Perturbed semigroup limit theorems with applications to discontinuous random evolutions

Abstract

For ε > 0 \varepsilon > 0 small, let U ε ( t ) {U^\varepsilon }(t) and S ( t ) S(t) be strongly continuous semigroups of linear contractions on a Banach space L L with infinitesimal operators A ( ε ) A(\varepsilon ) and B B respectively, where A ( ε ) = A ( 1 ) + ε A ( 2 ) + o ( ) A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o() as ε → 0 \varepsilon \to 0 . Let { B ( u ) ; u ⩾ 0 } \{ B(u);u \geqslant 0\} be a family of linear operators on L L satisfying B ( ε ) = B + ε Π ( 1 ) + ε 2 Π ε ( 2 ) + o ( ε 2 ) B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2}) as ε → 0 \varepsilon \to 0 . Assume that A ( ε ) + ε − 1 B ( ) A(\varepsilon ) + {\varepsilon ^{ - 1}}B() is the infinitesimal operator of a strongly continuous contraction semigroup T ε ( t ) {T_\varepsilon }(t) on L L and that for each f ∈ L , lim λ → 0 λ ∫ 0 ∞ e − λ t S ( t ) f d t ≡ P f 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 ε ( t ) {T_\varepsilon }(t) converges as → 0 \to 0 to the semigroup generated by the closure of P ( A ( 1 ) + Π ( 1 ) ) P({A^{(1)}} + {\Pi ^{(1)}}) on R ( P ) ∩ D ( A ( 1 ) ) ∩ D ( Π ( 1 ) ) \mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}}) . If P ( A ( 1 ) + Π ( 1 ) ) f = 0 , B h = − ( A ( 1 ) + Π ( 1 ) ) f P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f , and we let V ^ f = P ( A ( 1 ) + Π ( 1 ) ) h \hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h , then we show that T ε ( t / ε ) f {T_\varepsilon }(t/\varepsilon )f converges as ε → 0 \varepsilon \to 0 to the strongly continuous contraction semigroup generated by the closure of V ( 2 ) + V ^ {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.