Stochastic solutions of generalized time-fractional evolution equations

Abstract

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Saigo-Maeda generalized time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

References in corpus (15)

• "Diffusing diffusivity": A model for anomalous and "anomalous yet Brownian" diffusion
• Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
• Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion
• Characterizations and simulations of a class of stochastic processes to model anomalous diffusion
• Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics
• Non-Markovian diffusion equations and processes: analysis and simulations
• Stochastic calculus for uncoupled continuous-time random walks
• On the complete monotonicity of the three parameter generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma }(-x)$
• Fractional diffusions with time-varying coefficients
• Phase-space consistency of stellar dynamical models determined by separable augmented densities
• Integral Representation of Generalized Grey Brownian Motion
• Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles
• Time-changed fractional Ornstein-Uhlenbeck process
• Singularity of generalized grey Brownian motions with different parameters
• Form Factors for Generalized Grey Brownian Motion

Cited by in corpus (2)

• Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations
• Fractional Ito Calculus for Randomly Scaled Fractional Brownian Motion and its Applications to Evolution Equations