Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes

Abstract

Let ${L^\varepsilon } = \Sigma _{i,j = 1}^r{a^{ij,\varepsilon }}(x)\frac{{{\partial ^2}}}{{\partial {x^i}\partial {x^j}}} + \Sigma _{i = 1}^r{b^{i,\varepsilon }}(x)\frac{d}{{d{x^i}}}$ be a family of elliptic operators depending on a positive parameter ε. Denote by X ^ε _t the diffusion process in Re^r governed by L ^ε. Such a process exists, at least, if the coefficients a ^ij,ε(x), b ^i,ε(x) are regular enough functions of x ∈ Re^r. Denote by T ^ε _t f the semigroup corresponding to X ^ε _t T ^ε _t f(x) = E _x f(X ^ε _t ), with f belonging to the space Ĉ(Re^r) of continuous functions on Re^r vanishing as |x| → ∞ provided with the uniform topology. It is known (see [EK], Ch. 4, Th. 2.5) that the weak convergence of the processes X ^ε _t as ε ↓ 0 to a continuous Markow process X _t with the Feller property is equivalent to the convergence of the semigroups T ^ε _t to the semigroups corresponding to X _t .

This author was supported in part by ARO Grant DAAL03-92-G-0219.