Correctness and regularization of stochastic problems

Abstract

Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ ( t ) = A ⁢ u ⁢ ( t ) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ ( t ) , t > 0 , u ⁢ ( 0 ) = ξ . du(t)=Au(t)dt+BdW(t),\quad t>0,\qquad u(0)=\xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ ( t ) : t ≥ 0 } {\{W(t):t\geq 0\}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes.