Chernoff's theorem for evolution families
Evelina Shamarova

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 on a Banach space, a two-parameter family of operators satisfying the relation: , whose products are uniformly bounded for all subpartitions . The theorem states that converges to an evolution family solving a non-autonomous Cauchy problem. Furthermore, the theorem is formulated for a particular case when the…
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Taxonomy
Topicsadvanced mathematical theories · Advanced Mathematical Modeling in Engineering · Differential Equations and Boundary Problems
