The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an $α$-stable law with a constant intensity $\al$, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this paper we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes.
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law
Continuous Time Random Walk(CTRW) is a model where particle's jumps in space are coupled with waiting times before each jump. A Continuous Time Random Walk Limit(CTRWL) is obtained by a limit procedure on a CTRW and can be used to model anomalous diffusion. The distribution $p\left(dx,t\right)$ of a CTRWL $X_{t}$ satisfies a Fractional Fokker-Planck Equation(FFPE). Since CTRWLs are usually not Markovian, their one dimensional FFPE is not enough to completely define them. In this paper we find the FFPEs of the distribution of $X_{t}$ at multiple times , i.e. the distribution of the random vecto…
In this paper we study continuous time random walks (CTRWs) such that the holding time in each state has a distribution depending on the state itself. For such processes, we provide integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel. Particular attention is devoted to the case where the holding times have a power-law decaying density, whose exponent depends on the state itself, which leads to variable order fractional equations. A suitable limit yields a variable order fractional heat equation, which models anomalous diffu…
In this paper continuous time random walk models approximating fractional space-time diffusion processes are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is a Lévy's stable subordinator with the stability index $β\in (0,1).$ In the parer the convergence of constructed CTRWs to time-changed processes associated with the corresponding fractional diffusion equations are proved using a new analytic method.
We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian …
It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.
Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered.
We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^α- \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$α\in (1/2, 1)\cup(1, 2);$$ $\textit{B}$ is an uniform elliptic operator with variable coefficients and $\dot W$ is a Gaussian noise general in time with space covariance given by fractional, Riesz and Bessel kernel.
In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sob…
We study fractional smoothness of measures on $\mathbb{R}^k$, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.
