The Method of Chernoff Approximation

Chernoff approximations of Feller semigroups in Riemannian manifolds

“…This work in particular is devoted to the application of the Chernoff theorem (see Theorem 2.6) to the construction of an approximation formula for, on the one hand, the Feller semigroup and, on the other hand, the associated diffusion process and solutions to the evolution equation. This technique has been extensively implemented, for example, in the study of Chernoff approximations of Feller semigroups (and corresponding Feller processes) [9,10,12,13], in the construction of solutions to evolution equations [4,7,8], and in the construction of the Wiener measure on compact manifolds [3,57] (see for overviews [11,52,53]). Most of the results presented in literature are restricted to the case where either 𝑀 = ℝ 𝑑 or 𝑀 is compact.…”

“…They have continuous paths and can be constructed in terms of the (martingale) solution of stochastic differential equations of the form [19, 27, 28, 60] $$$$\begin{equation} \text{d}X(t)=\sum _{j=1}^r A_j(X(t))\circ \text{d}B^j(t) +A_0(X(t))\text{d}t. \end{equation}$$$$This work in particular is devoted to the application of the Chernoff theorem (see Theorem 2.6) to the construction of an approximation formula for, on the one hand, the Feller semigroup and, on the other hand, the associated diffusion process and solutions to the evolution equation. This technique has been extensively implemented, for example, in the study of Chernoff approximations of Feller semigroups (and corresponding Feller processes) [9, 10, 12, 13], in the construction of solutions to evolution equations [4, 7, 8], and in the construction of the Wiener measure on compact manifolds [3, 57] (see for overviews [11, 52, 53]). Most of the results presented in literature are restricted to the case where either $$M={\mathbb {R}}^d$$ or M is compact.…”

Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

“…(1.1) to give explicit representations of semigroups, see, e.g., [6,27,31]. For a survey of Chernoff approximations of operator semigroups we refer to [5], see also [17] for an overview on different kinds of approximations. A classical approach to nonlinear PDEs and the respective nonlinear semigroups is based on the theory of maximal monotone or maccretive operators, see [1,2,4,16,20].…”

Approximations of self-adjoint \(C_0\)-semigroups in the operator-norm topology

“…Instead of a double-iteration procedure of [7] we extend in this paper the Chernoff approximation formula [5] and the Trotter-Neveu-Kato approximation theorem [8], Theorem IX.2.16, to the operator-norm topology. Essentially we follow here the idea of lifting the strongly convergent Chernoff approximation formula to operator-norm convergence [9,11], whereas majority of results concerning this formula are about the strong operator topology, see, for example, review [2]. In the same vein we quote a recent book [1], where different aspects of semigroup convergence in the strong operator topology are presented in great details.…”