Representation of solutions of the stochastic Schrödinger equation in the form of a Feynman integral

Obrezkov, O. O.

doi:10.1007/s10958-008-0154-5

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…It is worth to mention that the method of Chernoff approximation has a wide range of applications. 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].…”

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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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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“…Recent works on the path integrals over the Lévy paths (e.g., [20]) lead to nonlocal Schrödinger equations. More physical investigations on fractional or nonlocal generalization of the Schrödinger equations may be found in, for example, [21,22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential

Li¹,

Yang²,

Duan³

2019

Preprint

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We consider the homogenization of a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. With help of a two-scale convergence technique, we establish a homogenization principle for this nonlocal stochastic partial differential equation. We explicitly derive the homogenized model. In particular, this homogenization principle holds when the nonlocal operator is the fractional Laplacian.

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“…• averaging of semigroups (see Sec. 2.7, [10,57,10,9]); Moreover, Chernoff approximations have been obtained for some stochastic Schrödinger type equations in [55,54,56,37]; for evolution equations with the Vladimirov operator (this operator is a p-adic analogue of the Laplace operator) in [68,69,67,66,65]; for evolution equations containing Lévy Laplacians in [2,1]; for some nonlinear equations in [58].…”

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confidence: 99%

The method of Chernoff approximation

Butko

2019

Preprint

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This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes, numerical schemes for PDEs and SDEs, path integrals. We discuss Chernoff approximations for operator semigroups and Schrödinger groups. In particular, we consider Feller semigroups in R d , (semi)groups obtained from some original (semi)groups by different procedures: additive perturbations of generators, multiplicative perturbations of generators (which sometimes corresponds to a random time-change of related stochastic processes), subordination of semigroups / processes, imposing boundary / external conditions (e.g., Dirichlet or Robin conditions), averaging of generators, "rotation" of semigroups. The developed techniques can be combined to approximate (semi)groups obtained via several iterative procedures listed above. Moreover, this method can be implemented to obtain approximations for solutions of some time-fractional evolution equations, although these solutions do not posess the semigroup property.

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Representation of solutions of the stochastic Schrödinger equation in the form of a Feynman integral

Cited by 4 publications

References 15 publications

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

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

Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential

The method of Chernoff approximation

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