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

Let denote the space of locally finite measures on Rd and let denote the space of probability measures on . Define the mean measure [pi][nu] of byFor such a measure [nu] with locally finite mean measure [pi][nu], let f be a nonnegative, locally bounded test function satisfying <f,[pi][nu]>=[infinity]. [nu] is said to satisfy the strong law of large numbers with respect to f if <fn,[eta]>/<fn,[pi][nu]> converges almost surely to 1 with respect to [nu] as n-->[infinity], for any increasing sequence {fn} of compactly supported functions which converges to f. [nu] is said to be mixing with respect to two sequences of sets {An} and {Bn} ifconverges to 0 as n-->[infinity] for every pair of functions f,g[set membership, variant]Cb1([0,[infinity])). It is known that certain classes of measure-valued diffusion processes possess a family of invariant distributions. These distributions belong to and have locally finite mean measures. We prove the strong law of large numbers and mixing for many such distributions.