Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions
Vladimir Igorevich Bogachev, Michael Röckner, Wilhelm Stannat · Sbornik Mathematics · 2002
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(ij)partial derivative(xi)partial derivative(xj) + b(i)partial derivative(xi) in local coordinates, where a(ij) is an element of H-loc(p,1), b(i) is an element of L-loc(p) for some p > d, and the matrix (a'j) 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: integralLphif dmu = 0 for all phi is an element of C-0(infinity)(M). In addition, the uniqueness of invariant probability measures for the corresponding semigroups (T-t(mu))tgreater than or equal to0 generated by the operator L is investigated. It is proved that if a probability measure it on M satisfies the equation L*mu = 0 and (L - I) (C-0(infinity)(M)) is dense in L-1 (M,p), then it is a unique solution of this equation in the class of probability measures. Examples are presented (even with a(ij) = delta(ij) and smooth b(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-t)(tgreater than or equal to0) generated by L has at most one invariant probability measure.