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

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, itis shown that the solution defines a Markov process. In general,this will be a time- and spaceinhomogeneous jump process. Tosolve the evolution equation, we combine a fixed-point methodwith the symbolic calculus for negative definite symbols developedby Hoh. The properties of the fundamental solution which ensurethe existence of a corresponding Markov process are proved alongthe lines of Eidelman, Ivasyshen and Kochubei. However, insteadof hyper-singular integral representations, we use the pseudo-differentialoperator representation together with the positive maximum principleto obtain the required properties.