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Construction of time‐inhomogeneous Markov processes via evolution equations using pseudo‐differential operators
Author(s) -
Böttcher Björn
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn043
Subject(s) - mathematics , markov process , operator (biology) , differential equation , representation (politics) , markov chain , differential (mechanical device) , jump process , jump , mathematical analysis , law , biochemistry , statistics , chemistry , physics , engineering , repressor , quantum mechanics , politics , political science , transcription factor , gene , aerospace engineering
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.