Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise

Abstract

Existence and uniqueness of approximate strong solutions of stochastic infinite-dimensional systems d u = [ A ( t ) u + B ( t , u ) ] d t + G ( t , u ) d W , u ( 0 , ⋅ ) = u 0 ∈ H , t ⩾ 0 with local Lipschitz-continuous, time-depending nonrandom operators A , B and G acting on a separable Hilbert space H are studied. For this purpose, some monotonicity conditions on those operators and an existing U-series expansion of the space–time Wiener process W ( U-valued, U ⊆ H , U Hilbert space) with ∑ n = 1 + ∞ α n 2 < + ∞ belonging to the trace of related covariance operator Q of W with local noise intensities α n 2 ∈ R 1 as eigenvalues of Q are exploited.