From non-Brownian Functionals to a Fractional Schrödinger Equation

Abstract

We derive backward and forward fractional Schrödinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.

5 pages

References in corpus (9)

• Fractional Quantum Mechanics
• Time fractional Schrodinger equation
• Stochastic Ergodicity Breaking: a Random Walk Approach
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• Weakly non-ergodic Statistical Physics
• Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories
• Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
• Investigation of a generalized Obukhov Model for Turbulence

Cited by in corpus (38)

• Lévy walks
• Boundary problems for the fractional and tempered fractional operators
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• On distributions of functionals of anomalous diffusion paths
• Tempered Fractional Feynman-Kac Equation
• A fractional Feynman-Kac equation for weak ergodicity breaking
• High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights
• Numerical algorithms for the forward and backward fractional Feynman-Kac equations
• Anomalous processes with general waiting times: functionals and multi-point structure
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• Numerical schemes of the time tempered fractional Feynman-Kac equation
• Branching exponential flights: travelled lengths and collision statistics
• Nonlocal Diffusion Operators for Normal and Anomalous Dynamics
• Cauchy's formulas for random walks in bounded domains
• Feynman-Kac equation revisited
• Fractional compound Poisson processes with multiple internal states
• Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data
• Transition in the decay rates of stationary distributions of Lévy motion in an energy landscape
• Strongly constrained stochastic processes: the multi-ends Brownian bridge
• Counting statistics: a Feynman-Kac perspective
• Semi-Markov processes in open quantum systems: Connections and applications in counting statistics
• Feynman-Kac Equations for Reaction and Diffusion Processes
• Testing Short Distance Anisotropy in Space
• High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Discrete Feynman-Kac formulas for branching random walks
• Random diffusivity processes in an external force field
• Aging Feynman-Kac Equation
• Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes
• Ergodic properties of functionals of Gaussian processes
• Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations
• Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation
• A fractional generalized Cauchy process
• Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation
• On the Sheffer-type polynomials related to the Mittag-Leffler functions: applications to fractional evolution equations
• A second-order difference scheme for the time fractional substantial diffusion equation