Integrable solutions of the stationary Kolmogorov equation
type: Item
instance of: scholarly article
title: Integrable solutions of the stationary Kolmogorov equation
zbMATH Open document ID: 1260.60133
DOI: 10.1134/S1064562412030027
author: A. I. Kirillov
author: Vladimir I. Bogachev
author: S. V. Shaposhnikov
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Consider the second-order elliptic operator \[ L = \sum_{1 \leq i,j \leq d} a^{ij} \partial_{x_i} \partial_{x_j} + \sum_{1 \leq i \leq d} b^i \partial_{x_i} \] and the corresponding stationary Kolmogorov equation (SKE) \[ L^{\star} \mu = 0, \] where \(L^{\star}\) is the (formal) adjoint operator of \(L\) and \(\mu\) is a bounded (i.e., finite) signed measure. In this short but interesting paper, the authors give a (sufficient) condition under which the nontrivial solutions of the SKE are of constant sign and unique up to a constant factor (hence, under their condition, there is a unique solution which is a probability measure). Since their condition (see Theorem 1) cannot be easily verified, they also give two broader conditions, which employ Lyapunov functions. Furthermore, the paper contains some interesting examples and recalls some relevant known results.
reviewed by: Vassilis G. Papanicolaou
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35R06
zbMATH DE Number: 6106449
zbMATH Keywords: stationary Kolmogorov equation
zbMATH Keywords: weak solution
zbMATH Keywords: bounded signed measure
zbMATH Keywords: probability measure
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
cites work: ON REGULARITY OF TRANSITION PROBABILITIES AND INVARIANT MEASURES OF SINGULAR DIFFUSIONS UNDER MINIMAL CONDITIONS
cites work: Elliptic and parabolic equations for measures
cites work: Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions
cites work: The nonuniqueness of solutions to elliptic equations for probability measures
cites work: On nonuniqueness of solutions to elliptic equations for probability measures
cites work: On probability and integrable solutions to the stationary Kolmogorov equation
cites work: A new topological approach to the \(L^{\infty }\)-uniqueness of operators and the \(L^{1}\)-uniqueness of Fokker--Planck equations
cites work: Q3525987
cites work: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators
cites work: Elliptic equations for invariant measures on finite and infinite dimensional manifolds
cites work: STOCHASTICALLY COMPLETE MANIFOLDS AND SUMMABLE HARMONIC FUNCTIONS
cites work: \(L^ p\) and mean value properties of subharmonic functions on Riemannian manifolds
full work available at URL: s1064562412030027
OpenAlex ID: W1998221207
description: scientific article; zbMATH DE number 6106449