Seminar 6. Chernoff Approximations of Resolvents

This session continues the discussion on the connection between semigroups and equations in Banach spaces, as well as applications to solving partial differential equations in finite-dimensional spaces. The informal meaning of the Chernoff theorem will be presented as an infinite-dimensional analogue of the fundamental limit from elementary calculus. It will be proven that the Laplace transform of the Chernoff approximations of a semigroup converges to the resolvent of the semigroup's generator. As a corollary, Chernoff approximations for solutions to differential equations with variable coefficients – both ordinary and elliptic partial differential equations – will be derived.