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 ν _h starting from a bounded a-harmonic function h based on self-duality property, and then we show the convergence to the invariant probability measure ν _h 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.