Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes
Mark Iosifovich Freidlin, Alexander D. Wentzell · Birkhäuser Boston eBooks · 1994
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 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 f the semigroup corresponding to X T f(x) = E x f(X ), 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 as ε ↓ 0 to a continuous Markow process X t with the Feller property is equivalent to the convergence of the semigroups T to the semigroups corresponding to X t .