Chernoff's theorem for evolution families

TL;DR

This paper extends Chernoff's theorem to time-inhomogeneous evolution families, providing a framework for approximating solutions to non-autonomous Cauchy problems and applying it to diffusions on manifolds.

Contribution

The paper generalizes Chernoff's theorem for evolution families with time-dependent generators, including differential operators, and demonstrates its application to manifold-valued diffusions.

Findings

Convergence of operator products to an evolution family solving a non-autonomous Cauchy problem.

Extension of Chernoff's theorem to time-inhomogeneous cases.

Application to constructing time-inhomogeneous diffusions on Riemannian manifolds.

Abstract

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 $A_t$ on a Banach space, a two-parameter family of operators $Q_{t,t+\Delta t}$ satisfying the relation: $\frac{\partial}{\partial \Delta t}Q_{t,t+\Delta t}|_{\Delta t = 0}=A_t$, 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.