Emergence of Freidlin-Wentzell's transmission conditions as a result of a singular perturbation of a semigroup
The semigroup setup is as follows. The operators \(\{A_n\}\) are infinitesimal generators of equibounded semigroups \(\{e^{t A_n}: t \geq 0\}\) in a Banach space \(X\) and \(Q\) generates a strongly continuous semigroup \(\{e^{tQ}: t \geq 0\}\) such that \[ \lim_{t \to \infty} e^{tQ}x = Px \quad (x \in X) \] exists (it follows that \(P\) is a bounded idempotent operator). Let \(A\) be an operator in \(X,\) \(D \subset D(X)\) such that for every \(x \in D\) there exists \(\{x_n\},\) \(x_n \in D(A_n)\), with \[ \lim_{n \to \infty} x_n = x \, , \quad \lim_{n \to \infty} A_n x_n = Ax \quad (x \in X). \] A result due to \textit{T. G. Kurtz} [J. Funct. Anal. 12, 55--67 (1973; Zbl 0246.47053)] shows that, under various additional conditions relating the \(A_n\) and \(Q\), we have \[ \lim _{n \to \infty} e^{tA_n}x = e^{t \overline{PA}}Px \quad (x \in X, \;t > 0). \] This result is applied to a sequence of diffusions which was one of the main motivations for an averaging principle due to \textit{M. I. Freidlin} and \textit{A. D. Wentzell} [Ann. Probab. 21, No. 4, 2215--2245 (1993; Zbl 0795.60042)].
differential operators with transmission conditions
Freidlin-Wentzell's transmission conditions
scientific article; zbMATH DE number 6581725
scientific article