On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Albeverio S, Bogachev V, Röckner M (1999)
Communications on Pure and Applied Mathematics 52(3): 325-362.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We prove uniqueness of "invariant measures," i.e., solutions to the equation L*mu = 0 where L = Delta + B . del on R-n with B satisfying some mild integrability conditions and mu being a probability measure on Rn. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L-1(mu) generates a strongly continuous semigroup having mu as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L-1(mu) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the "symmetric case") in particular is studied and conditions are identified ensuring that L*mu = 0 implies that L is symmetric on L-2(mu) Or L*mu = 0 has a unique solution. We also prove infinite-dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. (C) 1999 John Wiley & Sons, Inc.
Erscheinungsjahr
Zeitschriftentitel
Communications on Pure and Applied Mathematics
Band
52
Ausgabe
3
Seite(n)
325-362
ISSN
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Albeverio S, Bogachev V, Röckner M. On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Communications on Pure and Applied Mathematics . 1999;52(3):325-362.
Albeverio, S., Bogachev, V., & Röckner, M. (1999). On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Communications on Pure and Applied Mathematics , 52(3), 325-362. doi:10.1002/(SICI)1097-0312(199903)52:3<325::AID-CPA2>3.0.CO;2-V
Albeverio, S., Bogachev, V., and Röckner, M. (1999). On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Communications on Pure and Applied Mathematics 52, 325-362.
Albeverio, S., Bogachev, V., & Röckner, M., 1999. On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Communications on Pure and Applied Mathematics , 52(3), p 325-362.
S. Albeverio, V. Bogachev, and M. Röckner, “On uniqueness of invariant measures for finite- and infinite-dimensional diffusions”, Communications on Pure and Applied Mathematics , vol. 52, 1999, pp. 325-362.
Albeverio, S., Bogachev, V., Röckner, M.: On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Communications on Pure and Applied Mathematics . 52, 325-362 (1999).
Albeverio, Sergio, Bogachev, Vladimir, and Röckner, Michael. “On uniqueness of invariant measures for finite- and infinite-dimensional diffusions”. Communications on Pure and Applied Mathematics 52.3 (1999): 325-362.