Probability measure-valued polynomial diffusions

The Compact Support Property for Measure-valued Diffusions

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued diffusion processes corresponding to semi-linear equations of the form ut = Lu+ βu− αu p in R × (0,∞), p ∈ (1, 2]; u(x, 0) = f(x) in R; u(x, t) ≥ 0 in R × [0,∞). In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corres...

Hyperreal-Valued Probability Measures Approximating a Real-Valued Measure

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 n...

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ · ∇ + ν where ...

On dual processes of non-symmetric diffusions with measure-valued drifts

In this paper, we study properties of the dual process and Schrödinger-type operators of a non-symmetric diffusion with measure-valued drift. Let μ = (μ, . . . , μ) be such that each μ is a signed measure on R belonging to the Kato class Kd,1. A diffusion with drift μ is a diffusion process in R whose generator can be informally written as L + μ · ∇ where L is a uniformly elliptic differential ...