# The Poisson Equation and Estimates for Distances Between Stationary Distributions of Diffusions

The Poisson Equation and Estimates for Distances Between Stationary Distributions of Diffusions DOI 10.1007/s10958-018-3872-3 Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 THE POISSON EQUATION AND ESTIMATES FOR DISTANCES BETWEEN STATIONARY DISTRIBUTIONS OF DIFFUSIONS V. I. Bogachev Moscow State University, Moscow 119991, Russia National University Higher School of Economics 20, Myasnitskaya St., Moscow 101000, Russia St. Tikhon’s Orthodox Humanitarian University 23-5A, Novokuznetskaya St., Moscow 115184, Russia vibogach@mail.ru M. R¨ ockner Universit¨ at Bielefeld, Bielefeld 33501, Germany roeckner@math.uni-bielefeld.de S. V. Shaposhnikov Moscow State University, Moscow 119991, Russia National University Higher School of Economics 20, Myasnitskaya St., Moscow 101000, Russia St. Tikhon’s Orthodox Humanitarian University 23-5A, Novokuznetskaya St., Moscow 115184, Russia starticle@mail.ru UDC 517.95 We estimate distances between stationary solutions to Fokker–Planck–Kolmogorov equa- tions with diﬀerent diﬀusion and drift coeﬃcients. To this end we study the Poisson equation on the whole space. We have obtained suﬃcient conditions for stationary so- lutions to satisfy the Poincar´e and logarithmic Sobolev inequalities. Bibliography:35 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov 1 Introduction Let us consider two Borel probability measures μ and σ on R satisfying the stationary Fokker– ∗ ∗ ∗ ∗ Planck–Kolmogorov equations L μ =0 and L σ =0, where L and L are formally adjoint μ σ μ σ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Sciences Springer Journals

# The Poisson Equation and Estimates for Distances Between Stationary Distributions of Diffusions

, Volume OnlineFirst – Jun 2, 2018
29 pages

/lp/springer_journal/the-poisson-equation-and-estimates-for-distances-between-stationary-6kkzCH0kh1
Publisher
Springer US
Subject
Mathematics; Mathematics, general
ISSN
1072-3374
eISSN
1573-8795
D.O.I.
10.1007/s10958-018-3872-3
Publisher site
See Article on Publisher Site

### Abstract

DOI 10.1007/s10958-018-3872-3 Journal of Mathematical Sciences, Vol. 232, No. 3, July, 2018 THE POISSON EQUATION AND ESTIMATES FOR DISTANCES BETWEEN STATIONARY DISTRIBUTIONS OF DIFFUSIONS V. I. Bogachev Moscow State University, Moscow 119991, Russia National University Higher School of Economics 20, Myasnitskaya St., Moscow 101000, Russia St. Tikhon’s Orthodox Humanitarian University 23-5A, Novokuznetskaya St., Moscow 115184, Russia vibogach@mail.ru M. R¨ ockner Universit¨ at Bielefeld, Bielefeld 33501, Germany roeckner@math.uni-bielefeld.de S. V. Shaposhnikov Moscow State University, Moscow 119991, Russia National University Higher School of Economics 20, Myasnitskaya St., Moscow 101000, Russia St. Tikhon’s Orthodox Humanitarian University 23-5A, Novokuznetskaya St., Moscow 115184, Russia starticle@mail.ru UDC 517.95 We estimate distances between stationary solutions to Fokker–Planck–Kolmogorov equa- tions with diﬀerent diﬀusion and drift coeﬃcients. To this end we study the Poisson equation on the whole space. We have obtained suﬃcient conditions for stationary so- lutions to satisfy the Poincar´e and logarithmic Sobolev inequalities. Bibliography:35 titles. Dedicated to the memory of Vasilii Vasil’evich Zhikov 1 Introduction Let us consider two Borel probability measures μ and σ on R satisfying the stationary Fokker– ∗ ∗ ∗ ∗ Planck–Kolmogorov equations L μ =0 and L σ =0, where L and L are formally adjoint μ σ μ σ

### Journal

Journal of Mathematical SciencesSpringer Journals

Published: Jun 2, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations