Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a Riemannian manifold

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

Schilling

Smolyanov

2012

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.

Section: Feller and Lévy Semigroups And Their Generatorsmentioning

confidence: 99%

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

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

Schilling

Smolyanov

2012

“…Then the extension E U satisfies Assumption 3.1. The condition (10) actually means that the process (ξ t ) t≥0 is allowed to leave the domain G either continuously, or by a sufficiently large jump which brings the process even out of U . Note that if N (x, dy) corresponds to censored processes (i.e., N (x, dy) satisfies ( 5)), then the condition (10) is fulfilled.…”

Section: Distributed-order Fractional Derivativesmentioning

confidence: 99%

“…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: Introductionmentioning

confidence: 99%

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

2018

FCAA

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.

“…One can check (see [2]), that these families of operators are Chernoff equivalent to the semigroup e tA . Therefore, by the Chernoff theorem we obtain the following statement.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let functions V and f 0 belong to the class A. Suppose that the Cauchy problem(2) for the Schroedinger equation with the potential V and the initial condition f 0 has in K the unique solution f (t, z), which belongs to the class A 2 . Then f (t, x) can be represented by a functional integral over the Wienermeasure W x K : f (t, x) (x+ √ i(ξ(τ )−x))dτ f 0 (x + √ i(ξ(t) − x))W x K (dξ);by a functional integral over the external Smolyanov surface measure S x,E K :f (t, x) = c E (t, x) (x+ √ i(ξ(τ )−x))dτ f 0 (x + √ i(ξ(t) − x))S x,E K (dξ),(c E (t, x)) by a functional integral over the internal Smolyanov surface measure S x,I K :f (t, x) = c I (t, (x+ √ i(ξ(τ )−x))dτ f 0 (x + √ i(ξ(t) − x))S x,I K (dξ),(c I (t, x))…”

mentioning

confidence: 99%

Smolyanov surface measures and solutions of Schroedinger and heat type equations on a Riemannian manifold

DAYS on DIFFRACTION 2006

2006

In the paper we give solutions of Schroedinger and heat type equations on a Riemannian manifold in the form of functional integrals over Smolyanov surface measures on the set of trajectories in a manifold and over the Wiener measure, generated by Brownian motion in the manifold. These integrals are in fact limits of finite-dimensional integrals (over Cartesian products of some copies of the manifold) of elementary functions containing coefficients of equations, initial conditions and geometric characteristics of the manifold. In the proof, a substantial role is played by Smolyanov-Weizsaecker-Wittich asymptotic estimates for Gaussian integrals over a manifold, by the Chernoff theorem and by the method of transition from the Schroedinger to the heat equation going back to Doss.