Panqiu Xia, University of Kansas

Joint Holder Continuity of the Solutions to a Class of SPDEs Arising from Multi-dimensional Superprocesses in the Random Environment

Abstract

We consider a d-dimensional branching particle system in a random environment. Suppose the initial measure converges to a finite measure, which has a bounded Lebesgue density. Under some particular branching mechanism, we prove the equipped empirical measure converges almost surely to a finite measure-valued process in the weak topology and the limit process has a Lebesgue density. The density is also a weak solution to an SPDE. By using the techniques of Malliavin calculus and a conditional convolution representation, we prove that the density is jointly Holder continuous with time exponent less than 1/2, and spatial exponent less than 1.