Quasilinear Stochastic Cauchy Problem in Abstract Colombeau Spaces

Generalized solutions to the abstract Cauchy problem for a quasilinear equation with the generator of an integrated semigroup and with terms reflecting nonlinear perturbations and white noise type perturbations are under consideration. An abstract stochastic Colombeau algebra is constructed, and solutions in the algebra are studied.


Introduction
The paper is devoted to construction of solutions to the abstract quasilinear Cauchy problem where A is the generator of a C 0 semigroup or an integrated semigroup in a Hilbert space H, F is a nonlinear mapping from H to H, B is a linear bounded operator from a Hilbert space H to H, and W {W t , t ≥ 0} is a stochastic process of white noise type with values in H: W t W t, ω , ω ∈ Ω, B Ω , μ .Irregularity of white noise caused by independence of random variables W t 1 and W t 2 for t 1 / t 2 and by infinite variance necessitates to define the white noise in such a way that the problem 1.1 makes certain sense.

International Journal of Mathematics and Mathematical Sciences
One of the well-known ways to do this is to consider the corresponding integral equation replacing the white noise term by the integral with respect to a Wiener process W Ito integral , written as usually in Ito theory in the following form of differentials: dX t AX t dt F X dt BdW t , t ≥ 0, X 0 f.1.2 For the Cauchy problem 1.2 with generators of semigroups of class C 0 , with Lipschitz nonlinearities under some growth conditions mild solutions are constructed see, i.e., 1, 2 .
In this approach questions of whether the solutions obtained are differentiable, whether they satisfy the problem 1.1 , and whether the techniques can be applied for A generating regularized semigroups remain open.
Another approach, which we are going to use and generalize, is to consider 1.1 in spaces of abstract distributions, but here, due to nonlinearity of F in the equation, the problem of distribution products arises.A novel approach is to define an abstract stochastic Colombeau algebra G Ω, H a see definition in Section 1 and extend the distribution approach to the algebra.
Let H a be an algebra in a Hilbert space H, in particular, the subspace of continuous or a finitely many times differentiable functions in L 2 R closed under the topology of C k R , k 0, 1, . ... We consider the Cauchy problem 1.1 in the abstract stochastic Colombeau algebra G Ω, H a supposing that F is an infinitely differentiable mapping, B ∈ L H, H a , and H a ⊂ dom A. We define white noise W as an element in D H , the space of abstract distributions with values in H and supports in 0, ∞ , then by convolution with functions from D we transform W ∈ D H to infinitely differentiable with respect to t functions and as a result we obtain an element BW belonging to the algebra G Ω, H a .
As examples of A satisfying the conditions and generating different integrated semigroups one can take many of differential operators A P i ∂/∂x of correct in the sense of Petrovskiy systems 3 .The operators may be disturbed by bounded ones of any nature.For more examples see in 4, 5 .
In the paper we combine the multiplication theory in Colombeau algebras which has found applications in solving differential equations, mainly hyperbolic ones see, e.g., 6, 7 with the theory of regularized semigroups and the theory of stochastic processes in spaces of abstract distributions see, e.g., 8, 9 .This makes possible to solve nonlinear abstract stochastic equations with different types of white noise.

Definition of Abstract Colombeau Algebras
At the beginning we introduce Colombeau space of abstract Hilbert space valued generalized functions.For each q ∈ N 0 let A q be the set of all ϕ ∈ D such that 2.1 For algebra H a in a Hilbert space H we define the space of functions u ϕ : u ϕ t , t , ϕ ∈ A 0 , t ∈ R as follows:

2.2
According to the definition u ϕ is an infinitely differentiable H a -valued function of real argument t ∈ R for each ϕ ∈ A 0 .Thus u can be considered as a function of two variables: ϕ ∈ A 0 and t ∈ R, that is, and it is infinitely differentiable with respect to the second variable.Differentiation and multiplication are defined as follows: The space of the H a -valued distributions D H a being a subset of the abstract distributions space D H is embedded in E H a by the following mapping: Now let us introduce the functions ϕ ε t : 1/ε ϕ t/ε , t ∈ R, ε > 0, ϕ ∈ A 0 and define the linear manyfold of moderate elements E M H a consisting of all u ∈ E H a satisfying the following condition: M for each compact K ⊂ R and each n ∈ N 0 there exists q ∈ N such that To complete the definition of the Colombeau algebra of abstract generalized functions we introduce N H a consisting of all elements u ∈ E H a that satisfy the following condition: N for each compact K ⊂ R and each n ∈ N 0 there exists p ∈ N such that Elements of the space E M H a form a differential algebra, and N H a is the differential ideal in it.Now define the factor algebra Similarly to G R see, e.g., 6 , the algebra G H a is an associative and commutative H avalued differential one.Elements of G H a are classes of mappings.We denote them by capitals U, V, . . .and denote representatives of class U ∈ G H a by corresponding small letter u.
Due to the structure theorems for abstract distributions 10 , similarly to the R-valued case, we obtain that i maps the elements of D H a into E M H a and i −1 N H a consists of the null element of D H a .Thus, each element of D H a is imbedded in the corresponding class of G H a by the mapping i.
Support of an element U ∈ G H a is defined as follows.We say that U is equal to zero on an open set Λ ⊂ R if its restriction to G Λ H a is equal to zero in G Λ H a where algebra G Λ H a is defined in the same way as G Λ R n 6 .If w ∈ D H a , then, similarly to the case of D R , support of iw ∈ G H a coincides with that of w ∈ D H a .Now define G Ω, H a , the algebra of G H a -valued random variables {U U ω , ω ∈ Ω, B Ω , μ } as a mapping from Ω, B Ω , μ to G H a measurable in the following sense: there exists a representative u ∈ U such that for any ϕ ∈ A 0 u −1 ϕ, • maps any Borel subset of B C ∞ H a onto an element of B Ω , where the Borel σ-algebra B C ∞ H a on the space C ∞ H a is generated by the system of neighborhoods in C ∞ H a defined by the system of seminorms p n,k f sup t∈ −n,n f k t H .To complete the setting of the problem we define the generalized white noise process W W •, ω for each ω ∈ Ω, as an element in D H , and the space of abstract distributions with values in H and supports in 0, ∞ , and then transform it into an element of G Ω, H a .
One way to do this is based on the ideas of abstract stochastic distributions see, e.g., 8, 9 .Let S S R be the space of rapidly decreasing test functions.Denote by S H the space of H-valued distributions over S and consider a Borel σ-algebra B Ω generated by the weak topology of Ω : S H . Then by the generalization of the Bochner-Minlos theorem to the case of Hilbert space valued generalized functions 11 , there exist a unique probability measure μ on B Ω and a trace class operator Q satisfying the condition as follows Here and below, if it is not pointed out especially, • denotes the norm of L 2 R .It makes possible to define the white noise process on Ω, B Ω , μ with values in S H ⊂ D H by the identical mapping as follows:

2.10
The above-defined process is the generalization of the corresponding real-valued Gaussian process 12 , and it has zero mean and Cov W, θ θ 2 Q.Define W with support in 0, ∞ as follows:

2.11
Here W −k is a continuous function that according to the structure theorem is a primitive of W of an order k and χ is the Heaviside function.
Another way to define a generalized H-valued white noise on a Ω, B Ω , μ , more precisely Q-white noise, is via derivative of H-valued Q-Wiener process {W Q t , t ≥ 0} 9 continued by zero on −∞, 0 as follows: 2.12 Finally, we map the defined generalized white noise process into the Colombeau algebra G Ω, H a in the following manner.By convolution with a function from A 0 we transform W •, ω ∈ D H into the infinitely differential with respect to t ∈ R and measurable with respect to ω ∈ Ω function w as follows: Let B ∈ L H, H a , and then applying B to w we obtain that Bw ϕ, t, ω ∈ H a and the map

Solutions to the Cauchy Problem with Infinitely Differentiable Nonlinearities
Let H H L 2 R and the domain of A lie in the set of continuous functions of L 2 R .Let H a be the set of finitely many times differentiable functions of L 2 R and H a ⊆ dom A. Then multiplication of elements of L 2 R is well defined on the set L 2 R ∩ dom A as pointwise continuous functions multiplication.
In this section for the problem 1.1 , where nonlinearity F is infinitely differentiable, bounded with all its derivatives, and has the property F 0 0 and where stochastic term BW is constructed above, we will search a solution as an element of the abstract stochastic Colombeau algebra G Ω, H a .Since H a is chosen as the set of finitely many times differentiable functions in L 2 R , operator B ∈ L H, H a can be taken, for example, as convolution with a finitely many times differentiable function from L 2 R and with condition of the convolution existence.
Suppose at the beginning that A generates a C 0 -semigroup {V t , t ≥ 0} in L 2 R .Consider the question whether there is a solution to the problem where y t y ϕ, t, ω , ϕ ∈ A 0 , ω ∈ Ω.We will search a solution of this problem belonging to C ∞ −η; ∞ ; dom A for ω a.s.

International Journal of Mathematics and Mathematical Sciences
Consider the equation The introduced operator Q is a Volterra type one.Using the differentiability of F and boundedness of its derivative let us show that Q k where k k T is a contraction on the segment −η; T .Since F is differentiable, we have F μ − F λ F ξ μ − λ , ξ ∈ μ; λ , for any μ, λ ∈ R. Then for any y • and z • we get the pointwise equality where ξ is an appropriate point from y s ; z s and the following estimate holds: This and exponential boundedness of C 0 semigroups: for each t ∈ −η; ∞ imply that Qy t − Qz t ≤ CLe a t η t η max s∈ −η;t y s − z s .

3.7
For squares we have

3.8
Then we have and for every k ∈ N

3.11
The constant in this estimate can be made less then unity by choosing k k T .Thus Q k is the contraction, and the sequence of approximations y n t Q nk y 0 t has the limit in H: 12 uniform with respect to t ∈ −η; T .Note that if one takes an infinitely differentiable with respect to t function y 0 • as the first point for the approximating sequence, then function is also an infinitely differentiable with respect to t function, and consequently y 1 • Q k y 0 • has the same property as well as all subsequent iterations y n • .
It can be shown by the same arguments that the sequence y n • converges to its limit in H uniformly with respect to t ∈ −η; T ; hence y • is differentiable and y t lim n → ∞ y n t .Similarly it can be shown that y • is infinitely differentiable function with values in H. Now we show that y n t ∈ H a if y 0 t ∈ H a , t ≥ 0. Let t ≥ 0 be fixed.Note firstly that F α O α as α → 0. Really, due to the infinite differentiability of F and property F 0 0 F α can be repersented by the Taylor series with first term proportional to α.Then, since y 0 t ∈ H a , it is differentiable with respect to variable of L 2 R , and y 0 t → 0 as variable of L 2 R tends to infinity.Thus, F y 0 t O y 0 t as variable of L 2 R tends to infinity.
Further, semigroup operators V t map L 2 R into L 2 R , and, moreover, they map differentiable functions with respect to variable of L 2 R into the set of differentiable ones due to their boundness.It follows that y ϕ ε , s, ω ds.

3.18
Since Bw is a representative of class BW from G Ω, H a , the first term in the right-hand side of the inequality for every ϕ ∈ A q increases as ε → 0 not faster than ε −q for some q ∈ N.
H a 2.14 are representative of a class in G Ω, H a .The corresponding class we denote by BW.Since the support of W ∈ D H is 0; ∞ , by definition of support of an element of G Ω, H a we have supp BW 0; ∞ .That is the sense we attach to the stochastic term in 1.1 .
H a 7 .By definition of elements of G Ω, H a , for each fixed ϕ ∈ A 0 , Bw t is an infinitely differentiable function of t ∈ R with values in H a and measurable with respect to ω ∈ Ω.Let us take an arbitrary ϕ ∈ A 0 and consider the problem 1as an element of algebra G Ω, H a .To do this, for an arbitrary η > 0, we consider Bw t : Bw ϕ, t, ω , ϕ ∈ A 0 , ω ∈ Ω, with support in −η, ∞ as a representative of the white noise term BW ∈ G Ω, 3.14hence Q acts in H a and y 1 t Q k y 0 t ∈ H a as well as y n t for any n ∈ N. Thus, we obtain y n t ∈ H a , but in the general case lim n → ∞ y n t y t does not belong to H a since algebra H a is not closed in the sense of L 2 R convergence.If y t ∈ H a , then we show that it is a representative of a class from G Ω, H a .As it is known see, e.g., 4 , if F • is a differentiable function or F t ∈ dom A for any t ≥ 0, then solution of the inhomogeneous abstract Cauchy problem Since in the case under consideration F y t as well as any International Journal of Mathematics and Mathematical Sciences representative Bw t of white noise process are L 2 R -valued infinitely differentiable with respect to t functions, the solution of 3.3 is a solution to the problem 3.2 .Due to the property of C 0 semigroup V t 0 as t < 0, it follows from 3.3 that y t 0 as t ≤ −η; that is, support of the obtained solution lies in −η; ∞ .Now we show that y is a representative of a class Y ∈ G Ω, H a , that means that it satisfies the condition M almost everywhere.It follows from differentiability of F and the t 0 V t − s F s ds.
• can be checked up in the same manner using that F is infinitely differentiable and its derivatives are bounded.Now let us show that supp Y ⊆ 0; ∞ .To do this we consider two solutions of 3.2 y η 1 • and y η 2 • corresponding η 1 / η 2 and verify that difference y η 1 • − y η 1 • belongs to N H a .Note that η max{η 1 , η 2 }.Then we have H a as the difference of two representatives of the stochastic term BW whose support is in 0; ∞ .Then, similar to 3.18 , we obtain the following estimation:Since the first term satisfies to N , Gronwall-Bellman lemma implies thaty η 1 t − y η 2 t ∈ N H a , so supp Y ⊆ 0; ∞ .Now we show that the solution of 3.1 is unique in the algebra G Ω, H a .Let Y 1 , Y 2 ∈ G Ω, H awith support in 0; ∞ be two solutions of 3.1 .Then for any representatives y 1 , y 2 of these classes and each η > 0 the following relations hold:y 1 t − y 2 t A y 1 t − y 2 t F y 1 t − F y 2 t g t , t ≥ −η, y 1 t − y 2 t ∈ N H a , t ≤ −η, 3.22where g is an element of N H a .Then, as above, Gronwall-Bellman Lemma implies thaty 1 t − y 2 t ∈ N H a , that is Y 1 − Y 2 0 in G Ω, H a .Taking into account that the linear Cauchy problem, corresponding to 1.1 has the following form in spaces of distributions: solution to the Cauchy problem 1.1 in G Ω, H a is related with the obtained Y as follows: X t Y t V * iδ t f, t ≥ 0, f ∈ H a .In the general case since the limit of y n t / ∈ H a , we obtain only the approximated solutions of 3.3 -the fundamental sequence {y n } obtained by the following equalities Let A be the generator of a C 0 -semigroup {V t , t ≥ 0} in L 2 R .Let F be an infinitely differentiable function in R, bounded with all its derivatives and F 0 0. Let B ∈ L L 2 R , H a and BW be an element of G Ω, H a with representative Bw defined by 2.13 .Then for any η > 0 and ϕ ∈ A 0 there exists the unique solution of 3.2 y ∈ C ∞ −η; ∞ ; H .If y ∈ C ∞ −η; ∞ ; H a , then 3.1 has the unique solution in algebra G Ω, H a .In this case the solution to the Cauchy problem1.1 in G Ω, H a is X Y V * iδ f for any f ∈ H a .Now consider the case of A generating an integrated semigroup.If operator A generates an exponentially bounded n-times integrated semigroup {V n t , t ≥ 0}, then y η 1 t − y η 2 t A y η 1 t − y η 2 t F y η 1 t − F y η 2 t g t , t ≥ −η, y η 1 t − y η 2 t 0, t ≤ −η,3.20whereg ∈ N