xsup E | u 1  s , x  |2r ∈  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 ThenP  | ∑ 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 ||2K2××r ∈  s , t sup | E | C  r , ξ  r   − x |2 t − s  ≤ R 1 t − s 2Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  21. 21. where the positive constant R 1is easily defined by formula (1.3). Estimate the remainedterms of the o 3  t − s  . We haveP i∑ |  u 1  s, ξ  t i  b  t i, ξ  t i  − u 1  s, x  b  s, x  ++ 12∑ 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  | ++ 12Tr 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− 1R 2  t − s 12+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 observethatλ → 0lim I 1 + I 2 + I 3 == u 1  s, x   b  s, x   t − s  + C  s, x   w  s  − w  t   + 12∑ i Tr C∗ s, x   u 2  s, x   C  s, x   t − s  + o  t − s whereCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  22. 22. P.t → slim  t − s − 1o  t − s  = 0 .Let 0 = s 0< s 1< ... s N= t . Applying the estimates obtained above to eachinterval  s i , s i + 1  we findN − 1i = 0∑ u  si + 1, x  − u  si, x  =N − 1i = 0∑ u  si, ξ  si ; si + 1, x   − u  si, x  ++s i + 1s i∫ f  s, ξ  s ; s i + 1 , x  , u  s, ξ  s ; s i + 1, x    d s −−s i + 1s i∫ g  s , ξ  s ; s i + 1, x  d←w  s  =N − 1i = 0∑  u 1  s i, x  b  s i, x  ++ 12Tr C∗ s i, x   u 2  s i, x   C  s i, x    s i + 1− s i ++N − 1i = 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 → 0lim | o  λ  | = 0 . Passing to the limitin 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 Wienerprocesses taking values in Ed i, i = 1 , 2 , ... l . Denote by ϜΔ i the σ −algebraCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  23. 23. generated by the increments of the w i  t  , t ∈ Δ and Ϝ Δ =Ii = 1∪ ϜΔ i . Considerthe systemξ  s ; t, x  = x +ts∫ b  r, ξ  r ; t, x  d r −i = 1I∑ts∫ Ci  r, ξ  r ; t, x  d←wi  r u  t, x  = ϕ  ξ  s ; t, x   + #+ts∫ f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r −ts∫ 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 nonrandommatrices with elements c k j i  t , x  ; k = 1, 2, ... d i, j = 1, 2 , .. m . It is easyto 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 oftheorem 1.5. Then the function v  t , x , ω  is a ”classical” solution of the Cauchyproblemv  t , x  = ϕ  x  +t0∫   b  s , x  , ∇  v  s , x  ++ 12 C  s, x  C∗ s, x  ∇ , ∇  v  s, x  + f  s, x, v    d s + #+t0∫  C 1  s, x  d w 1  s  , ∇  v  s, x  +t0∫ g  s, x  d w 1  s where ,C  s , x  C∗ s , x  =Ii = 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. Thethe proof of the theorem 1.6 follows that of theorem 1.5 in the main. Therefore we underlineonly the moments, which differ. We haveCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  24. 24. u  t , x  − ϕ  x  =N − 1j = 0∑ u  s j + 1 , x  − u  s j , x  ==N − 1j = 0∑Ii = 1∑  C i  s j , x  Δ w i  s j  ,∇  u  s j , x  +N − 1j = 0∑   b  s j, x  , ∇  u  s j, x  ++ 12 C∗ s , x  C  s , x  ∇ , ∇   u  s , x   s j + 1− s j ++N − 1j = 0∑ f  s j, x, v  s j, x    s j + 1− s j ++N − 1j = 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 theconditional expectation with respect to σ − algebra Ϝ  0 , t  1and replacing the operator ∇and the conditional expectation, we haveE   C i s j, x  Δ w i s j , ∇  u  s j, x  / Ϝ  0 , t  1 =Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  25. 25. = 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 thisequality. SetϜ 1= Ϝ  s j , t  1, Ϝ 2= Ϝ  0 , s J  1, Ϝ 3=Ii = 2∪ Ϝ  s j , s j + 1  i ∪ Ϝ  0 , s j It is easy to show that σ − algebras Ϝ 1and Ϝ 3are conditionally independent withrespect to σ −algebra Ϝ 2. Let Y kare arbitrary nonnegative random variables , measurablewith respect to σ −algebra Ϝ k , k = 1, 3 . Taking into account that Ϝ 2 ⊂ Ϝ 3 andthat σ −algebra Ϝ 1does not depend on σ −algebras Ϝ 2and Ϝ 3, we haveE  Y 1Y 3/ Ϝ 2 = E  Y 3E  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 ofσ −algebras Ϝ 1and Ϝ 3. Hence, (see[32]), with probability 1E  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 haveE  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  = 0In this case i = 1 we assume thatCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  26. 26. 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) wehaveE  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 obtainE  u  s j, x  / Ϝ  0 , t  1  = E  u  s j, x  / Ϝ  0 , s j  1  = v  s j, x ThereforeE 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 − 1j = 0∑   b  s j , x  , ∇  + 12 C∗ s j , x  C  s j , x  ∇ , ∇   ×× u  s j , x  Δ s j / Ϝ  0 , t  1 =N − 1j = 0∑   b  s j , x  , ∇  ++ 12 C∗ s j , x  C  s j , x  ∇ , ∇   v  s j , x  Δ s jPassing 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 equationof particle trajectories. Moreover the uniqueness of the solution of the Cauchy problem (1.6) isCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  27. 27. not supposed, but it follows from the uniqueness of the solution of the correspondingtrajectory- problems.Let B 22 + γdenote the Banach space of Ϝ t − measurable separable randomfunctions V  t , x , ω  with the norm|| V || 2 , 2 + γ= max || ∇iV || 2 , γ.Theorem 1.7. Assume that V  t , x , ω  is an ’classical ’ solution of equation(1.6) belonging to the space B 22 + γwith 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 22 + γwe haveQ  s, ξ  s ; t, x   − Q  t, x  ==Ni = 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:Ni = 1∑ Q  s , ξ  t i − 1   − Q  s , ξ  t i   ==Ni = 1∑  ξ  t i − 1 − ξ  t i  ∇ Q  s , ξ  t i  ++ 12Tr  ξ  t i − 1 − ξ  t i ∗Δ Q  s , ξ  t i   ξ  t i − 1 − ξ  t i  ++10∫  1 − θ  12Tr  ξ  t i − 1  − ξ  t i  ∗ Δ Q  s, ξ θ  t i   −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  28. 28. − 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 functionQ  t , x  ∈ B 22 + γ, analogously to theorem 5 we obtainP  | I 3 | > ε  ≤ R ε− 1 t − s  λγ2,where R is a constant independent of x , ε , λ . For the expressions I 1 and I 2 usingthe methods applied above, we findI 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 = 12i∑ 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 whereλ → 0lim E | o 1 λ  | + | o 2 λ  | = 0 .Hence, we can note thatI 1+ I 2+ I 3++tr∫ f  l, ξ  l ; t, x , Q  l, ξ  l ; t, x    d l −tr∫ 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  +Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  29. 29. + 12i∑ 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 → slim | o  t − s  | = 0 . Consequently,Q  s, x  +  b  s, x    t − s  + C  s, x    w  t  − w  s    ∇ Q  s, x  ++  12Tr 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 −−ts∫ 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 22 + γ, we choosea given ’classical’ solution V  t , x , ω  of the Cauchy problem (1.6) with initial conditionV  0 , x , ω  = ϕ  x  . Let 0 = t 0< t 1< t 2< .... < t N= t be apartition of the interval  0 , t  , λ = max Δ t i. Then applying the equality (1.12) to t 0 , t 1  , we findV  t 1, x  = ϕ  ξ  t 0; t 1, x   ++t 1t 0∫ f  l, ξ  l ; t 1 , x , V  l, ξ  l ; t 1 , x    d l −−t 1t 0∫ g  l, ξ  l ; t 1, x   d←w  r  + o  λ  + γ  t 1− t 0, x  ,whereCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  30. 30. γ  t 1− t 0, x  =t 1t 0∫   b  s , x   ∇ V  s , x  ++ 12Tr C∗ s, x    Δ V  s, x   C  s, x   + f  s, x, V  s, x    −−  b  t0 , x   ∇V  t0 , x  + 12Tr C∗ t0 , x    ΔV  t0, x   C  t0 , x   ++ f  t 0 , x, V  t 0 , x     d s +t 1t 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 holdsV  t k − 1, x  = ϕ  ξ  t 0; t k − 1, x   ++t k − 1t 0∫ f  l, ξ  l ; t k − 1 , x  , V  l, ξ  l ; t k − 1 , x    d l −−t k − 1t 0∫ g  l, ξ  l ; t k − 1, x   d←w  l  ++  k − 1  o  λ  +k − 1i = 1∑ γ  t i− t i − 1, x We show that the similar representation also holds for the moment t k, and then estimatethe sum of variables γ. By (1.12) we haveV  t k , x  = V  t k − 1 , y  +Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  31. 31. +t kt k − 1∫ f  l , ξ  l ; t k , x  , V  l , ξ  l ; t k , x    d l −−t kt 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 − 1t 0∫ f  l, ξ  l ; t k − 1, y , V  l, ξ  l ; t k − 1, y    d l −−t 1t 0∫ g  l, ξ  l ; t k − 1, x   d←w  l  ++k − 1i = 1∑ γ  t i− t i − 1, y  y = ξ  tk − 1; tk, x + #+t k − 1t 0∫ f  l , ξ  l ; t k, x  , V  l , ξ  l ; t k, x    d l −−t kt k − 1∫ g  l, ξ  l ; t k, x   d←w  l  + k o  λ  + γ  t k− t k − 1, x Since,k − 1i = 1∑ γ  t i− t i − 1, y  |y = ξ  tk − 1; tk, x + γ  t k− t k − 1, x  = 0then, it is sufficient to show, thatCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  32. 32. λ → 0limNi = 1∑ γ  t i − t i − 1 , x  = 0Both the ordinary integrals and the stochastic Ito integrals are contained in the sum underconsideration. Ordinary integrals are estimated by standard methods. So, as an example, weshall estimate one of the sums of stochastic integralsE |Ni = 1∑t iti − 1∫  C  r, x  ∇V  r, x  − C  ti − 1, x  ∇V  ti − 1, x   d w  r  |2≤≤ t | t  − t  | < λmax | C  t, x  − C  t, x  |2maxxE | ∇ V  t , x  |2++tmax | C  t , x  |2| t  − t  | < λmax E | V  t, x  − V  t |2From the condition of theorem 1.7 it follows that the expression in brackets tends to 0 asλ → 0 . Passing to the limit in equality ( 1. 12 ) as and supposing k = N we can verifythat the function v  t, x  = v  t , x ; 0, ϕ  being a solution of equation (1.6) is also asolution of equation (1.4). Thus by conditions of theorem 1.7 each ”classical” solution of theCauchy problem (1.6) is also a solution of the equation (1.4) and vice versa. By virtue oftheorem 1.2 the solution of the problem (1.4) is unique and consequently the solution of theproblem (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 tothe space B 22 + γmeasurable with the flow of σ −algebras Ϝ  0 , t  1 independent of theWiener processes w j t  , j = 2, 3, ..., l contained in the system (1.8) and satisfyingthe equality (1.9) for all t and x at once with probability is called the ”classical” solution ofthe problem (1.9). It is rather difficult to apply the methods analogous to those in the proof ofTheorem 1.7 directly. Therefore another technique of proving the uniqueness of the solution ofequation (1.9) will be used.Theorem 1.8. Let the conditions of the theorem 1.6 hold and assume that the functionsb  t , x  , C i t , x  i = 1, 2, ..l for any t ∈  0 , T  are equal 0 outsidesome 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  . ThenCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  33. 33. h q t , x  =t0∫ mi = 1∑ b i s , x ∂ h q s , x ∂ x i++mi , j = 1∑Ik = 1∑d kν = 1∑ c i , ν k  s, x  c j , ν k  s, x ∂2h  s, x ∂ x i ∂ x j+ f q s, x, v 1 s, x   −− f q s, x, v2  s, x    d s +mi = 1∑d 1ν = 1∑t0∫ c i , ν 1  s, x ∂ hq s, x ∂ xid w ν 1 s q = 1, 2, ..., m . Treating the variable x in the preceding equality as a parameter, weapply Ito’s formula to the function h q2 s , x . Thenh q2 s, x  =t0∫ 2 h q s, x  mi = 1∑ b i s, x ∂ h q  s, x ∂ x i++mi , j = 1∑Ik = 1∑d kν = 1∑ c i , ν k  s, x  c j , ν k  s, x ∂2h  s, x ∂ x i ∂ x j++  f q s, x, v 1 s , x   − f q s, x, v 2 s, x     d s ++t0∫mi , j = 1∑d 1ν = 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 ++mi = 1∑d 1ν = 1∑t0∫ c i , ν 1  s , x ∂ h q2 s , x ∂ x id w ν 1  s We shall now take the mathematical expectation from the equality , and integrate theequality 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, thatintegrand function be absolutely integrable with respect to measure dx × dt with theCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  34. 34. probability 1. For the order of integration to be reversed in the stochastic integral it is sufficient[39] , that with probability1T0∫E m∫ | C 1  t , x  ∇ h2 t , x  d x d t < ∞ #Since the function h  ∗  belongs to the space B 22 + γ, and the diffusion coefficientC 1  ∗  is equal to 0 outside some compact , then condition (1.14) holds. Integratingby parts in the inner integral in the first two terms on the right-hands side, we obtainE m∫ E h q2 s , x  d x =t0∫ d sE m∫  mi = 1∑∂ b i  s , x ∂ x i++mi , j = 1∑Ik = 1∑d kν = 1∑∂2c i , ν k  s , x  c j , ν k  s , x ∂ x i∂ x j E h q2 s, x  + #+mi , j = 1∑Ik = 1∑d kν = 1∑ c i , ν k  s, x  c  s, x ∂ h q2 s, x ∂ x i∂ h q2 s, x ∂ x j++mi , j = 1∑d 1ν = 1∑ c i , ν 1  s, x  c j , ν 1  s, x ∂ h q2 s, x ∂ x i∂ h q s, x ∂ x ij++ k f E h q2 s , x   d x .Here, k f= sup t , x , v| ∇ vf  t , x , v  | . Summing over all q we establishestimates for the function h  t , x  = v 1 t , x  − v 2 t , x  :E m∫ E h q2 t , x  d x ≤t0∫ d sE m∫  div b  s , x  ++ 12 ∇ , ∇ C  s, x  C∗ s, x   + k fn  E | h  s , x  | d sBy Granules’s lemma,Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  35. 35. E m∫ E | v 1 t , x  − v 2 t , x  |2d x = 0.Since subintegral function is continuous, the solutions coincide for all t , x at once withprobability 1, which is what had to be proved.Now we are going to study other types of stochastic dynamic system. Note, that one of thepossible applications of the theory introduced above is design of the methods of resolving theproblem Cauchy for nonlinear stochastic parabolic equation. Turn back to the Theorem 1.5 wewould 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 isgenerated 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 alongthe trajectories of the characteristics ξ  ∗ . Now we represent a scheme where theseparameters will changed along the distribution of the characteristics. This is the directextension of Kolmogorov’s equations and can be considered as a generalization of probabilisticmethods of studying deterministic nonlinear parabolic equations [ 4 ]. Given the conditions ofthe theorem (1.5) we introduce the problemv  t, x  = E ϕ  ξ  0 ; t, x   ++t0∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r d r ++t0∫ 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 sameas before. Some insignificant modifications involved based on the new term in (1.16). To usecontraction mapping principle we need to substitute random functionsv  ,  , ξ  ∗ ; t, x  into conditional expectation.We will show that after substitution correspondent stochastic integral remain measurablefunction. First, let B be a set of Borel’s bounded random functions depending on t , x andv  t , x, ω  be an arbitrary function belonging to B . FunctionG  s, t, x, ω  = g  r , ξ  r ; t, x  , v  r , ξ  r ; t, x   is measurable overs, t, x, ω as a composition of measurable functions andCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  36. 36. s , t , xsup E | G  s, t, x, ω  | < ∞Now , we can exhibit that there exist measurable modification of the conditionalexpectation E  G  s, t, x, ω  | Ϝ s  .Lemma 1.2. Let G  s, t, x, ω  be a measurable random field ,andsup s , t , x E | G  s, t, x, ω  | < ∞. Then there exist s , t , x ω − measurablemodification 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 as, t, x, ω − measurable set of a type A = A 1× A 2× A 3× A 4, whereA 1 , A 2 , A 3 are Borel sets from  0 , t  ,  0 ,T  , Emrespectively, andA 4 ∈ Ϝ . ThenE 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 ofright hand side of the equality. Set U is an algebra and monotone class and contains indicatorsof the measurable sets of chosen type. Hence, U contains indicators of alls, t, x, ω − measurable sets. Evidently, that U is linear and closed with respect tomonotone limit transition. So that U contains all measurable nonnegative functions. Since, anymeasurable function can be represented as a difference of two nonnegative functions one gets aproof of the lemma.Turn back to the problem (1.16). The solution of the problem exist, unique and satisfiesrelationshipv  t, x  = E v  ξ  s ; t, x   / Ϝ s ++ts∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r d r + #+ts∫ 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 othersterms can be establish without any difficulties. Since, the random variable ξ  s ; t, x  isindependent on Ϝ sthenCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  37. 37. 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 noteE ϕ ξ 0 ; t, x   +0s∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r d r ++s0∫ 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   ++s0∫ E  f  r, ξ  r ; s, y , v  r, ξ  r ; s, y    / Ϝr  d r ++s0∫ 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 thesystem (1.16). Granting these we are able to formulateTheorem 1.9. Under conditions of the theorem 1.5 the solution of the equation (1.16) isa solution of the Cauchy problemv  t , x  = ϕ  x  +t0∫ mi = 1∑ ∂ v  s , x ∂ x ib i s , x  ++ 12i , j = 1m∑ ∂2v  s, x ∂ x i∂ x jk = 1d∑ ci k  s, x  cj k  s, x  + f  s, x, v  s, x    d s +Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  38. 38. +t0∫ 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 fromequalityv  t, x  − v  s, x  = E  v  ξ  s ; t, x   − v  s, x  / Ϝ s  ++ts∫ E  f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    / Ϝ r  d r ++ts∫ 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 systemsIn this section we will consider quasilinear stochastic parabolic systems. Two significantdifferences between a quasilinear and a semilinear systems should be noted. The firstdifference is that the solution of quasilinear systems is local in time and the solution ofsemilinear systems exists on an arbitrary finite interval of time. The second one is that thesolution of quasilinear systems with smooth coefficients is continuous in x and therefore thesolution must be interpreted as generalized. For semilinear systems differentiability of thesolution in x corresponds to the differentiability of the coefficients in x . Some types ofstochastic partial differential equations were studied in [12,36,41].We introduce the needed notation. Let the w j  t  be mutually independent Weinerprocesses with values in d j-dimensional Euclidean space Ed j, d j≥ 1 , j = 1, 2 ,and let w  t  = w 1 t  , w 2 t  . For an arbitrary intervalΔ ⊂  0 , T  letϜ Δ j = σ  w j  t 1  − w j  t 2  ; t 1, t 2 ∈ Δ  and Ϝ Δ =2j = 1∪ Ϝ Δ j .Consider the quasilinear systemCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  39. 39. ξ  s ; t, x  = x +ts∫ b  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r −−ts∫ 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   +t0∫ f  r, ξ  r ; t, x , v  r, ξ  r ; t, x    d r ++t0∫ 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 arrowhere over the Wiener process w  t  means that the stochastic integral is to be interpreted asIto’s inverse integral . A solution of the system (2.1) on the interval  0 , T  is understoodto be a pair of random functions ξ  s ; t, x  , u  t , x  that are defined fors ≤ t < T and satisfy the estimate (2.2), are measurable in all their arguments, aremeasurable 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 samesymbol | ∗ | will be used for the norm of a vector in Euclidean space and for the norm of thetrace of a matrix.THEOREM 2.1 Suppose that the nonrandom functions ϕ  x  , f  t , x , u  withvalues in Enand Borel measurable in all their arguments, the function b  t , x , u  withvalues in Em, the matrix-function g  t , x  of dimension d 1× n and matrix- functionc  t , x , u  =  c 1 t , x , u  , c 2 t , x , u   , where c j t , x , u  is ofdimension d j × m, are uniformly bounded by a positive constant K and satisfy a Lipschitzcondition in x and u with a positive constant L. Then there exist a time interval  0 , T  inwhich the system (2.1) has unique solution u  t , x  , ξ  s ; t , x  in the class of thefunctionsx , t 1 ≠ t 2sup E |ξ  s ; t 1 , x  − ξ  s ; t 2 , x | t 1− t 2|0 . 5| p++x , s 1 ≠ s 2sup E |ξ  s 1; t , x  − ξ  s 2; t , x | s 1− s 2|0 . 5|p+Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  40. 40. +xsup E | u  t, x  |p+x ≠ ysup E |u  t , x  − u  t , y x − y|p+ #s , xsup E |ξ  s ; t, x 1 + | x ||p+s , x ≠ ysup E |ξ  s ; t, x  − ξ  s ; t, y x − y|p++t ≠ s , xsup 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 uniquein the class of function that satisfy ( 2.2 ).Proof. Before proceeding to prove the theorem , we point out that the conditionalexpectation in the integrand of stochastic integral has a predictable modification. It can beproved the same way as Lemma 2 . Rewriting the system (2.1) in operator formξ  s ; t , x  = L  s ; t , x   ξ , u u  t , x  = L  t , x   ξ , u we form successive approximationsξ 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 theseilinear 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 letL ndenote the constant on right -hand of (2.2). Show that there is a constant L n + 1thatmajorized the left-hand side of (2.2), whereξ  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 notCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  41. 41. terminate. We now indicate the time-interval in which the sequence of numbers L nisuniformly bounded. Let Δ be an arbitrary interval of time p ≥ 2 , and R  t , ξ , u  bethe expression of the right hand side of ( 2.2 ). By definition we putR n=t ∈ Δsup R  t , ξ n , u n We will establish the constant R n + 1 for whichR n + 1 =t ∈ Δsup R  t , ξ n + 1 , u n + 1 We have 1 + | x |p− 1E | ξ n  s ; t , x  |p≤ 3p − 1 1 + Kpα p t  E | u n + 1  t , x  |p≤ 3p − 1Kp 1 + α p  t  where α p  t  = p2 p − 1  tp2− 1+ tp − 1. Denoteu t n p =x ≠ ysup E |u  t , x  − u  t , y x − y|p,ξ s , t n p=x ≠ ysup E |ξ  s ; t , x  − ξ  s ; t , y x − y|pThenu t n + 1 p ≤ 3p − 1Lp ξ 0 , t n p ++ α p  t t0∫  1 + u s n p  ξ 0 , t n p d s With Gronwall’s lemma, we obtainCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  42. 42. t ≤ Tsup u t n + 1 p ≤ U n , pwhereU n , p= 3p − 1 L R np 1 + T α p T   exp 3p − 1α  T   L R npTSimilarly ,ξ s , t n p≤ V n , p,t ≠ s , xsup E |u n + 1  t, x  − u n + 1  s, x | t − s |0 . 5|p≤ 3p − 1 L R np××  1 + 2p − 1T α p T   1 + U n , p + 2p − 1T α p T   ,x , t 1 ≠ t 2sup E |ξ n + 1  s ; t 1, x  − ξ n + 1  s ; t 2, x | t 1− t 2|0 . 5|p≤≤  43p − 1 V n , p+ KpTp+ α p T  ,x , s 1 ≠ s 2sup E |ξ n + 1  s 1; t, x  − ξ n + 1  s 2 ; t, x | s 1 − s 2 |0 . 5|p≤≤ 2p − 1Kp1 + α p T whereV n , p= 3p − 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 uniformlyCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  43. 43. bounded. We haveE | u n + 1  t , x  − u n + 1  t , y  |p≤≤ β p t ssup E | ξ n  s ; t, x  − ξ n  s ; t, y  |p;ssup E | ξ n  s ; t, x  − ξ n  s ; t, y  |p≤ 3p − 1 | x − y |p++ μ p  t ssup E | ξ n − 1  s ; t, x  − ξ n − 1  s ; t, y  |pwhereβ p  t  = 3p − 1Lp 1 + t   1 + α p  t   exp 3p − 1Lpt α p  t  ,μ p  t  = t  1 + α p  t    1 + β p  t   LpThe function μ p t  is continuous and monotone increasing and μ p 0  = 0. Lett 1denote the root of equation μ p t  = 31 − p. Then, for any t < t 1ssup E | ξ n  s ; t, x  − ξ n  s ; t, y  |p≤≤3p − 11 − 3p − 1μ p t | x − y |p#E | u n  t, x  − u n  t, y  |p≤3p − 1β p t 1 − 3p − 1μ p  t | x − y |pUsing these estimates, we can prove that the successive approximations converge. LetB pt, N ptbe Banach spaces of the random functionsϜ t , Ϝt, t ∈  0 , T  with normsCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  44. 44. || u || t , pp=x , s ≤ tsup E | u  s , x  |p,||| ξ ||| t , pp=x , s ≤ tsup E |ξ  s ; t , x 1 + | x ||p,respectively. Putting for simplicity p = 2 we have|| u n + 1 − u n || t , 22≤ 3 L2  1 + t α 2  t   ++ 2α 2  t t0∫ || u n + 1 − u n || s , 22d s + 2α 2  t  ××t0∫xsup 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 sApplying the formulas (2.3) we find thatt0∫xsup 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 , 22Applying Gronwall’s lemma, we obtain|| u n + 1 − u n || t , 22≤ γ  t  ||| ξ n − ξ n − 1 ||| t , 22where γ  t  = 3 L2 1 + t α 2  t   + 6 t β 2  t 1 − 3 μ 2  t  exp 6 L2t α 2 t  .Similarly to the preceding||| ξ n − ξ n − 1 ||| t , 22≤ 2 L2α 2 t   t ||| ξ n − 1 − ξ n − 2 ||| t , 22+Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  45. 45. + 2t || u n − u n − 1 || t , 22+ 2β 2 t t0∫xsup 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 , 22where λ  t  = 2 L2α 2  t  1 + 2 γ  t   + 6 β 2  t 1 − 3 μ 2  t . We point out thatλ  t  is continuous and monotone increasing, andλ  0  = 0 . Thus, λ  t 2 = 1 for some positive t 2and therefore||| ξ n − ξ n − 1 ||| t , 22≤ λn t  ||| ξ 1 − ξ 0 ||| t , 22|| u n − u n − 1 || t , 22≤ γ  t  λn t  ||| ξ 1 − ξ 0 ||| t , 22Let T = min  t 1, t 2 . Then, for t < T , the preceding inequalities lead ton , m → ∞lim  || u n − u m ||| t , 22+ ||| ξ n − ξ m ||| t , 22 = 0This means that there exist processes ξ  s ; t , x  , u  t , x  for whichm → ∞lim  || u − u m ||| t , 22+ ||| ξ − ξ m ||| t , 22 = 0when t < T .It easy to verify thatE | ξ n  s 1 ; t , x  − ξ n  s 2 ; t , x  |p≤≤ 2pKp 1 + α p  t   | s 1 − s 2 |p2,Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  46. 46. E | ξ n  s ; t 1 , x  − ξ n  s ; t 2 , x  |p≤ k 1  t∗ | t 1 − t 2 |p2,E | u n  t 1, x  − u n  t , x  |p≤ k 2  t∗ | t 1− t 2|p2,where t∗= max  t 1 , t 2  andk 1  t  = 4p − 1Kp 1 + α p t   exp 4p − 1Lp 1 + α p t   t ;k 2  t  = 6p − 1Kp 1 + α p t   + 3p − 1Lp 1 ++ 2p − 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 processesξ  s ; t , x  , u  t , x  belong to the spaces B pt, N pt, satisfy (2.2) and thereforehave 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 itsuffices to show possibility to pass to the limit in each of the terms in the system of successiveapproximations. Justifying the passage to the limit in each of the terms occurring in the systemis basically the same and so we shall explain it just for one of the stochastic integrals. Applyingthe properties of conditional expectations and using (2.2) for t < T we noteE |ts∫ 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    / Ϝr d w← r  | 2≤≤ 2 L2ts∫ E  | ξ n − 1  r ; t, x  − ξ  r ; t, x  |2 1 + || v n || r2 ++ || u n − u || r2 d r ≤ N 1  ||| ξ n − 1 − ξ ||| t2+ || u n − u || t2Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  47. 47. In what follows, N i = N i t  , i = 1 , 2 , ... will denote continuous increasingfunctions on  0 , T  . By means of analogous reasoning , by passing to the limit in thesystem of successive approximations as n → ∞ we can show that the functionsξ  s ; t , x  , u  t , x  give a solution to the system (2.1) on  0 , T  for which theinequality (2.2) holds. To prove uniqueness of the solution let suppose thatξ 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. ThenE | ξ 1  s ; t, x  − ξ 2  s ; t, x  |2≤ 2  1 + t  L2××ts∫ 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 2ts∫ E  | ξ 1 r ; t, x  − ξ 2 r ; t, x  |2++xsup E | u 1  r, x  − u 2  r, x  |2 d r .By Grownwall’s lemma , we find thatE | ξ 1 s ; t, x  − ξ 2 s ; t, x  |2≤≤ N 2ts∫xsup E | u 1 r, x  − u 2 r, x  |2d r .In exactly the same way one can establishxsup E | u 1 t, x  − u 2 t, x  |2≤ N 4||| ξ 1− ξ 2||| t2.Taking these inequalities into account, we havexsup E | u 1 t, x  − u 2 t, x  |2≤ .Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  48. 48. ≤ N 5ts∫xsup E | u 1 r, x  − u 2 r, x  |2d rFrom this, we easily deduce thatxsup E | u 1 t , x  − u 2 t , x  |2= 0 ,and hences , xsup 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 quasilinearsystems. First we find estimates for the derivatives of u  t , x  The system determiningξ 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 theoriginal one, has more complicated form and we have not succeeded in proving its solvabilityby the same method as Theorem 2 1. More precisely, the right-hand side of the systemcontains an integral of the product of u 1 and ξ 1 which is not allowed by thehypotheses of the theorem 2.1. Therefore when speaking of estimates of derivatives of thesolution 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  oru 1  t , x  have been shown to exist , then proving the existence of the high orderderivatives is of no difficulty, since the equations for them satisfy the hypotheses of Theorem2.1 and the proof is similar to the theorem on the continuous dependence of the solution to thesystem (2.1) on the coefficients. Formula (2.2) may be used to obtain a priori estimates forξ 1  s ; t , x  and u 1  t , x  . However, it is inadequate for obtaining estimates forthe higher derivatives. Assume for arbitrary p ≥ 2 that there exists a functionu 1  t , x  such thath → 0limxsup E |u  t , x + h  − u  t , x h− u 1  t , x  |p= 0for any t in some subinterval of  0 , T  introduced in Theorem 2.1. All of theCurrent address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  49. 49. subsequence discussion will be carried out on this subinterval.THEOREM 2.2 Assume that the functionsϕ 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 andn → ∞limxsup | ϕ n  x  − ϕ  x  | +T0∫x , usup  | 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 = 0Then , solutions u n t, x  , ξ n s ; t, x  converge to u  t, x  , ξ  s ; t, x n → ∞limt , xsup E | u n t, x  − u  t, x  |2+ | ξ n s ; t, x  − ξ  s ; t, x  |2 = 0Here 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 andso we just briefly remind the main stand points. One hass , xsup E | ξn  s ; t, x  − ξ  s ; t, x  |2≤≤ 8t0∫u , xsup  t | b n r, x, u  − b  r, x, u  |2++ | c n t, x, u  − c  t, x, u  |2 d s + 4 L2 1 + t   1 + 2 || u || t2 ××t0∫xsup  E  | u n  r, x  − u  r, x  |2+ | ξ n  r ; t, x  − ξ  r ; t, x  |2 d r ;Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  50. 50. t , xsup E  | u n t, x  − u  t, x  |2≤ 12t0∫u , xsup  | g n s, x  − g  s, x  |2++ | f n s, x, u  − f  s, x, u  |2 d s + 3xsup | ϕ n x  − ϕ  x  |2++ 12 L2 1 + t  1 + 2 || u || t2t0∫xsup  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 theproof. Consider the equationη  s ; t, x  = I +ts∫ b 1  r ; t, x, η  r ; t, x   d r − #−ts∫ E  c 1  r ; t, x, η  r ; t, x   / Ϝ  d←w  r whereb 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 ;Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  51. 51. 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) bydifferentiating with respect to parameter x .Lemma 2.1. Suppose that the hypotheses of Theorem 2.1 hold and that the functionsb  t, x, u  and c  t, x, u  have bounded continuous first derivatives in x and u . Thenthe system (2.4) has a unique solution with η  r ; t, x  = = ξ 1  r ; t, x  , andthe derivative is understood in the sense of mean square.Proof. Since the proof of the lemma is similar to the corresponding one for ordinarystochastic differential equations, we shall state only main steps. Denote the right hand side of(2.4) byL η  s ; t, x  . ThenE | Lη  s ; t, x  |2≤ 3  1 +  1 + T  K2×× || 1 + u 1 || t2ts∫ E | η  r ; t, x  |2d r  ;E | L η  s ; t, x  − L ζ  s ; t, x  |2≤≤ 2  1 + T  K2|| 1 + u 1 || t2ts∫ E | η  r ; t, x  − ζ  r ; t, x  |2d rFrom this inequalities, it is easy to see that some power of operator L is a contractingoperator in the Banach space of random functions with finite second order moments. Thisimplies the first assertion of the lemma. We then haveE | η  s ; t, x  −ξ  s ; t, x + Δ x  − ξ  s ; t, x Δ x|2≤≤ 2ts∫ E  t | B 1  r ; t, x  − B Δ 1  r ; t, x  |2+Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  52. 52. + | C 1  r ; t, x  − C Δ 1  r ; t, x  |2 d rTo simplify notation we put B 1  r ; t, x  = b 1  r ; t, x, η  r ; t, x   , andwe have taken for B Δ 1  r ; t, x  a similar expression corresponding to the driftcoefficient of the processξ  s ; t, x + Δ x  − ξ  s ; t, x Δ xand 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 thatE | B 1  r ; t, x  − B Δ 1  r ; t, x  |2≤≤ 4 E | ∇ xb  r , ξ  r ; t, x , v  r, ξ  r ; t, x    −−10∫  1 − θ  ∇xb  r, ξθ  r ; t, x , v  r, ξ  r ; t, x + Δ x    d θ |2||| η ||| t2++ 4 K2E | 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    −Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  53. 53. −10∫  1 − θ  ∇ ub  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 K2E || u 1 || t2E | η  s ; t, x  −ξ  s ; t, x + Δ x  − ξ  s ; t, x Δ x|2++ 6 K2E | ξ 1  r ; t, x  |2| v 1  r , ξ  r ; t, x   −−t0∫  1 − θ  v 1  r , ξ θ  r ; t, x    d θ |2where ξ θ  r ; t, x  = θ ξ  r ; t, x  +  1 − θ  ξ  r ; t, x + Δ x . ApplyingHolder’s inequality and using the conditions of the Lemma and finiteness of the moments ofthe random functions we easily find thatxsup E | η  s ; t, x  −ξ  s ; t, x + Δ x  − ξ  s ; t, x Δ x|2≤   Δ x  ++ R0t∫xsup E | η  r ; t, x  −ξ  r ; t, x + Δ x  − ξ  r ; t, x Δ x|2d rwhereΔ x → 0lim   Δ x  = 0 and R = R  || v 1 || , K  . The proof of Lemma 2.1is easily completed by resorting to Gronwall’s lemma.Remark. Taking lemma 2.1 as our starting assumption and applying the theorem oncontinuous 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) bydifferentiating formally with respect to x . Therefore the systemu 1  t, x  = Φ 1  0 ; t, x  +t0∫ F 1 r ; t, x  d r +Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  54. 54. +t0∫ G 1 r ; t, x  d←w 1  r  #ξ 1  s ; t, x  = I +ts∫ B 1 r ; t, x  d r ++ts∫ 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 functionu  t , x  . Here Φ 1  0 ; t , x  = ∇ ϕ  ξ  0 ; t, x   ξ 1  0 ; t, x  andF 1 r ; t , x  and G 1 r ; t , x  are given by expressions similar toB 1 r ; t , x  in Lemma 2.1.LEMMA 2.2. Under hypotheses of Lemma2.1, suppose that the partial derivatives of thecoefficients of the system (2.1) satisfy a Holder condition in x and u with exponentγ ∈  0 , 1 . Then, for p ≥ 2x ≠ ysup 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 haveE | ξ 1  s ; t , x  − ξ 1  s ; t , y  |p≤≤ 2p − 1st∫ E  tp − 1| B 1  r ; t, x  − B Δ 1  r ; t, x  |p++p2 p − 1  t12 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≤Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  55. 55. ≤ 3p − 1E  | Φ 1  0 ; t, x  − Φ 1  0 ; t, y  |p++ ∫ stE  tp − 1| F 1  r ; t, x  − F Δ 1  r ; t, x  |p++p2 p − 1  t12 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:t0∫ E | F 1  r ; t, x  − F Δ 1  r ; t, x  |p≤≤ 2p − 1Et0∫  | ∇ xf  r, ξ  r ; t, x , v  r, ξ  r ; t, x    ξ 1  r ; t, x  −− ∇ xf  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  −− ∇ uf  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 = 2p − 1Et0∫  I 1  r  + I 2  r   d r .In the subsequent computations, N i, i = 1,2... , denote positive constants notdepending on x or y . Applying Hölder inequality, we easily obtain the estimateI 2 r  ≤ N 1E | 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.Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202
  56. 56. SinceE | Φ 1  0 ; t, x  − Φ 1  0 ; t, y  |p≤≤ 2p − 1E  G γp| ξ 1  r ; t, x  − ξ 1  r ; t, y  |p γ++ Kp| 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 7t0∫x ≠ ysup 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 rApplying 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 thecorrespondent coefficients of the system ( 2.1 ). For instanceΦ 2  0 ; t, x  == Tr   ξ 1  0 ; t, x  ∗ ∂2ϕ  ξ  0 ; t, x  ∂ x2ξ 1  0 ; t, x  ++∂ ϕ  ξ  0 ; t , x  ∂ xξ 2  0 ; t , x   ,Current address: PLLS,49 East Fourth Street, Cincinnati, OH 45202

×