Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions

Journal Article · Sat Aug 31 04:00:00 EDT 2002 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2002V193N07ABEH000665; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21205707

Bogachev, V I ^[1]; Rockner, M; Stannat, W ^[2]

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1. M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
2. Universitat Bielefeld, Bielefeld (Germany)

Let M be a complete connected Riemannian manifold of dimension d and let L be a second order elliptic operator on M that has a representation L=a{sup ij}{partial_derivative}{sub x{sub i}}{partial_derivative}{sub x{sub j}}+b{sup i}{partial_derivative}{sub x{sub i}} in local coordinates, where a{sup ij} element of H{sub loc}{sup p,1}, b{sup i} element of L{sub loc}{sup p} for some p>d, and the matrix (a{sup ij}) is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation L*{mu}=0 for probability measures {mu}, which is understood in the weak sense: {integral}L{phi}f d{mu}=0 for all {phi} element of C{sub 0}{sup {infinity}}(M). In addition, the uniqueness of invariant probability measures for the corresponding semigroups (T{sub t}{sup {mu}}){sub t{>=}}{sub 0} generated by the operator L is investigated. It is proved that if a probability measure {mu} on M satisfies the equation L*{mu}=0 and (L-I)C{sub o}{sup {infinity}}(M)) is dense in L{sup 1}(M,{mu}), then {mu} is a unique solution of this equation in the class of probability measures. Examples are presented (even with a{sup ij}={delta}{sup ij} and smooth b{sup i}) in which the equation L*{mu}=0 has more than one solution in the class of probability measures. Finally, it is shown that if p>d+2, then the semigroup (T{sub t}){sub t{>=}}{sub 0} generated by L has at most one invariant probability measure.

• OSTI ID: 21205707
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 7 Vol. 193; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COORDINATES
DIFFUSION
EQUATIONS
MATHEMATICAL SOLUTIONS
MATRICES
PROBABILITY