%0 Journal Article %1 bogachev2002uniqueness %A Bogachev, V I %A Rockner, M %A Stannat, W %D 2002 %J Sbornik: Mathematics %K PDE diffusions %N 7 %P 945 %T Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions %U http://stacks.iop.org/1064-5616/193/i=7/a=A01 %V 193 %X 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_x_i\partial_x_j+b^i\partial_x_i$ in local coordinates, where $ a^ijH^p,1_loc$, $ b^iL^p_loc$ for some $ p>d$, and the matrix $ (a^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: $ \displaystyleLf\,d\mu=0$ for all $ \varphiC_0^ınfty(M)$. In addition, the uniqueness of invariant probability measures for the corresponding semigroups $ (T_t^\mu)_t0$ 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)\big(C^ınfty_0(M)\big)$ is dense in $ L^1(M,\mu)$, then $ \mu$ 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)_t0$ generated by $ L$ has at most one invariant probability measure.

@article{bogachev2002uniqueness, abstract = {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_{x_i}\partial_{x_j}+b^i\partial_{x_i}$ in local coordinates, where $ a^{ij}\in H^{p,1}_{\text{loc}}$, $ b^i\in L^p_{\text{loc}}$ for some $ p>d$, and the matrix $ (a^{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: $ \displaystyle\int L\varphi f\,d\mu=0$ for all $ \varphi\in C_0^\infty(M)$. In addition, the uniqueness of invariant probability measures for the corresponding semigroups $ (T_t^\mu)_{t\geqslant 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)\big(C^\infty_0(M)\big)$ is dense in $ L^1(M,\mu)$, then $ \mu$ 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)_{t\geqslant 0}$ generated by $ L$ has at most one invariant probability measure. }, added-at = {2011-10-31T05:49:36.000+0100}, author = {Bogachev, V I and Rockner, M and Stannat, W}, biburl = {https://www.bibsonomy.org/bibtex/2e07af8c5bfbcfa00bd7f96393daa9ce5/peter.ralph}, interhash = {5cb92ae311a8c017e787d29d46a8ad6d}, intrahash = {e07af8c5bfbcfa00bd7f96393daa9ce5}, journal = {Sbornik: Mathematics}, keywords = {PDE diffusions}, number = 7, pages = 945, timestamp = {2011-10-31T05:49:36.000+0100}, title = {Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions}, url = {http://stacks.iop.org/1064-5616/193/i=7/a=A01}, volume = 193, year = 2002 }