ABSTRACT
In this chapter, we make use of the Green function constructed in Chapter 4 to study the invariant measure of the semigroup generated by the integrodifferential operator L− I and the boundary operator B, when the coefficients are independent of t and a0 = 0. In fact, we can prove that a unique positive Ho¨lder continuous functionm exists such that the probability measure µ := mdx can be interpreted as the invariant measure of the corresponding Markov-Feller process. This, together with the crucial ergodic property, allows us to study the asymptotic behavior of the stationary problem when the zero-order coefficient λ vanishes. Moreover, we can obtain the solution of the elliptic problem by studying the asymptotic behavior of the solution of the parabolic problem when t goes to infinity. This is based on our paper Garroni and Menaldi [43].
