Some boundedness properties of certain stationary diffusion semigroups. (English) Zbl 0562.60078
Let \((P_ t)\) be some conservative diffusion semigroup with uniformly hypoelliptic smooth diffusion operator and let m be a \((P_ t)\)- invariant probability measure on \(R^ N\). Using Malliavin calculus the authors estimate \(\| P_ t\|_{L^ p(m)\to L^ q(m)}(1<p<q\leq \infty,t>0)\). They explain what advantages their results have in comparison to E. B. Davies and B. Simon’s, Ultracontractive semigroups and some problems in analysis. To appear in: Aspects of mathematics and its applications, North-Holland.
Reviewer: R.Mikulevičius
MSC:
60J35 | Transition functions, generators and resolvents |
47D07 | Markov semigroups and applications to diffusion processes |
Keywords:
conservative diffusion semigroup; hypoelliptic smooth diffusion operator; Malliavin calculusReferences:
[3] | Donsker, M.; Varadhan, S. R.S, Asymptotic evaluation of certain Markov process expectations for large time, I, Comm. Pure Appl. Math., 28, 1-47 (1975) · Zbl 0323.60069 |
[4] | Glimm, J., Boson fields with nonlinear self-interaction in two dimensions, Comm. Math. Phys., 8, 12-25 (1968) · Zbl 0173.29903 |
[5] | Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus, (Itô, K., Stochastic Analysis. Stochastic Analysis, Proc. of 1982 Taniguchi Internätl. Symp. at Katata and Kyoto, 32 (1984), North-Holland), 271-306, Part I · Zbl 0568.60059 |
[7] | Revuz, D., Markov Chains (1975), North-Holland: North-Holland Amsterdam/Oxford/New York · Zbl 0332.60045 |
[8] | Stroock, D.; Varadhan, S. R.S, On the support of diffusion processes with applications to the strong maximum principle, (Proceedings, Sixth Berkeley Symp. on Math. Stat. and Prob., Vol. III (1970)), 333-360 · Zbl 0255.60056 |
[9] | Stroock, D.; Varadhan, S. R.S, Multidimensional Diffusion Processes, (Grundlehren Series No. 233 (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 1316.60124 |
[10] | Stroock, D., Introduction to the Theory of Large Deviations, (Universitext Series (1984), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg) · Zbl 0552.60022 |
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