Spatial Interactions

Abstract

We continue our study of measure-valued processes which behave locally like (A, γ, g)-DW superprocesses, i.e., where A, γ, and g may depend on the current state, X_t, of the process. In this Chapter we allow the generator governing the spaticl motion, A, to depend on X_t. These results are taken from Perkins (1992), (1995). To simplify the exposition we set γ = 1 and g = 0, although as discussed below (in Section V.5) this restriction may be relaxed. Our approach may be used for a variety of dependencies of A_x, on X_t but we focus on the case of state dependent diffusion processes. Let

$$ \sigma :M_F (\mathbb{R}^d ) x \mathbb{R}^d \to \mathbb{R}^{dxd} , b: M_F (\mathbb{R}^d )x\mathbb{R}^d \to \mathbb{R}^d , a = \sigma \sigma ^* , $$

and set

$$ A_\mu \not o(x) = \sum\limits_i {\sum\limits_j {a_{ij} (\mu ,x)\not o_{ij} (x)} + \sum\limits_i {b_i (\mu ,x)\not o_i (x)} , for \not o \in C_b^2 (\mathbb{R}^d )} . $$