Luciana Angiuli, Giorgio Metafune and Chiara Spina

Feller semigroups and invariant measures

Pages: 347-406
Received: 1 March 2010   
Accepted: 14 May 2010
Mathematics Subject Classification (2010): 35JXX, 35KXX, 47D07, 60JXX.

Keywords: Markov processes, elliptic operators with unbounded coefficients, invariant measures.

Abstract: This paper is devoted to present both classical and recent results on Markov semigroups associated with elliptic second order operators with (possibly unbounded) coefficients, using analytic methods. We consider second order elliptic operators like $A=\sum_{i,j=1}^{N}a_{ij}D_{ij}u+ \sum_{i=1}^{N}b_i D_i u+ cu$ defined in $\R^N$; we study the invariant measures associated with the semigroup $\{T(t)\}$ generated by $A$ in $C_b(\R^N)$ and the regularity properties of $\{T(t)\}$ in $L^p$-spaces related to this measure. A concrete example of an elliptic operator with unbounded coefficients in $\R^N$ is given by the Ornstein-Uhlenbeck operator whose main properties are also presented.