# The method of Chernoff approximation

## TL;DR

This survey explores Chernoff approximation methods for operator semigroups, connecting theoretical results with applications in stochastic processes, PDEs, and fractional evolution equations, highlighting recent advances and techniques.

## Contribution

It provides a comprehensive overview of Chernoff approximation techniques for various operator semigroups and their applications, including recent developments and methodological insights.

## Key findings

- Chernoff approximations effectively model operator semigroups.
- Techniques apply to Schrödinger groups and Feller semigroups.
- Approximations extend to solutions of time-fractional evolution equations.

## Abstract

This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes, numerical schemes for PDEs and SDEs, path integrals. We discuss Chernoff approximations for operator semigroups and Schr\"{o}dinger groups. In particular, we consider Feller semigroups in $\mathbb{R}^d$, (semi)groups obtained from some original (semi)groups by different procedures: additive perturbations of generators, multiplicative perturbations of generators (which sometimes corresponds to a random time-change of related stochastic processes), subordination of semigroups / processes, imposing boundary / external conditions (e.g., Dirichlet or Robin conditions), averaging of generators, "rotation" of semigroups. The developed techniques can be combined to approximate (semi)groups obtained via several iterative procedures listed above. Moreover, this method can be implemented to obtain approximations for solutions of some time-fractional evolution equations, although these solutions do not posess the semigroup property.