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This paper presents conditions for solutions and for some functions of the solutions to parabolic equation with" white noise" on the right to vanish with probability 1 as t++=. The paper is to continue paper [1], where conditions have been established for stabilization with probability 1 of solutions to one-dimensional parabolic equation, and [2] which dealt with the behavior of the solutions of abstract stochastic equations. The stabilization problems involved in Solving partial differential equations are of significant interest, a comprehensive survey of papers devoted to deterministic problems can be found in [3]. The behavior of solutions of stochastic equations is studied not so well; along with the papers described in [I, 2] we would like to mention the paper [4] where stabilization conditions for statistic solutions of hyperbolic equations are given.

I. Equation with Constant Operator. Assume that H and X are separable Hilbert …