The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

Obrezkov, O. O.

doi:10.1142/s0219025703001109

Cited by 11 publications

(4 citation statements)

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“…Recently, this method has been successfully applied to obtain Feynman formulae for different classes of problems for evolutionary equations on different geometric structures, see, e.g. [5]- [7], [26], [27], [32] and also to construct some surface measures on infinite dimensional manifolds (see [34]- [38]). This method is based on Chernoff's theorem (see [12] and [33] for the version used here), which is a generalization of the well-known Trotter formula.…”

Section: Introductionmentioning

confidence: 99%

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

Butko

Schilling

Smolyanov

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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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.

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Section: Introductionmentioning

confidence: 99%

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

Butko

Schilling

Smolyanov

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…Interpreting Eq (2.5) as the path integral quantization of the hamiltonian H with Remarks. This paper views Dσ as a volume form on W (M ) and approximates Wiener measure on the piecewise geodesic path space H P (M ); an approach which started with [3] and has further been explored by [1,2,4,5,6,7,11,13,14,19,29,32,33,34,35,36,37,42,43,45]. A wealth of literature pertaining to Eq.…”

Section: Resultsmentioning

confidence: 99%

An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature

Laetsch

2013

Journal of Functional Analysis

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Abstract. This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A L 2 Riemannian metric G P is given on the space of piecewise geodesic paths H P (M ) adapted to the partition P of [0, 1], whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as mesh(P) → 0, the approximate Wiener measure converges in a L 1 sense to the measure exp{− 2+ √ 3 20 √ 3 1 0 Scal(σ(s))ds}dν(σ) on the Wiener space W (M ) with Wiener measure ν. This gives a possible prescription for the path integral representation of the quantized Hamiltonian, as well as yielding such a result for the natural geometric approximation schemes originating in [3] and followed by [34].

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“…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].…”

Section: Feynman Formula Solving the Cauchy-dirichlet Problem For A Cmentioning

confidence: 99%

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

Butko

2018

FCAA

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Semigroups, generated by Feller processes killed upon leaving a given domain, are considered. These semigroups correspond to Cauchy-Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation converts into a Feynman formula, i.e. into a limit of n-fold iterated integrals of certain functions as n → ∞. Representations of solutions of evolution equations by Feynman formulae can be used for direct calculations and simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional (including distributed order time-fractional) evolution equations is suggested. This method is based on connections between time-fractional and time-nonfractional evolution equations as well as on Chernoff approximations for the latter ones. Moreover, this method leads to Feynman formulae for solutions of time-fractional evolution equations. To illustrate the method, a class of distributed order time-fractional diffusion equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy-Dirichlet problems are obtained.

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The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

Cited by 11 publications

References 1 publication

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

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

An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature

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

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