Non-Markovian diffusion equations and processes: analysis and simulations

Abstract

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.

43 pages, 19 figures, in press on Physica A (2008)

References in corpus (14)

• Fractional calculus and continuous-time finance
• The fundamental solution of the space-time fractional diffusion equation
• Fractional calculus and continuous-time finance II: the waiting-time distribution
• Analytical properties and applications of the Wright function
• Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
• Continuous time random walk and parametric subordination in fractional diffusion
• CTRW Pathways to the Fractional Diffusion Equation
• Brownian subordinators and fractional Cauchy problems
• On solutions of a class of non-Markovian Fokker-Planck equations
• Fractional Calculus as a Macroscopic Manifestation of Randomness
• Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories
• Mellin transform and subordination laws in fractional diffusion processes
• Non-Markovian Levy diffusion in nonhomogeneous media
• Evolution, its Fractional Extension and Generalization

Cited by in corpus (20)

• Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion
• Stochastic calculus for uncoupled continuous-time random walks
• Fractional kinetics emerging from ergodicity breaking in random media
• Generalised geometric Brownian motion: Theory and applications to option pricing
• Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion
• Subdiffusion equation with Caputo fractional derivative with respect to another function
• Universal spectral features of different classes of random diffusivity processes
• Fractional diffusions with time-varying coefficients
• Is Fickian Yet Non-Gaussian Diffusion Ubiquitous?
• On the kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts
• Non-Gaussian displacement distributions in models of heterogeneous active particle dynamics
• Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles
• Stochastic solutions of generalized time-fractional evolution equations
• From Random Times to Fractional Kinetics
• Singularity of Generalized Grey Brownian Motion and Time-Changed Brownian Motion
• Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
• The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations
• Fractional Ito Calculus for Randomly Scaled Fractional Brownian Motion and its Applications to Evolution Equations
• Path-averaged Kinetic Equation for Stochastic Systems
• Cesaro Limits for Fractional Dynamics