Related papers: Renewal processes linked to fractional relaxation …

Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse,…

We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit…

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for…

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic…

Digital System Research has pioneered the mathematics and design for a new class of computing machine using residue numbers. Unlike prior art, the new breakthrough provides methods and apparatus for general purpose computation using several…

In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. (2016). For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the…

We relate the convergence of time-changed processes driven by fractional equations to the convergence of corresponding Dirichlet forms. The fractional equations we dealt with are obtained by considering a general fractional operator in…

We propose Fractional Policy Gradients (FPG), a reinforcement learning framework incorporating fractional calculus for long-term temporal modeling in policy optimization. Standard policy gradient approaches face limitations from Markovian…

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population…

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…

The general scheme for the treatment of relaxation processes and temporal autocorrelations of dynamical variables for many particle systems is presented in framework of the recurrence relations approach. The time autocorrelation functions…

For a generalized step reinforced random walk, starting from the origin, the first step is taken according to the first element of an innovation sequence. Then in subsequent epochs, it recalls a past epoch with probability proportional to a…

Non-linear renewal theory is extended to include random walks perturbed by both a slowly changing sequence and a stationary one. Main results include a version of the Key Renewal Theorem, a derivation of the limiting distribution of the…

We consider a time fractional differential equation of order $\alpha$, $0<\alpha<1$, $$ \frac{\partial c(x,t)}{\partial t}={}^C_0\mathcal{D}_t^{\alpha}[(Ac)(x,t)]+q(x,t) ,\quad x > 0, t > 0, \quad c(x,0)=f(x). $$ where…

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out of equilibrium condition, the former being based on a stationary although very slow correlation function and…

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW) model for diffusion, including the possibility of anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to…

We study the process of suitably normalized successive return times to rare events in the setting of infinite-measure preserving dynamical systems. Specifically, we consider small neighborhoods of points whose measure tends to zero. We…

We present a unified approach to those observables of stochastic processes under reset that take the form of averages of functionals depending on the most recent renewal period. We derive solutions for the observables, and determine the…

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…

This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It…