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Journal articles on the topic 'Generalized Feynman-Kac formula'

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Published: 1 February 2025

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1

CHEN, CHUAN-ZHONG, ZHI-MING MA, and WEI SUN. "ON GIRSANOV AND GENERALIZED FEYNMAN–KAC TRANSFORMATIONS FOR SYMMETRIC MARKOV PROCESSES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 02 (June 2007): 141–63. http://dx.doi.org/10.1142/s0219025707002671.

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Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form [Formula: see text]. For [Formula: see text], the extended Dirichlet space, we have the classical Fukushima's decomposition: [Formula: see text], where [Formula: see text] is a quasi-continuous version of u, [Formula: see text] the martingale part and [Formula: see text] the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by [Formula: see text] and the generalized Feynman–Kac transform induced by [Formula: see text]. For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process [Formula: see text]. Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process [Formula: see text]. For the generalized Feynman–Kac transform, we give a necessary and sufficient condition for the generalized Feynman–Kac semigroup to be strongly continuous.

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2

OUERDIANE, HABIB, and JOSÉ LUIS SILVA. "GENERALIZED FEYNMAN–KAC FORMULA WITH STOCHASTIC POTENTIAL." Infinite Dimensional Analysis, Quantum Probability and Related Topics 05, no. 02 (June 2002): 243–55. http://dx.doi.org/10.1142/s0219025702000808.

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In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space [Formula: see text].

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3

Ettaieb, Aymen, Narjess Turki Khalifa, and Habib Ouerdiane. "Quantum white noise Feynman–Kac formula." Random Operators and Stochastic Equations 26, no. 2 (June 1, 2018): 75–87. http://dx.doi.org/10.1515/rose-2018-0007.

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Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

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4

Herzog, Bodo. "Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion." Mathematics 10, no. 3 (January 23, 2022): 340. http://dx.doi.org/10.3390/math10030340.

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The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

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5

Pardoux, Etienne, and Aurel Răşcanu. "Continuity of the Feynman–Kac formula for a generalized parabolic equation." Stochastics 89, no. 5 (January 16, 2017): 726–52. http://dx.doi.org/10.1080/17442508.2016.1276911.

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6

HIROSHIMA, FUMIO, TAKASHI ICHINOSE, and JÓZSEF LŐRINCZI. "PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN." Reviews in Mathematical Physics 24, no. 06 (June 17, 2012): 1250013. http://dx.doi.org/10.1142/s0129055x12500134.

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Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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7

Sun, Hui, and Yangyang Lyu. "Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions." AIMS Mathematics 9, no. 12 (2024): 34838–62. https://doi.org/10.3934/math.20241659.

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<p>In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.</p>

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8

Caffarel, Michel, and Pierre Claverie. "Treatment of the Schrödinger equation through a Monte Carlo method based upon the generalized Feynman-Kac formula." Journal of Statistical Physics 43, no. 5-6 (June 1986): 797–801. http://dx.doi.org/10.1007/bf02628305.

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9

Caffarel, Michel, and Pierre Claverie. "Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism." Journal of Chemical Physics 88, no. 2 (January 15, 1988): 1088–99. http://dx.doi.org/10.1063/1.454227.

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10

Caffarel, Michel, and Pierre Claverie. "Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. II. Applications to simple systems." Journal of Chemical Physics 88, no. 2 (January 15, 1988): 1100–1109. http://dx.doi.org/10.1063/1.454228.

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11

Zhao, Huaizhong. "The travelling wave fronts of nonlinear reaction–diffusion systems via Friedlin's stochastic approaches." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 2 (1994): 273–99. http://dx.doi.org/10.1017/s030821050002847x.

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In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using then-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

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12

ROGERS, ALICE. "SUPERSYMMETRY AND BROWNIAN MOTION ON SUPERMANIFOLDS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, supp01 (September 2003): 83–102. http://dx.doi.org/10.1142/s0219025703001225.

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An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-Itô formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. After a discussion of the BFV approach to the quantization of theories with symmetry, it is shown how the quantization of the topological particle leads to the supersymmetric model introduced by Witten in his study of Morse theory.

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13

Wang, Hanxiao. "Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula." Stochastics and Dynamics 21, no. 01 (March 11, 2020): 2150004. http://dx.doi.org/10.1142/s0219493721500040.

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This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

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14

Akhtari, Bahar, Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller. "Generalized Feynman–Kac formula under volatility uncertainty." Stochastic Processes and their Applications, December 2022. http://dx.doi.org/10.1016/j.spa.2022.12.003.

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15

Lyu, Yangyang, and Hui Sun. "Spatial Hölder continuity for the parabolic Anderson model with the singular initial conditions." Journal of Mathematical Physics 65, no. 11 (November 1, 2024). http://dx.doi.org/10.1063/5.0172994.

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Under the singular initial conditions, we consider the parabolic Anderson model driven by the time-independent generalized Gaussian fields, which include some frequently-used non-stationary Gaussian fields. When the initial value u0 belongs to the weighted Besov space with negative regularity Bq,∞−κ,wλ, it is proved that there exists a unique pathwise solution for the model in the Young sense. Moreover, if u0 also satisfies the measure-valued initial condition, by the Feynman-Kac formula based on Brownian bridge, we find that the solution owns a spatially Hölder continuous modification.

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16

Datta, Sumita. "Quantum simulation of non-Born–Oppenheimer dynamics in molecular systems by path integrals." International Journal of Modern Physics B, August 19, 2023. http://dx.doi.org/10.1142/s0217979224503132.

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A numerical algorithm based on the probabilistic path integral approach for solving Schrödinger equation has been devised to treat molecular systems without Born–Oppenheimer approximation in the nonrelativistic limit at zero temperature as an alternative to conventional variational and perturbation methods. Using high-quality variational trial functions and path integral method based on generalized Feynman–Kac method, we have been able to calculate the non-Born–Oppenheimer energy for hydrogen molecule for the [Formula: see text] state and hydrogen molecular ion. Combining these values and the value for ionization potential for atomic hydrogen, the dissociation energy and ionization potential for hydrogen molecules have been determined to be 36 113.672(3)[Formula: see text]cm[Formula: see text] and 124 446.066(10)[Formula: see text]cm[Formula: see text], respectively. Our results favorably compare with other theoretical and experimental results and thus show the promise of being a nonperturbative alternative for testing fundamental physical theories.

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17

"Approximate travelling waves for generalized KPP equations and classical mechanics." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 446, no. 1928 (September 8, 1994): 529–54. http://dx.doi.org/10.1098/rspa.1994.0119.

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We consider the existence of approximate travelling waves of generalized KPP equations in which the initial distribution can depend on a small parameter μ which in the limit μ → 0 is the sum of some δ -functions or a step function. Using the method of Elworthy & Truman (1982) we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears. By the Feynman–Kac formula and the Maruyama–Girsanov–Cameron–Martin transformation we obtain an identity from which, with a late caustic assumption, we see the propagation of the global wave front and the shape of the trough. Our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave. Finally, we prove a Huygens principle for KPP equations on complete riemannian manifolds without cut locus, with some bounds on their volume element, in particular Cartan–Hadamard manifolds.

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18

Datta, Sumita. "Computing quantum correlation functions by importance Sampling method based on path integrals." International Journal of Modern Physics B, September 26, 2022. http://dx.doi.org/10.1142/s0217979223500248.

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In this paper, an importance sampling method based on the Generalized Feynman–Kac (GFK) method has been used to calculate the mean values of quantum observables from quantum correlation functions for many-body systems with the Born–Oppenheimer approximation in the nonrelativistic limit both at zero and finite temperature. Specifically, the expectation values [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for the ground state of the lithium and beryllium and the density matrix, the partition function, the internal energy and the specific heat of a system of quantum harmonic oscillators are computed, in good agreement with the best nonrelativistic values for these quantities. Although the initial results are encouraging, more experimentation will be needed to improve the other existing numerical results beyond chemical accuracies specially for the last two properties for lithium and beryllium. Also more work needs to be done to improve the trial functions for finite temperature calculations. Although these results look promising, more work needs to be done to achieve the spectroscopic accuracy at zero temperature and to estimate the finite temperature effects from the non-Born–Oppenheimer calculations. Also more experimentation will be needed to study the convergence criteria for the inverse properties for atoms at zero temperature.

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19

Bender, Christian, Marie Bormann, and Yana A. Butko. "Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations." Fractional Calculus and Applied Analysis, August 19, 2022. http://dx.doi.org/10.1007/s13540-022-00082-8.

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AbstractWe consider a class of generalized time-fractional evolution equations containing a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

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20

Bressloff, Paul C. "Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator." Journal of Physics A: Mathematical and Theoretical, September 25, 2023. http://dx.doi.org/10.1088/1751-8121/acfcf3.

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Abstract In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time $\ell(t)$. If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density $\psi(\ell)$. The marginal probability density for particle position $\X(t)$ prior to absorption depends on $\psi$ and the joint probability density for the pair $(\X(t),\ell(t))$, also known as the local time propagator. In the case of normal diffusion one can use a Feynman-Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW).We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at $n=0$. We derive an evolution equation for the joint probability density of the particle location $N(t)\in \{n\in {\mathbb Z},n\geq 0\}$ and the amount of time $\chi(t)$ spent at the origin. The continuum limit involves rescaling $\chi(t)$ by a factor $1/\Delta x$, where $\Delta x$ is the lattice spacing. In the limit $\Delta x \rightarrow 0$, the rescaled functional $\chi(t)$ becomes the Brownian local time at $x=0$. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval $[0,L]$ with a reflecting boundary at $x=L$. In particular, we determine how the choice of function $\psi$ affects the large-$t$ power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

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