The heat equation with random initial conditions from Orlicz spaces

Abstract

Abstract. Conditions for justification of the Fourier method for parabolic equations with random initial conditions from Orlicz spaces of random variables are obtained. Bounds for the distribution of the supremum of solutions of such equations are found.We study conditions justifying the application of the Fourier method for parabolic equations with random initial conditions and obtain bounds for the distribution of the supremum of solutions of these equations. Similar problems for hyperbolic equations are considered in [1,2]. A survey of the corresponding results can be found in [3,4]. In what follows we consider random initial conditions from the Orlicz spaces of random variables.The paper is organized as follows. Section 1 contains necessary definitions and results of the theory of the Orlicz space. The setting of the problem as well as statements of the main results of the paper is given in Section 2. Conditions for the convergence of stochastic processes in C(T ) and bounds for the distribution of the supremum of solutions of the corresponding equations are presented in Section 3. The proofs of the main results are placed in Section 4. Stochastic processes belonging to an Orlicz spaceDefinition 1.1 ([3]). An even, continuous, convex function U (x) such that U (x) > 0 for x = 0 is called a C-function.Let {Ω, , P} be a standard probability space.The Orlicz space L U (Ω) is a Banach space with respect to the norm, t ∈ T } be a stochastic process. We say that X belongs to the Orlicz space L U (Ω) if, for all t ∈ T , the random variable X(t) belongs to the space L U (Ω). Lemma 1.1 ([3]).Let ξ ∈ L U (Ω) and let E U (ξ/r) ≤ a for some r > 0 and a > 0. Then ξ L U ≤ r max{0; a}.2000 Mathematics Subject Classification. Primary 60G60, 60G17.

Justification of Application of the Fourier Method for Parabolic Equations with Random Boundary Conditions from Orlicz Spaces

“…The following results are generalization of results described in the paper by Kozachenko and Veresh (2010).…”

“…  , conditions (3.4) cover all known boundary value conditions except boundary value conditions from the paper by Kozachenko and Veresh (2010). The following results are generalization of results described in the paper by Kozachenko and Veresh (2010).…”

“…The next theorem is a particular case of the theorem proved by Kozachenko and Veresh (2010). ( ) → ( ) as → ∞, ∈ , in probability; 3.…”

“…( ) → ( ) as → ∞, ∈ , in probability; 3. Lemma 2.1 (Kozachenko and Veresh, 2010). Let ( ), > 0, ∈ , ∈ (0, ∞), be a function such that:…”

The heat equation on line with random right part from Orlicz space

“…Several researchers have investigated solutions of the heat equation depending on various types with random conditions [1,7,[10][11][12]. In this paper we consider the Cauchy problem for the heat equation on line with random right part.…”