Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

Abstract

A Feynman formula is a representation of the semigroup, generated by an initialboundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.In the present paper we formulate and prove the Hamiltonian Feynman and FeynmanKac formulae for heat semigroups describing the diffusion of particles with the position-

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].…”

“…Since ε > 0 was chosen arbitrary, the statement follows. (20). Assume that the semigroup (T t ) t≥0 , whose generator L stands in the right hand side of the equation, corresponds to a Markov process (ξ(t)) t≥0 .…”

Contractivity and ground state domination properties for non-local Schrödinger operators

“…In many interesting cases the specific models involve non-local Schrödinger operators based on generators of Lévy processes with jumps. Recent investigations include heat trace and spectral gap estimates [1,5,21], gradient estimates of harmonic functions [31], properties of radial solutions, ground states, eigenfunctions and eigenvalues [32,33,23,13,26,35,18], smoothing properties of evolution semigroups [25,9,22], properties of the associated transformed jump processes [24,28], as well as applications in quantum theory [34,17,16,19,15,3].…”

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

“…Later on, evolution semigroups e −t H have been treated by the same approach in papers [12], [13], [6]. In [12] the identity (1.1) has been established for the case of qp-quantization of a function H(q, p), which corresponds to a particle with variable mass in a potential field. The semigroup e −t H has been considered on the Banach space C ∞ (R d ) of continuous, vanishing at infinity functions.…”

“…The semigroup e −t H (again on C ∞ (R d )) generated by τ -quantization of a function H(q, p), which is polynomial with respect to p with variable, depending on q coefficients, has been approximated in [6] by a family of pseudo-differential operators with some qp-symbols. The obtained Hamiltonian Feynman formula has been interpreted as a phase space Feynman path integral with respect to the Feynman pseudomeasure defined in [12].…”