Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

Abstract

We consider stochastic reaction-diffusion equations with colored noise and prove Schauder type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup, defined on the Banach space of bounded and uniformly continuous functions.1.1) where γ ∈ [0, 1]. For any x ∈ E := Dom(A) (the closure is taken in C(O), with respect the uniform norm), under suitable hypotheses (see [15, Proposition 6.2.2]), equation (1.1) has a unique Evalued pathwise continuous mild solution {X(t, x)} t≥0 (see Definition 2.1) and we can define the