Time fractional equations and probabilistic representation
TLDR
In this article, the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts, were studied.Abstract:
In this paper, we study the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts. An explicit relation between occupation measure for Markov processes time-changed by inverse subordinator in open sets and that of the original Markov process in the open set is also given.read more
Citations
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Some Compactness Criteria for Weak Solutions of Time Fractional PDEs
Lei Li,Jian-Guo Liu +1 more
TL;DR: In this article, the Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs, and the authors aim to develop some compactness criteria that are an...
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A generalized definition of Caputo derivatives and its application to fractional ODEs
Lei Li,Jian-Guo Liu +1 more
TL;DR: The definition of Caputo derivatives of order in $(0,1)$ is extended to a certain class of locally integrable functions using a convolution group and allowed to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which enables the general Gronwall inequality with the most general conditions.
Journal ArticleDOI
Heat kernel estimates for time fractional equations
TL;DR: In this paper, the authors established existence and uniqueness of weak solutions to general time fractional equations and gave their probabilistic representations, and derived sharp two-sided estimates for fundamental solutions of a family of time fractionsal equations in metric measure spaces.
Posted Content
Some compactness criteria for weak solutions of time fractional PDEs
Lei Li,Jian-Guo Liu +1 more
TL;DR: This work develops some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs and establishes some time regularity estimates of the functions provided that the weak Caputo derivatives are in certain spaces.
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Semi-Markov Models and Motion in Heterogeneous Media
Costantino Ricciuti,Bruno Toaldo +1 more
TL;DR: In this article, the authors studied continuous time random walks such that the holding time in each state has a distribution depending on the state itself, and provided integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel.
References
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Journal ArticleDOI
The random walk's guide to anomalous diffusion: a fractional dynamics approach
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
Journal ArticleDOI
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes as mentioned in this paper, and a large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker-Planck equation.
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Stochastic Models for Fractional Calculus
TL;DR: In this article, the traditional diffusion model was extended to the vector fractional diffusion model, which is the state-of-the-art diffusion model for the problem of diffusion.
Journal ArticleDOI
Limit theorems for continuous-time random walks with infinite mean waiting times
TL;DR: In this article, the scaling limit of a continuous-time random walk is shown to be an operator Levy motion subordinated to the hitting time process of a classical stable subordinator.
Book
Fractional calculus an introduction for physicists
TL;DR: Fractional Derivatives Friction Forces Fractional Calculus The Fraction Harmonic Oscillator Wave Equations and Parity Nonlocality and Memory Effects Quantum Mechanics Fractionals Spin Factorization Symmetries The Fractal Symmetric Rigid Rotor Fraction Spectroscopy of Hadrons Higher Dimensional Fractionally Rotation Groups
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