Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part I

Abstract

The paper continues the studies in the equivalence of the measures generated by the solutions of nonlinear evolution differential equations with unbounded linear operators perturbed by random Gaussian processes in a Hilbert space, in particular I. Two different nonlinear evolution differential equations perturbed by the same random Gaussian process in the right-hand side are considered in the space I. The sufficient existence and uniqueness conditions are established for the solutions of these equations, the equivalence of the measures generated by the solutions is proved, and explicit formulas for the Radon---Nikodym density of the respective measures calculated in terms of the coefficients of the considered equations are written.