We consider the Laplacian generated by the Gaussian measure on a separable Hilbert space and prove the ergodic theorem for the corresponding one-parameter semigroup.
Similar content being viewed by others
References
Yu. V. Bogdanskii, “Laplacian with respect to a measure on a Hilbert space and an L2-version of the Dirichlet problem for the Poisson equation,” Ukr. Mat. Zh., 63, No. 9, 1169–1178 (2011); English translation : Ukr. Math. J., 63, No. 9, 1339–1348 (2012).
V. I. Bogachev, Differentiable Measures and Malliavin Calculus [in Russian], Regulyar. Khaotich. Dinam., Moscow (2008).
M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in a Hilbert Space [in Russian], Leningrad University, Leningrad (1980).
Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983).
Yu. V. Bogdanskii and Ya. Yu. Sanzharevskii, “The Dirichlet problem with Laplacian with respect to a measure in the Hilbert space,” Ukr. Mat. Zh., 66, No. 6, 733–739 (2014); English translation: Ukr. Math. J., 66, No. 6, 818–826 (2014).
A.V. Skorokhod, Integration in Hilbert Space [in Russian], Nauka, Moscow (1975).
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York (2000).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 9, pp. 1172–1180, September, 2015.
Rights and permissions
About this article
Cite this article
Bogdanskii, Y.V., Sanzharevskii, Y.Y. Laplacian Generated by the Gaussian Measure and Ergodic Theorem. Ukr Math J 67, 1316–1326 (2016). https://doi.org/10.1007/s11253-016-1155-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1155-z