Construction of time-inhomogeneous Markov processes via evolution equations using pseudo-differential operators

Abstract

For a pseudo-differential operator with symbol which is time- and space-dependent, elliptic and continuous negative definite, the corresponding evolution equation is solved. Further, it is shown that the solution defines a Markov process. In general, this will be a time- and spaceinhomogeneous jump process. To solve the evolution equation, we combine a fixed-point method with the symbolic calculus for negative definite symbols developed by Hoh. The properties of the fundamental solution which ensure the existence of a corresponding Markov process are proved along the lines of Eidelman, Ivasyshen and Kochubei. However, instead of hyper-singular integral representations, we use the pseudo-differential operator representation together with the positive maximum principle to obtain the required properties