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Journal of the London Mathematical Society Advance Access published online on July 11, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn043
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© 2008 London Mathematical Society

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

Björn Böttcher

Fakultät Mathematik und Naturwissenschaften
Institut für mathematische Stochastik
D-01062 Dresden
Germany

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 space-inhomogeneous 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.

2000 Mathematics Subject Classification 60J35, 35S10, 47G30, 60J75, 35K90.

Received October 25, 2007;
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