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 Rer 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 ∈ Rer. 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(Rer) of continuous functions on Rer 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.
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© 1994 Springer Science+Business Media New York
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Freidlin, M.I., Wentzell, A.D. (1994). Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes. In: Freidlin, M.I. (eds) The Dynkin Festschrift. Progress in Probability, vol 34. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0279-0_4
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DOI: https://doi.org/10.1007/978-1-4612-0279-0_4
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