Perturbed semigroup limit theorems with applications to discontinuous random evolutions

Abstract:

For $\varepsilon > 0$ small, let ${U^\varepsilon }(t)$ and $S(t)$ be strongly continuous semigroups of linear contractions on a Banach space $L$ with infinitesimal operators $A(\varepsilon )$ and $B$ respectively, where $A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()$ as $\varepsilon \to 0$. Let $\{ B(u);u \geqslant 0\}$ be a family of linear operators on $L$ satisfying $B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})$ as $\varepsilon \to 0$. Assume that $A(\varepsilon ) + {\varepsilon ^{ - 1}}B()$ is the infinitesimal operator of a strongly continuous contraction semigroup ${T_\varepsilon }(t)$ on $L$ and that for each $f \in L,{\lim _{\lambda \to 0}}\lambda \int _0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf$ exists. We give conditions under which ${T_\varepsilon }(t)$ converges as $\to 0$ to the semigroup generated by the closure of $P({A^{(1)}} + {\Pi ^{(1)}})$ on $\mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}})$. If $P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f$, and we let $\hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h$, then we show that ${T_\varepsilon }(t/\varepsilon )f$ converges as $\varepsilon \to 0$ to the strongly continuous contraction semigroup generated by the closure of ${V^{(2)}} + \hat V$. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.