A Hölder regularity index at given points for density states of (α, 1, β)-superprocesses with α > 1 + β is determined. It is shown that this index is strictly greater than the optimal index of local Hölder continuity for those density states.

We establish Hölder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.

We prove the interior Hölder continuity of weak solutions to parabolic systems ∂uj ∂t −Dαa α j (x, t, u,∇u) = 0 in Q (j = 1, . . . , N) (Q = Ω×(0, T ),Ω ⊂ R2), where the coefficients aj (x, t, u, ξ) are measurable in x, Hölder continuous in t and Lipschitz continuous in u and ξ.

We prove that the solution of stochastic differential equations with deterministic diffusion coefficient admits a Hölder continuous density via a condition on the integrability of the Fourier transform of the drift coefficient. In our result, the integrability is an important factor to determine the order of Hölder continuity of the density. Explicit examples and some applications are given.

In a setting, where only “exit measures” are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant Hölder continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling proper...

We show that every measurable invariant conformal structure for a Hölder continuous linear cocycle over a subshift of finite type coincides almost everywhere with a continuous invariant conformal structure. We use this result to establish Hölder continuity of a measurable conjugacy between Hölder continuous cocycles where one of the cocycles is assumed to be uniformly quasiconformal. As a speci...

We prove partial Hölder continuity for the gradient of minimizers u ∈W (Ω,R ), Ω ⊂ R a bounded domain, of variational integrals of the form