Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions

Abstract

STOCHASTIC EQUATIONS II. SOLUTIONS IN SPACES OF ABSTRACT STOCHASTIC DISTRIBUTIONS I. V. Melnikova, A. I. Filinkov, and M. A. Alshansky UDC 519.219 Introduction In Part II, we study stochastic evolution equations with additive noise using semigroup methods in the framework of white noise calculus. White noise analysis has been developed during the past three decades by many authors (see, e.g., [1–4] and the references therein). In the present survey, we use the approach of [4], where the theory of R -valued stochastic distributions is presented, and of [5], where the theory of Hilbert space valued stochastic distributions is developed. Our aim is to apply the theory of semigroups of linear operators to stochastic (ordinary and partial) differential equations in cases more general than that described in Part I. Consider, for example, the stochastic heat equation dX(t, x) dt = ∆xX(t, x) +W(t, x), t ∈ [0, T ], x ∈ O = { x ∈ R ; 0 1 in the space S(H)−0 of stochastic distributions with values in H The central point of Part II is the construction of the spaces of H-valued test functions S(H)ρ, ρ ∈ [0, 1], and stochastic distributions S(H)−ρ: S(H)1 ⊂ S(H)ρ ⊂ S(H)0 ⊂ L2(S ;H) ⊂ S(H)−0 ⊂ S(H)−ρ ⊂ S(H)−1 for a separable Hilbert space H. Then we develop the calculus in these spaces and consider basic examples of H-valued stochastic processes: the H-valued weak Wiener process {W (t)} and the H-valued singular white noise process {W(t)}. Generally, both of them take values in S(H)−0 for any fixed t. One can expect that the Q-Wiener processes discussed in Part I belong to L2(S ′;H). It is crucial for our main results on evolution equations that the white noise be (infinitely) differentiable with respect to t (t plays the role of a parameter) in S(H)−1, and in this space, we can give a description of the convergence similar to the convergence in S(R )−1. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 96, Funktsional’nyi Analiz, 2001. 362