Chernoff's theorem for evolution families

Simple item page
dc.creatorShamarova, Evelina
dc.date2007-06-27
dc.date.accessioned2026-07-07T08:12:43Z
dc.date.available2026-07-07T08:12:43Z
dc.descriptionA generalized version of Chernoff's theorem has been obtained. Namely, the version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov, Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff's theorem for evolution families, deals with a family of time-dependent generators of semigroups $A_t$ on a Banach space, a two-parameter family of operators $Q_{t,t+Δt}$ satisfying the relation: $\frac{\partial}{\partial Δt}Q_{t,t+Δt}|_{Δt = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$ are uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$. The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges to an evolution family $U(s,t)$ solving a non-autonomous Cauchy problem. Furthermore, the theorem is formulated for a particular case when the generators $A_t$ are time dependent second order differential operators. Finally, an example of application of this theorem to a construction of time-inhomogeneous diffusions on a compact Riemannian manifold is given. Keywords: Chernoff's theorem, evolution family, strongly continuous semigroup, evolution families generated by manifold valued stochastic processes.
dc.identifierhttps://arxiv.org/abs/0706.4079
dc.identifierhttp://arxiv.org/abs/0706.4079
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/132553
dc.subjectFunctional Analysis
dc.subjectProbability
dc.subject47D06; 47D07
dc.titleChernoff's theorem for evolution families
dc.typetext

Files

Collections

arXiv