We consider a discrete-time Markovian random walk with resets on connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent renewal process. Some nodes of network target and we focus statistics first hitting these nodes. In non-Markov case process, both light- fat-tailed inter-reset distributions. derive propagator matrix terms discrete backward recurrence time PDFs light-tailed show existence non-equilibrium steady...

In this paper we study some properties of the Prabhakar integrals and derivatives their extensions such as regularized derivative or Hilfer-Prabhakar derivative.Some Opial-and Hardy-type inequalities are derived.In last section point out on relationships with probability theory.

In this paper we introduce a novel Mittag--Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then the utility of these results solve equations by means operational methods. show advantage our approach through examples. Among these, an application modified Lamb--Bateman integral equation is presented.

We consider a fractional version of the classical nonlinear birth process which Yule–Furry model is particular case. Fractionality obtained by replacing first order time derivative in difference-differential equations govern probability law with Dzherbashyan–Caputo derivative. derive distribution number $\mathcal{N}_{\nu}(t)$ individuals at an arbitrary $t$. also present interesting representation for $t$, form subordination relation $\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))$, where...

We present the stochastic solution to a generalized fractional partial differential equation (fPDE) involving regularized operator related so-called Prabhakar and admitting as specific cases, among others, diffusion telegraph equation. The is expressed Lévy process time-changed with inverse linear combination of (possibly subordinated) independent stable subordinators different indices. Furthermore (SDE) derived discussed.

In this paper, we introduce and examine a fractional linear birth–death process $N_ν(t), t>0$, whose fractionality is obtained by replacing the time derivative with in system of difference-differential equations governing state probabilities $p_k^ν(t), t>0, k≥0$. We present subordination relationship connecting classical $N(t), means $T_{2ν}(t), distribution related to time-fractional diffusion equation. obtain explicit formulas for extinction probability $p_0^ν(t)$ k≥1$, three relevant...

The space-fractional Poisson process is a time-changed homogeneous where the time change an independent stable subordinator. In this paper, further generalization discussed that preserves Lévy property. We introduce generalized by suitably time-changing superposition of weighted processes. This can be related to specific subordinator for which it possible explicitly write characterizing measure. Connections are highlighted Prabhakar derivatives, convolution-type integral operators. Finally,...

In this paper we analyse the fractional Poisson process where state probabilities p k ν ( t ), ≥ 0, are governed by time-fractional equations of order 0 < ≤ 1 depending on number events that have occurred up to time . We able obtain explicitly Laplace transform ) and various representations probabilities. show with intermediate waiting times differs from constructed (in case = ν, for all , they coincide process). also introduce a different form state-dependent as weighted sum homogeneous...

This paper is concerned with the fractionalized diffusion equations governing law of fractional Brownian motion BH(t). We obtain solutions these which are probability laws extending that Our analysis based on McBride operators generalizing hyper-Bessel L and converting their power Lα into Erdélyi–Kober integrals. study also probabilistic properties random variables whose distributions satisfy space-time involving Caputo Riesz derivatives. Some results emerging from time-varying coefficients...

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of derivatives in connection a recently introduced generalization Le Roy function. We also able to prove general fractional hyper-Bessel-type Hadamard operators functions.

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This decays as power law exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for expected intensity and number events counting process. These are used validate algorithms that numerically invert Laplace transform well Monte Carlo simulations Finally, derive full distribution events. The this paper available at {\tt...

Real-world networks may exhibit detachment phenomenon determined by the cancelling of previously existing connections. We discuss a tractable extension Yule model to account for this feature. Analytical results are derived and discussed both asymptotically finite number links. Comparison with original is performed in supercritical case. The first-order asymptotic tail behavior two models similar but differences arise second-order term. explicitly refer World Wide Web modeling we show...

<p style='text-indent:20px;'>In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov hitting times independent subordinators. Such are governed integro-differential convolution equations generalized fractional type. The aim this paper is to develop discrete-time counterpart such theory and show relationships differences with respect continuous time case. We present chains which can be constructed as time-changed we obtain...