2007/06/27 by Evelina Shamarova, Shamarova, Evelina
Computer Science · Mathematics · #47D06 #47D07 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #advanced mathematical theories #math.FA #math.PR #msc:47D06 #msc:47D07
paper · pdf · doi:10.48550/arxiv.0706.4079
arxiv created 2007/06/27 · openalex publication_date 2007/06/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 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 At on a Banach space, a two-parameter family of operators Qt,t+Δt satisfying the relation: (∂)/(∂ Δt)Qt,t+Δt|Δt = 0=At, whose products Q_ti,ti+1... Q_tk-1,tk are uniformly bounded for all subpartitions s = t0 < t1 < >... < tn = t. The theorem states that Qt0,t1... Q_tn-1,tn 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 At 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.