Some Ergodic Theorems for a Parabolic Anderson Model

In this paper, we study some ergodic theorems of a class of linear systems of interacting diffusions, which is a parabolic Anderson model. First, under the assumption that the transition kernel a = （a（i,j））i,j∈s is doubly stochastic, we obtain the long-time convergence to an invariant probability measure Vh starting from a bounded a-harmonic function h based on self-duality property, and then we show the convergence to the invariant probability measure Uh holds for a broad class of initial distributions. Second, if （a（i, j））i,j∈s is transient and symmetric, and the diffusion parameter c remains below a threshold, we are able to determine the set of extremal invariant probability measures with finite second moment. Finally, in the case that the transition kernel （a（i,j））i,j∈s is doubly stochastic and satisfies Case I （see Case I in [Shiga, T.： An interacting system in population genetics. J. Math. Kyoto Univ., 20, 213-242 （1980）]）, we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.