Chernoff’s theorem for evolution families

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, Weizsäcker, and Wittich [1] 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+∆t satisfying the relation: ∂ ∂∆tQt,t+∆t ∣∣ ∆t=0 = At, whose products Qti,ti+1 . . . Qtk−1,tk are uniformly bounded for all subpartitions s = t0 < t1 < · · · < tn = t. The theorem states that Qt0,t1 . . . Qtn−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.