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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 509026, 11 pages
Quasilinear Stochastic Cauchy Problem in Abstract Colombeau Spaces
Institute of Mathematics and Computer Sciences, Ural Federal University, 51 Lenina Avenue, Ekaterinburg 620083, Russia
Received 30 March 2012; Accepted 9 September 2012
Academic Editor: Palle E. Jorgensen
Copyright © 2012 Irina V. Melnikova and Uljana A. Alekseeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The paper is devoted to construction of solutions to the abstract quasilinear Cauchy problem where is the generator of a semigroup or an integrated semigroup in a Hilbert space , is a nonlinear mapping from to , is a linear bounded operator from a Hilbert space to , and is a stochastic process of white noise type with values in : .
Irregularity of white noise caused by independence of random variables and for and by infinite variance necessitates to define the white noise in such a way that the problem (1.1) makes certain sense.
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 (Ito integral), written as usually in Ito theory in the following form of differentials: For the Cauchy problem (1.2) with generators of semigroups of class , 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 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 in the equation, the problem of distribution products arises. A novel approach is to define an abstract stochastic Colombeau algebra (see definition in Section 1) and extend the distribution approach to the algebra.
Let be an algebra in a Hilbert space , in particular, the subspace of continuous or a finitely many times differentiable functions in closed under the topology of , . We consider the Cauchy problem (1.1) in the abstract stochastic Colombeau algebra supposing that is an infinitely differentiable mapping, , and . We define white noise as an element in , the space of abstract distributions with values in and supports in , then by convolution with functions from we transform to infinitely differentiable with respect to functions and as a result we obtain an element BW belonging to the algebra .
As examples of satisfying the conditions and generating different integrated semigroups one can take many of differential operators of correct in the sense of Petrovskiy systems . 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.
2. Definition of Abstract Colombeau Algebras
At the beginning we introduce Colombeau space of abstract (Hilbert space valued) generalized functions. For each let be the set of all such that For algebra in a Hilbert space we define the space of functions as follows: According to the definition is an infinitely differentiable -valued function of real argument for each . Thus can be considered as a function of two variables: and , that is, and it is infinitely differentiable with respect to the second variable.
Differentiation and multiplication are defined as follows: The space of the -valued distributions being a subset of the abstract distributions space is embedded in by the following mapping: Now let us introduce the functions , , and define the linear manyfold of moderate elements consisting of all satisfying the following condition:
(M) for each compact and each there exists such that To complete the definition of the Colombeau algebra of abstract generalized functions we introduce consisting of all elements that satisfy the following condition:
(N) for each compact and each there exists such that Elements of the space form a differential algebra, and is the differential ideal in it. Now define the factor algebra Similarly to (see, e.g., ), the algebra is an associative and commutative -valued differential one. Elements of are classes of mappings. We denote them by capitals and denote representatives of class by corresponding small letter .
Due to the structure theorems for abstract distributions , similarly to the -valued case, we obtain that maps the elements of into and consists of the null element of . Thus, each element of is imbedded in the corresponding class of by the mapping .
Support of an element is defined as follows. We say that is equal to zero on an open set if its restriction to is equal to zero in (where algebra is defined in the same way as ). If , then, similarly to the case of , support of coincides with that of .
Now define , the algebra of -valued random variables as a mapping from to measurable in the following sense: there exists a representative such that for any maps any Borel subset of onto an element of , where the Borel -algebra on the space is generated by the system of neighborhoods in defined by the system of seminorms .
To complete the setting of the problem we define the generalized white noise process for each , as an element in , and the space of abstract distributions with values in and supports in , and then transform it into an element of .
One way to do this is based on the ideas of abstract stochastic distributions (see, e.g., [8, 9]). Let be the space of rapidly decreasing test functions. Denote by the space of -valued distributions over and consider a Borel -algebra generated by the weak topology of . Then by the generalization of the Bochner–Minlos theorem to the case of Hilbert space valued generalized functions , there exist a unique probability measure on and a trace class operator satisfying the condition as follows (Here and below, if it is not pointed out especially, denotes the norm of ).
It makes possible to define the white noise process on with values in by the identical mapping as follows: The above-defined process is the generalization of the corresponding real-valued Gaussian process , and it has zero mean and Cov. Define with support in as follows: Here is a continuous function that according to the structure theorem is a primitive of of an order and is the Heaviside function.
Another way to define a generalized -valued white noise on a , more precisely -white noise, is via derivative of -valued -Wiener process  continued by zero on as follows:
Finally, we map the defined generalized white noise process into the Colombeau algebra in the following manner. By convolution with a function from we transform into the infinitely differential with respect to and measurable with respect to function as follows: So, a.s., and .
Let , and then applying to we obtain that and the map are representative of a class in . The corresponding class we denote by . Since the support of is , by definition of support of an element of we have . That is the sense we attach to the stochastic term in (1.1).
3. Solutions to the Cauchy Problem with Infinitely Differentiable Nonlinearities
Let and the domain of lie in the set of continuous functions of . Let be the set of finitely many times differentiable functions of and . Then multiplication of elements of is well defined on the set as pointwise continuous functions multiplication.
In this section for the problem (1.1), where nonlinearity is infinitely differentiable, bounded with all its derivatives, and has the property and where stochastic term is constructed above, we will search a solution as an element of the abstract stochastic Colombeau algebra . Since is chosen as the set of finitely many times differentiable functions in , operator can be taken, for example, as convolution with a finitely many times differentiable function from and with condition of the convolution existence.
Suppose at the beginning that generates a -semigroup in .
Consider the question whether there is a solution to the problem as an element of algebra . To do this, for an arbitrary , we consider , , , with support in as a representative of the white noise term . By definition of elements of , for each fixed , is an infinitely differentiable function of with values in and measurable with respect to . Let us take an arbitrary and consider the problem where . We will search a solution of this problem belonging to for a.s.
Consider the equation The introduced operator is a Volterra type one. Using the differentiability of and boundedness of its derivative let us show that (where ) is a contraction on the segment .
Since is differentiable, we have , for any . Then for any and we get the pointwise equality where is an appropriate point from and the following estimate holds: This and exponential boundedness of semigroups: for each imply that For squares we have Then we have and for every hence The constant in this estimate can be made less then unity by choosing . Thus is the contraction, and the sequence of approximations has the limit in : uniform with respect to .
Note that if one takes an infinitely differentiable with respect to function as the first point for the approximating sequence, then function is also an infinitely differentiable with respect to function, and consequently has the same property as well as all subsequent iterations .
It can be shown by the same arguments that the sequence converges to its limit in uniformly with respect to ; hence is differentiable and . Similarly it can be shown that is infinitely differentiable function with values in .
Now we show that if , . Let be fixed. Note firstly that as . Really, due to the infinite differentiability of and property can be repersented by the Taylor series with first term proportional to . Then, since , it is differentiable with respect to variable of , and as variable of tends to infinity. Thus, as variable of tends to infinity.
Further, semigroup operators map into , and, moreover, they map differentiable functions (with respect to variable of ) into the set of differentiable ones due to their boundness. It follows that hence acts in and as well as for any .
Thus, we obtain , but in the general case does not belong to since algebra is not closed in the sense of convergence.
If , then we show that it is a representative of a class from . As it is known (see, e.g., ), if is a differentiable function or for any , then solution of the inhomogeneous abstract Cauchy problem with generating a semigroup of operators exists and is defined by the formulae . Since in the case under consideration as well as any representative of white noise process are -valued infinitely differentiable with respect to functions, the solution of (3.3) is a solution to the problem (3.2).
Due to the property of semigroup as , it follows from (3.3) that as ; that is, support of the obtained solution lies in .
Now we show that is a representative of a class , that means that it satisfies the condition (M) almost everywhere. It follows from differentiability of and the condition that for each the following equality where , , , holds. Now for an arbitrary compact from (3.3) we obtain that and due to boundedness of we have Since is a representative of class from , the first term in the right-hand side of the inequality for every increases as not faster than for some . Then, due to Gronwall-Bellmann inequality the left-hand side behaves in the same way, that proves the condition (M) with . Let us remind that the Gronwall-Bellmann inequality states that if for positive continuous functions and , than . Behavior of derivatives of can be checked up in the same manner using that is infinitely differentiable and its derivatives are bounded.
Now let us show that supp . To do this we consider two solutions of (3.2) and corresponding and verify that difference belongs to . Note that . Then we have where as the difference of two representatives of the stochastic term whose support is in . Then, similar to (3.18), we obtain the following estimation: Since the first term satisfies to (N), Gronwall-Bellman lemma implies that , so .
Now we show that the solution of (3.1) is unique in the algebra . Let with support in be two solutions of (3.1). Then for any representatives , of these classes and each the following relations hold: where is an element of . Then, as above, Gronwall-Bellman Lemma implies that , that is in .
Taking into account that the linear Cauchy problem, corresponding to (1.1) has the following form in spaces of distributions: and the required solution to the Cauchy problem (1.1) in is related with the obtained as follows: , , .
In the general case since the limit of , we obtain only the approximated solutions of (3.3)—the fundamental sequence obtained by the following equalities
So, we get the following result.
Theorem 3.1. Let be the generator of a -semigroup in . Let be an infinitely differentiable function in , bounded with all its derivatives and . Let and be an element of with representative defined by (2.13). Then for any and there exists the unique solution of (3.2) . If , then (3.1) has the unique solution in algebra . In this case the solution to the Cauchy problem (1.1) in is for any .
Now consider the case of generating an integrated semigroup. If operator generates an exponentially bounded -times integrated semigroup , then solution operators of homogeneous Cauchy problem are defined as follows: , , and instead of (3.3) we have the following equation: Here all derivatives (in ) exist due to infinite differentiability of and . Using the equality, similarly to the case of semigroups of class , we obtain the corresponding approximations and the solution in if the limit of belongs to .
In conclusion we note that the present paper is only the beginning of researches of abstract stochastic equations with nonlinearities in Colombeau algebras. Among important questions that remain open and are supposed to be investigated in future are convergence of solutions as , equations with generators of more general regularized semigroups, and equations in arbitrary Hilbert spaces.
This work was completed with the support of the Programme of the Ministry of Education and Science of Russian Federation 1.1016.2011 and RFBR 10-01-96003 Ural.
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