Hölder continuity of the solutions to a class of SPDEs arising from multidimensional superprocesses in random environment

2018/10/18 by Hu, Yaozhong, Nualart, David, Xia, Panqiu
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1810.08120

Abstract

We consider a d-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure X_tⁿ converges weakly in the Skorohod space D([0,T];M_F(ℝ^d)) and the limit has a density u_t(x), where M_F(ℝ^d) is the space of finite measures on ℝ^d. We also derive a stochastic partial differential equation u_t(x) satisfies. By using the techniques of Malliavin calculus, we prove that u_t(x) is jointly Hölder continuous in time with exponent (1)/(2)-ε and in space with exponent 1-ε for any ε>0.

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