LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS AND NONAUTONOMOUS DYNAMICAL SYSTEMS

Abstract

We prove that the linear stochastic equation dx(t) = (Ax(t)+f(t))dt+g(t)dW(t) (*) with linear operator A generating a C0-semigroup{U(t)}t≥0 and Levitan almost periodic forcing termsf and g admits a unique Levitan almost periodic [3,ChIV] solution in distrution sense if it has at least one precompact solution on R+and the semigroup{U(t)}t≥0is asymptotically stable.