Langevin equations for continuous time Lévy flights

Abstract

We consider the combined effects of a power law Lévy step distribution characterized by the step index $f$ and a power law waiting time distribution characterized by the time index $g$ on the long time behavior of a random walker. The main point of our analysis is a formulation in terms of coupled Langevin equations which allows in a natural way for the inclusion of external force fields. In the anomalous case for $f<2$ and $g<1$ the dynamic exponent $z$ locks onto the ratio $f/g$. Drawing on recent results on Lévy flights in the presence of a random force field we also find that this result is {\em independent} of the presence of weak quenched disorder. For $d$ below the critical dimension $d_c=2f-2$ the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.

10 pages, Latex, IFA Report No. 94/10

Cited by in corpus (117)

• Fractional calculus and continuous-time finance
• Lévy walks
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Levy Flight Superdiffusion: An Introduction
• Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
• Fractional Fokker--Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises
• First passage and arrival time densities for Lévy flights and the failure of the method of images
• Levy Anomalous Diffusion and Fractional Fokker--Planck Equation
• Levy flights from a continuous-time process
• Boundary problems for the fractional and tempered fractional operators
• Levy Flights in External Force Fields: From Models to Equations
• Markovian embedding of non-Markovian superdiffusion
• Joint Probability Distributions for a Class of Non-Markovian Processes
• On distributions of functionals of anomalous diffusion paths
• Continuous time random walk with correlated waiting times
• First-passage and first-hitting times of Levy flights and Levy walks
• Levy flights in quenched random force fields
• Fluctuation Analysis of Time-Averaged Mean-Square Displacement for Langevin Equation with Time-Dependent and Fluctuating Diffusivity
• Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories
• Generalised geometric Brownian motion: Theory and applications to option pricing
• Non-equilibrium Phase Transitions with Long-Range Interactions
• Fractional advection-diffusion-asymmetry equation
• Fractional dynamics from the ordinary Langevin equation
• Fluctuation relations for anomalous dynamics
• On relation between generalized diffusion equations and subordination schemes
• Subordinated Langevin Equations for Anomalous Diffusion in External Potentials - Biasing and Decoupled Forces
• Subordination Pathways to Fractional Diffusion
• Markovian embedding of fractional superdiffusion
• Anomalous processes with general waiting times: functionals and multi-point structure
• A fractional diffusion equation for two-point probability distributions of a continuous-time random walk
• Time transformation for random walks in the quenched trap model
• Subordinated diffusion and CTRW asymptotics
• Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times
• Particle Dispersion on Rapidly Folding Random Hetero-Polymers
• Non-Gaussian diffusion of mixed origins
• Lévy-walk-like Langevin dynamics
• Correlated continuous-time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics
• Weak subordination breaking for the quenched trap model
• Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
• Subdiffusion in an external force field
• Ergodic properties of heterogeneous diffusion processes in a potential well
• The Levy diffusion as an effect of sporadic randomness
• Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator
• Fractional Fokker-Planck subdiffusion in alternating fields
• Weak Galilean invariance as a selection principle for coarse-grained diffusive models
• Should I stay or should I go? Zero-size jumps in random walks for Lévy flights
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• Integro-differential diffusion equation for continuous time random walk
• Anomalous diffusion for overdamped particles driven by cross-correlated white noise sources
• Lévy walk dynamics in an external harmonic potential
• Generalized Central Limit Theorem and Renormalization Group
• A topological approximation of the nonlinear Anderson model
• A mixed SOC-turbulence model for nonlocal transport and Levy-fractional Fokker-Planck equation
• Nonlocal Diffusion Operators for Normal and Anomalous Dynamics
• Feynman-Kac equation revisited
• Epidemic spreading with long-range infections and incubation times
• Continuous Time Random Walks with Internal Dynamics and Subdiffusive Reaction-Diffusion Equations
• Onset of anomalous diffusion from local motion rules
• Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes
• Ergodic property of random diffusivity system with trapping events
• Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations
• Langevin formulation of a subdiffusive continuous time random walk in physical time
• A self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity
• Superstatistical approach of the anomalous exponent for scaled Brownian motion
• Lévy walk revisited: Hermite polynomial expansion approach
• Integral decomposition for the solutions of the generalized Cattaneo equation
• Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion
• Langevin dynamics for Lévy walk with memory
• On the fractional Poisson process and the discretized stable subordinator
• Langevin picture of Lévy walk in a constant force field
• Lévy noise-driven escape from arctan potential wells
• Two-point correlation function of the fractional Ornstein-Uhlenbeck process
• Continuous time multidimensional Markovian description of Levy walks
• Lévy flights in inhomogeneous environments and 1/f noise
• Anomalous diffusion in nonhomogeneous media: Time-subordinated Langevin equation approach
• From Quenched Disorder to Continuous Time Random Walk
• Lévy flights on a comb and the plasma staircase
• Classical intermittency and quantum Anderson transition
• Langevin picture of anomalous diffusion processes in expanding medium
• Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift
• Noisy voter model for the anomalous diffusion of parliamentary presence
• An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps
• Escape from bounded domains driven by multi-variate $\alpha$-stable noises
• Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Stationarity-conservation laws for certain linear fractional differential equations
• Space-time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model
• Non-Debye relaxations: The ups and downs of the stretched exponential vs Mittag-Leffler's matchings
• Nonequilibrium phase transition in a spreading process on a timeline
• Volterra-Prabhakar derivative of distributed order and some applications
• Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
• 1/f noise from point process and time-subordinated Langevin equations
• Escape from the potential well: competition between long jumps and long waiting times
• The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model
• Random diffusivity processes in an external force field
• Lévy walk dynamics in non-static media
• Non-Markovian Stochastic Liouville equation and its Markovian representation. Extensions of the continuous time random walk approach
• Competition between ac driving-forces and Lévy flights in a nonthermal ratchet
• Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps
• Fractional Klein-Gordon Equation on AdS${}_{2+1}$
• The Transient Case of The Quenched Trap Model
• Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach
• Dynamical response function of the disordered kinetic Ising model
• Fluctuation Effects in the Pair-Annihilation Process with Lévy Dynamics
• Simulation of the continuous-time random walk using subordination schemes
• Non-Debye relaxations: The characteristic exponent in the excess wings model
• Equivalent continuous and discrete realizations of Levy flights: Model of one-dimensional motion of inertial particle
• Effect of a magnetic flux on the critical behavior of a system with long range hopping
• Sub-diffusive behavior in the Standard Map
• The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations
• Active Lévy Matter: Anomalous Diffusion, Hydrodynamics and Linear Stability
• Driven particle in a cloud of mobile impurities
• Generalized Khinchin Theorem for a Class of Aging Processes
• A fractional generalized Cauchy process
• Drifted escape from the finite interval