Feynman-Kac equation revisited

Abstract

Functionals of particles' paths have diverse applications in physics, mathematics, hydrology, economics, and other fields. Under the framework of continuous time random walk (CTRW), the governing equations for the probability density functions (PDFs) of the functionals, including the ones of the paths of stochastic processes of normal diffusion, anomalous diffusion, and even the diffusion with reaction, have been derived. Sometimes, the stochastic processes in physics and chemistry are naturally described by Langevin equations. The Langevin picture has the advantages in studying the dynamics with an external force field and analyzing the effect of noise resulting from a fluctuating environment. We derive the governing equations of the PDFs of the functionals of paths of Langevin system with both space and time dependent force field and arbitrary multiplicative noise; and the backward version is proposed for the system with arbitrary additive noise or multiplicative Gaussian white noise together with a force field. For the newly built equations, their applications of solving the PDFs of the occupation time and area under the trajectory curve are provided, and the results are confirmed by simulations.

arXiv comment13 pages, 5 figures

References in corpus (30)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Langevin equations for continuous time Lévy flights
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Local fractional Fokker-Planck equation
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• Towards deterministic equations for Levy walks: the fractional material derivative
• From the area under the Bessel excursion to anomalous diffusion of cold atoms
• From non-Brownian Functionals to a Fractional Schrödinger Equation
• Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
• On distributions of functionals of anomalous diffusion paths
• Generalized Fokker-Planck equation: Derivation and exact solutions
• Levy flights in quenched random force fields
• Tempered Fractional Feynman-Kac Equation
• Generalized arcsine laws for fractional Brownian motion
• Hitting probability for anomalous diffusion processes
• A fractional Feynman-Kac equation for weak ergodicity breaking
• Functionals of the Brownian motion, localization and metric graphs
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Steady-State Lévy Flights in a Confined Domain
• Anomalous processes with general waiting times: functionals and multi-point structure
• Probability distributions for one component equations with multiplicative noise
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• Investigation of a generalized Obukhov Model for Turbulence
• Fractional compound Poisson processes with multiple internal states
• Exit and Occupation times for Brownian Motion on Graphs with General Drift and Diffusion Constant
• Feynman-Kac Equations for Reaction and Diffusion Processes
• Aging Feynman-Kac Equation

Cited by in corpus (8)

• Lévy-walk-like Langevin dynamics
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Subdiffusion in an external force field
• Nonlocal Diffusion Operators for Normal and Anomalous Dynamics
• Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative
• Random diffusivity processes in an external force field
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Statistical formulation of Onsager-Machlup variational principle