Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

Abstract

This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number τ ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity (such representations are called Feynman formulae). Some of these representations are constructed with the help of another pseudo-differential operators, obtained by the same procedure of quantization (such representations are called Hamiltonian Feynman formulae). Some representations are based on integral operators with elementary kernels (these ones are called Lagrangian Feynman formulae and are suitable for computations). A family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented also as phase space Feynman path integrals with respect to these Feynman pseudomeasures. The obtained Lagrangian Feynman formulae allow to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.Keywords Feynman formulae; Phase space Feynman path integrals, Hamiltonian Feynman path integrals, symplectic Feynman path integrals, Feynman-Kac formulae, functional integrals; Hamiltonian (symplectic) Feynman pseudomeasure, Chernoff theorem, pseudo-differential operators, approximations of semigroups, 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

“…For example, this method has been used to investigate Schrödinger type evolution equations in [71,66,74,41,30,84,81,83]; stochastic Schrödinger type equations have been studied in [58,57,59,34]. 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].…”

“…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…”

“…for each ϕ ∈ Y , each x 0 ∈ G and each t > 0. The convergence in the Feynman formula (19) is locally uniform with respect to x 0 ∈ G and uniform with respect to t ∈ (0, t * ] for all t * > 0.…”

On numerical density approximations of solutions of SDEs with unbounded coefficients

“…To prove the convergence results in (ii), we use the Chernoff product formula which is a generalisation of the well-known Kato-Trotter product formula. Note that our approach to (ii) is similar to the approach for Feynman formulas in [8].…”