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

Tanaka formula and local time for a class of interacting branching measure-valued diffusions

“…Remark: Recently, using Malliavin Calculus, Hu et al [13] proved Hölder continuity for a related class of processes, namely SDSM with the classical Dawson-Watanabe branching mechanism replaced by the more cohesive and therefore asymptotically smoother Mytnik-Sturm branching, under an initial measure μ0 with a bounded Radon-Nikodym derivative with respect to Lebesgue measure λ 0 . More precisely they have shown that, in the case where c is the identity matrix and h is smooth enough (and matrix valued), this regularized SDSM {μ t } has a density ft = dμ t /dλ 0 which is almost surely jointly Hölder continuous, with exponent β 1 ∈ (0, 1) in space and β 2 ∈ (0, 1/2) in time.…”