Strong law of large numbers and mixing for the invariant distributions of measure-valued diffusions

Let M(Rd) denote the space of locally finite measures on Rd and let M1(M(Rd)) denote the space of probability measures on M(Rd). Define the mean measure πν of ν∈M1(M(Rd)) byπν(B)=∫M(Rd) η(B) dν(η), for B⊂Rd.For such a measure ν with locally finite mean measure πν, let f be a nonnegative, locally bounded test function satisfying 〈f,πν〉=∞. ν is said to satisfy the strong law of large numbers with respect to f if 〈fn,η〉/〈fn,πν〉 converges almost surely to 1 with respect to ν as n→∞, for any increasing sequence {fn} of compactly supported functions which converges to f. ν is said to be mixing with respect to two sequences of sets {An} and {Bn} if∫M(Rd) f(η(An))g(η(Bn)) dν(η)−∫M(Rd) f(η(An)) dν(η)∫M(Rd) g(η(Bn)) dν(η)converges to 0 as n→∞ for every pair of functions f,g∈Cb1([0,∞)). It is known that certain classes of measure-valued diffusion processes possess a family of invariant distributions. These distributions belong to M1(M(Rd)) and have locally finite mean measures. We prove the strong law of large numbers and mixing for many such distributions.