On the martingale problem for pseudo-differential operators of variable order
Consider parabolic pseudo-differential operators L = ∂t − p(x,Dx) of variable order α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is not always differentiable with respect to x. We will show the uniqueness of Markov processes with the generator L. The essential point in...
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| Datum: | 2008 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2008
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| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/4551 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the martingale problem for pseudo-differential operators of variable order / T. Komatsu // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 42–51. — Бібліогр.: 10 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | Consider parabolic pseudo-differential operators L = ∂t − p(x,Dx) of variable order α(x) ≤ 2. The function α(x) is assumed to be smooth, but the symbol p(x, ξ) is not always differentiable with respect to x. We will show the uniqueness of Markov processes with the generator L. The essential point in our study is to obtain the Lp-estimate for resolvent operators associated with solutions to the martingale problem for L. We will show that, by making use of the theory of pseudo-differential operators and a generalized Calderon–Zygmund inequality for singular integrals. As a consequence of our study, the Markov process with the generator L is constructed and characterized. The Markov process may be called a stable-like process with perturbation. |
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