Semi-Markov Models and Motion in Heterogeneous Media

Abstract

Abstract. 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 diffusions in heterogeneous media.

Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

“…In this situation, the space-dependent variable-order model is more suitable to describe location-dependent diffusion processes, see e.g. [3][4][5][6][7][8][9][10][11]. We mention also the existence of time-dependent variable-order models when diffusion behaviour changes with the time evolution [12][13][14][15][16] and some other recent interesting applications of fractional calculus [17][18][19][20].…”

A Semi-Markov Leaky Integrate-and-Fire Model

“…Equations (17) and (18) give us the fact that the involved integrals are finite, hence we can split the integral in two parts and then use the aforementioned equations.…”

“…We considered only a simple linear model, i.e., the Leaky Integrate-and-Fire model, to give an easy example of application of this fractionalization procedure and how such procedure produces both a weighted covariance structure that, together with the non-Markov property, gives us correlation of the spiking times, and a delay in the firing activity due to the introduction of a sort of stochastic clock. We refer to fractionalization procedure since this random time-change of Markov processes introduces semi-Markov processes governed by fractional equations (see, for example [14][15][16][17]), which are very popular in applications, and thus we establish a connection of our model with fractional equations. This procedure can be adapted to various processes.…”

Goodness-of-fit test for multistable Lévy processes

“…The study of its generator and some related dierential equations for multistable Markov processes are also major subjects (see e.g. Beghin and Ricciuti 2018;Orsingher, Ricciuti, and Toaldo 2016;Ricciuti and Toaldo 2017).…”