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)) by πν(B) = ∫ M(Rd) η(B)dν(η), for B ⊂ R. 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 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 ∈ C1 b ([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. In this paper we prove a strong law of large numbers and a mixing result for the invariant probability distributions of certain spatially dependent measure-valued diffusions. Because of the spatial dependence, these distributions will not be stationary. Let M(Rd) denote the space of locally finite measures on R, equipped 1991 Mathematics Subject Classification. 60J60.