Generalized solutions for stochastic problems in the Ito form in Gelfand–Shilov spaces

V. A. Bovkun

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 2, pp. 5-13

Résumé

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 infinitedimensional 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: $$ X(t,x)-\zeta(x)=\int_0^tA\left(i\frac\partial{\partial x}\right)X(s,x)ds+\int_0^tBdW(s,x), \quad t\in[0;T],\ x\in\mathbb{R}, $$ where the operator $A\left(i\frac\partial{\partial x}\right)$ — a linear differential operator-matrix of finite order, which is a generator of $R$-semigroup in the space $L_2^m(\mathbb{R})$, $\zeta\in L_2^m$, $W(t,x)$ — $\mathcal{Q}$-a Wiener process. Formulation of the problem in the space $L_2^m(\mathbb{R})$ 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 $L_2^m(\mathbb{R})$ in particular. Results of the study of the Cauchy problem for deterministic systems with $A\left(i\frac\partial{\partial x}\right)$ operator show that, in general, the operator in the space $L_2^m(\mathbb{R})$ 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 $A\left(i\frac\partial{\partial x}\right)$ is built.

Détail

Keywords: stochastic Cauchy problem, Wiener process, generalized Fourier transform, generalized solution, Gelfand–Shilov spaces.

Bibliographie

[1] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, 1992, 482 pp. | DOI | MR | Zbl

[2] Melnikova I. V., Filinkov A. I., “Weak and generalized solutions of abstract stochastic equations”, Doklady Mathematics, 62:3 (2000), 373–377 | MR | MR | Zbl

[3] I. V. Melnikova, A. I. Filinkov, U. A. Anufrieva, “Abstract stochastic equations I: Classical and Generalized Solutions”, Journal of Mathematical Sciences, 111:2 (2002), 3430–3475 | DOI | MR | Zbl

[4] I. V. Melnikova, A. I. Filinkov, M. A. Alshansky, “Abstract Stochastic Equations. II: Solutions in Spaces of Abstract Stochastic Distributions”, Journal of Mathematical Sciences, 116:5 (2003), 3620–3656 | DOI | MR | Zbl

[5] I.V. Melnikova, A.I. Filinkov, “Abstract Stochastic Problems with Generators of Regularized Semigroups”, J. Communications in Applied Analysis, 13:2 (2009), 195–212 | MR | Zbl

[6] Alshansky M. A., Melnikova I. V., “Regularized and generalized solutions of infinite-dimensional stochastic problems”, Mat. Sbornik, 202:11 (2011), 3–30 | DOI | DOI | MR

[7] I. V. Melnikova, U. A. Alekseeva, “Weak regularized solutions to stochastic Cauchy problems”, Chaotic modeling and simulations, 2014, no. 1, 49–56

[8] Gel'fand I. M., Shilov G. E., Generalized function, v. 3, Theory of differential equations, Academic Press Inc., 1967, 222 pp. | MR

[9] Gel'fand I. M., Shilov G. E., Generalized Functions, v. 2, Spaces of Fundamental and Generalized Functions, Academic Press Inc., 1968, 261 pp. | MR | Zbl