Hölder continuity for spatial and path processes via spectral analysis

Abstract

For ν(dθ), a σ-finite Borel measure on R d , we consider L 2(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ t 0 e −λ(θ)( t − s ) dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ t 0 g(s,θ)ds. We prove timewise Holder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and Holder continuity results are also provenfor the path process t (τ)≗Y(τt∧t).