Holder continuity for spatial and path processes via spectral analysis

For v(d theta), a sigma -finite Borel measure on R-d, we consider L-2(v(d theta))-valued stochastic processes Y(t) with the property that Y(t) = y(t, (.)) where y(t, theta) = integral (l)(0) e(-lambda(theta)(t-s)) dm (s, theta) and m (t, theta) is a continuous martingale with quadratic variation [m](t) = integral (t)(0) g(s, theta) ds. We prove timewise Holder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superprocesses 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 proven for the path process Y-l (tau) (=) over circle Y(tau t boolean AND t).