The Rate of Convergence of Chernoff Approximations to Operator Semigroups and Approximate Solution of Differential Equations

We will discuss a (surprisingly simple) proof of a theorem on the rate of convergence of Chernoff approximations to a strongly continuous operator semigroup exp(tL) parameterized by a non-negative real number t. A theorem similar to this has been sought for over 50 years, since the publication of Chernoff's theorem in 1968. Colleagues from all over the world used a wide range of tools of functional analysis to find it, but a result was only recently obtained, and with very simple techniques.
The one-dimensional analogue of Chernoff's theorem states the following: if S is a real-valued function of a real variable, S(0) = 1, S'(0) = L, then for every real number t, the numbers (S(t/n))^n tend to exp(tL) as n tends to infinity. This simple fact is easily proven using the "second remarkable limit theorem" from a course in mathematical analysis. The infinite-dimensional version of this statement is called Chernoff's theorem. In it, L is a closed, densely defined linear operator on a Banach space, exp(tL) is a C0-semigroup of operators with generator L, and S is called the Chernoff function for L. Thus, to approximately find a semigroup, it suffices to find at least one Chernoff function for the semigroup's generator. This simplifies the problem, as it is much easier to find a Chernoff function than, for example, the generator's resolvent. This is because if an operator is the generator of a semigroup, then this semigroup is unique, and this operator's resolvent is also uniwue. However, there are always many Chernoff functions for the generator, so finding one of the Chernoff functions is easier than finding a unique semigroup or a unique resolvent. Given this Chernoff function, we can use Chernoff's theorem to first obtain a semigroup and then a resolvent — since the resolvent is obtained from the semigroup using the Laplace transform.
Chernoff's original theorem doesn't specify any properties of the Chernoff function that influence the rate of convergence of Chernoff approximations as n tends to infinity, and even in the one-dimensional case, this is a nontrivial problem. The general solution to this problem will be discussed in the talk.
We will also discuss how to use Chernoff approximations to find the resolvent of a semigroup generator, and how to use it to find solutions to differential equations with variable coefficients — ordinary and elliptic partial differential equations.
Note that the theory of operator semigroups initially arose from the need to express solutions to linear evolution partial differential equations (parabolic and Schrödinger) in the language of operator theory. Chernoff's theorem allows, in many cases, to express arbitrarily accurate approximations to the solution of the Cauchy problem in terms of the coefficients of these equations and the initial condition, as well as to mathematically justify the correctness of representing the solution as a Feynman integral. Many works and results in this area are attributed to the distinguished professor of Moscow State University Oleg Georgievich Smolyanov, the teacher of the report's co-authors, who introduced them to this topic.