H\"{o}lder Continuity of the Solution for a Class of Nonlinear SPDE Arising from One Dimensional Superprocesses

The H"older continuity of the solution to a nonlinear stochastic partial differential equation arising from one dimensional super process is obtained. It is proved that the H"older exponent in time variable is as close as to 1/4, improving the result of 1/10 in a recent paper by Li et al [3]. The method is to use the Malliavin calculus. The H"older continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This H"older continuity result is sharp since the corresponding linear heat equation has the same H"older continuity.