Schauder-type estimates for higher-order parabolic SPDEs

Abstract

In this paper, we consider the Cauchy problem for 2m-order stochastic partial differential equations of parabolic type in a class of stochastic Holder spaces. The Holder estimates of solutions and their spatial derivatives up to order 2m are obtained, based on which the existence and uniqueness of solution are proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when m is odd or even: the condition for odd m coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even m it depends on the integrability index p. The sharpness of the new-found coercivity condition is demonstrated by an example.