相关论文: Random-time processes governed by differential equ…
In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations…
We analyze here different forms of fractional relaxation equations of order {\nu}\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing…
We study here a heat-type differential equation of order n greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess…
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson…
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is…
This paper introduces a discrete-time fractional Poisson process defined as a renewal process, where the waiting times follow a discrete Mittag-Leffler distribution. We investigate its fundamental properties by explicitly deriving the…
In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{N}, whose distribution is related to space-time fractional n-dimensional…
The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$ of events…
This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial…
This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is…
Fractional Poisson processes, a rapidly growing area of non-Markovian stochastic processes, are useful in statistics to describe data from counting processes when waiting times are not exponentially distributed. We show that the fractional…
The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of…
We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which,…
In this paper, we introduce and examine a fractional linear birth--death process $N_{\nu}(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential…
In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^\alpha(t)$, $t>0$, $\alpha \in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -\lambda^\alpha (1-B)p_k^\alpha(t)$, where…
Given a stochastic structure with a filtration $\mathbb{F}$, the class of all random times whose conditional distribution functions are differentiable with respect to some $\mathbb{F}$ adapted non decreasing processes is considered. The…
We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing…
In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0…
The aim of this paper is to analyze a class of random motions which models the motion of a particle on the real line with random velocity and subject to the action of the friction. The speed randomly changes when a Poissonian event occurs.…
We consider stochastic dynamical systems defined by differential equations with a uniform random time delay. The latter equations are shown to be equivalent to deterministic higher-order differential equations: for an $n$-th order equation…