Lagrangian and Hamiltonian Feynman Formulae for Some Feller Semigroups and Their Perturbations

Abstract

Abstract. A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman-Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.Keywords Feynman formulae; Feynman-Kac formulae; approximations of functional integrals, approximations of transition densities.MSC 2010: 47D07, 47D08, 35C99, 60J35, 60J51, 60J60.

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

“…Analogous results hold true for distributed order time-fractional Fokker-Planck-Kolmogorov equations with non-local operators L considered in Subsection 3.2 and in [16,21] uniformly in x ∈ G and locally uniformly in t ∈ [0, ∞). Here the family (F o (t)) t≥0 has been constructed from the family (F (t)) t≥0 given in (17)…”

“…Second order parabolic equations related to diffusions in different geometrical structures (e.g., in Eucliean spaces and their subdomains, Riemannian manifolds and their subdomains, metric graphs, Hilbert spaces) have been studied, e.g., in [19,15,69,14,67,82,70,7,20,90,18,89,17,13,12,86,11,10,85,56]. Evolution equations with non-local operators generating some Markov processes in R d have been considered in [16,19,21,22]. Evolution equations with the Vladimirov operator (this operator is a p-adic analogue of the Laplace operator) have been investigated in [79,80,78,77,76].…”

“…Then, by Thm. 3.1 in [21] and Remark 15 in [19], the following family (F (t)) t≥0 on X is Chernoff equivalent to (T t ) t≥0 : F (0) = Id and for all t > 0, all ϕ ∈ X and all…”

“…To this aim, we need some preparations. 4 Note, that some families (F (t)) t≥0 , which are Chernoff equivalent to certain Feller semigroups, are constructed, e.g., in [16,21,22].…”

Chernoff approximation of subordinate semigroups

“…In this case the generator L η 0 of the corresponding semigroup S η 0 t is a bounded linear operator given as in (4). The generator L f0 of the semigroup (T f0 t ) t 0 subordinate to (T t ) t 0 with respect to (η 0 t ) t 0 is given by (9) and is also bounded. Therefore, the semigroup (T f0 t ) t 0 can be constructed, e.g., via Taylor series representation.…”