AN L-p-THEORY FOR DIFFUSION EQUATIONS RELATED TO STOCHASTIC PROCESSES WITH NON-STATIONARY INDEPENDENT INCREMENT

Abstract

Let X = (X-t)(t >= 0) be a stochastic process which has a (not necessarily stationary) independent increment on a probability space (Omega, P). In this paper, we study the following Cauchy problem related to the stochastic process X: (*) partial derivative u/partial derivative t(t,x) = A(t)u(t, x) + f(t, x), u(0, .) = 0, (t,x) is an element of (0, T) x R-d, where f is an element of L-p((0, T);L-p(R-d)) = L-p((0,T);L-p) and A(t)u(t, x) = lim(h down arrow 0) E[u(t, x + Xt+h -X-t) - u(t, x)]/h. We provide a sufficient condition on X (see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in L-p([0, T]; H-p(phi)), where H-p(phi) is a phi-potential space on R-d (see Definition 2.9). Furthermore we show that for this solution, parallel to u parallel to L-p ([0, T]; H-p(phi)) (<= N parallel to f parallel to L)(p) ([0, T]; L-p), where N is independent of u and f.