Generalized Solutions for Stochastic Problems in the Ito Form in Gelfand–Shilov Spaces

Abstract

The paper is devoted to one of the modern trends of research in mathematics, which is the study of problems with regard to the impact of random factors. One of the integral parts among such problems is taken by the models with differential equations containing heterogeneity of white noise in the infinite-dimensional spaces.

The main subject of research in the article is a stochastic Cauchy problem for systems of differential equations of the Gelfand–Shilov of order m in the form of Ito:

 

where the operator A(i∂/∂x) – a linear differential operator-matrix of finite order, which is a generator of R–semigroup in the space L^m₂(R), ξ $\in$ L^m₂(R), W(t,x) – Q- a Wiener process.

Formulation of the problem in the space L^m₂(R)is motivated by the fact that in modern applications that lead to models in the form of abstract stochastic problems, the process W takes values in a Hilbert space and in the space L^m₂(R) in particular.

Results of the study of the Cauchy problem for deterministic systems with A(i∂/∂x)operator show that, in general, the operator in the space L^m₂(R) generates ill-posed problem [8]. Solution of the problem for various classes of systems is determined in the generalized sense in the corresponding Gelfand-Shilov spaces using generalized Fourier transform technique.

Taking into account the results of the study of deterministic problems, the solution in the considering case of stochastic problem will be a generalized random process on the variable x. More precisely, in this paper a generalized according to spatial variable solution for stochastic problem in Gelfand-Shilov spaces corresponding to the classes of systems of Petrovskii well-posedness, conditional well-posedness and ill-posedness defined by a differential operator behavior A(i∂/∂x) is built.