We consider a stochastic boundary value elliptic problem on a bounded domain $D\subset \mathbb{R}^k$, driven by a fractional Brownian field with Hurst parameter $H=(H_1,...,H_k)\in[{1/2},1[^k$. First we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove existence and uniqueness of solution, along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.
In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the result available on fractional Poisson processes in several directions.
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse $α$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto in \cite{sfpp}, is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NSTFPP). We compute their {\it pmf} and generating function and investigate the associated differential equation. The limit theorems and the law of iterated logarithm for the NSTFPP process are studied. We study the distributional proper…
Dynamical scaling is an asymptotic property typical for the dynamics of first-order phase transitions in physical systems and related to self-similarity. Based on the integral-representation for the marginal probabilities of a fractional non-homogeneous Poisson process introduced by Leonenko et al. (2017) and generalising the standard fractional Poisson process, we prove the dynamical scaling under fairly mild conditions. Our result also includes the special case of the standard fractional Poisson process.
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.
We prove that bounded harmonic functions of anisotropic fractional Laplacians are Hölder continuous under mild regularity assumptions on the corresponding Lévy measure. Under some stronger assumptions the Green function, Poisson kernel and the harmonic functions are even differentiable of order up to three.
We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha
The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study non-homogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with non-stationary increments), denoted by H. Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by us…
We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.
In this paper, we show that the mixed fractional Poisson process (MFPP) exhibits the long-range dependence (LRD) property. It is proved by establishing an asymptotic result for the covariance of inverse mixed stable subordinator. Also, it is shown that the increments of the MFPP, namely, the mixed fractional Poissonian noise (MFPN) has the short-range dependence (SRD) property.
We obtain the state probabilities of various fractional versions of the classical homogeneous Poisson process using an alternate and simpler method known as the Adomian decomposition method (ADM). Generally these state probabilities are obtained by evaluating probability generating function using Laplace transform. A generalization of the space and time fractional Poisson process involving the Caputo type Saigo differential operator is introduced and its state probabilities are obtained using ADM.
We introduce two non-homogeneous processes: a fractional non-homogeneous Poisson process of order $k$ and and a fractional non-homogeneous Pólya-Aeppli process of order $k$. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.
In this paper we localize some of Watanabe's results on fractional Wiener functionals, and use them to give a precise estimate of the difference between two Donsker's delta functionals even with fractional differentiability. As an application, the convergence rate of the density of the Euler scheme for non-Markovian stochastic differential equations is obtained.
We introduce a non-homogeneous fractional Poisson process by replacing the time variable in the fractional Poisson process of renewal type with an appropriate function of time. We characterize the resulting process by deriving its non-local governing equation. We further compute the first and second moments of the process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from the specialization of the non-homogeneous process.
We consider a fractional counting process with jumps of amplitude $1,2,\ldots,k$, with $k\in \mathbb{N}$, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same dis…
In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^α(t)$, $t>0$, $α\in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -λ^α(1-B)p_k^α(t)$, where $(1-B)^α$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^α(t)$, the probability generating functions $G_α(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\ uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the m…
We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities.
In this paper we study finite velocity planar random motions with an infinite number of possible directions, where the number of changes of direction is randomized by means of an inhomogeneous fractional Poisson distribution. We first discuss the properties of the distributions of the generalized fractional inhomogeneous Poisson process. Then we show that the explicit probability law of the planar random motions where the number of changes of direction is governed by this fractional distribution can be obtained in terms of Mittag-Leffler functions. We also consider planar random motions with r…
We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.
We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in …
In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.
We study the long-range dependence (LRD) of the increments of the fractional Poisson process (FPP), the fractional negative binomial process (FNBP) and the increments of the FNBP. We first point out an error in the proof of Theorem 1 of Biard and Saussereau (2014) and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index $β$ satisfies $0<β<\frac{1}{3}$. We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.
The term "moderate deviations" is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with integer IID interarrival times which converge in well-scaled continuous-time limits to the Prabhakar-generalized fractional Poisson process. These processes exhibit non-Markovian features and long-time memory effects. We recover for special choices of para…
We consider some fractional extensions of the recursive differential equation governing the Poisson process, by introducing combinations of different fractional time-derivatives. We show that the so-called "Generalized Mittag-Leffler functions" (introduced by Prabhakar [20]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t tending to infinite. On the other hand, near the origin the behavior of the law of the interarri…
In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc.…
The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{ν_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<ν_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{ν_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $ν_k$ differs from that constructed from the fractional state equations (in the case $ν_k = ν$, for all $k$, they coincide with the time-fra…
We consider a system of $d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle $S^1$. We obtain sharp results on the Hölder continuity in time of the paths of the solution $u=\{u(t, x)\}_{t \in \mathbb{R}_+, x \in S^1}$. We then establish upper and lower bounds on hitting probabilities of $u$, in terms of respectively Hausdorff measure and Newtonian capacity.
In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_α(t)$, $N_β(t)$, $t>0$, we show that $N_α(N_β(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_β(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns c…
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson process, where the Poisson counting process is replaced by the Mittag-Leffler counting process also known as fractional Poisson process. This process is no longer Markovian and Lévy. However, several analytical results are available and some of them are discussed here. The functional…
In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.
We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.
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.
In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poiss…
We present a method that allows, under suitable equivariance and regularity conditions, to determine the Poisson boundary of a diffusion starting from the Poisson boundary of a sub-diffusion of the original one. We then give two examples of application of this dévissage method. Namely, we first recover the classical result that the Poisson boundary of Brownian motion on a rotationally symmetric manifolds is generated by its escape angle, and we then give an "elementary" probabilistic proof of the delicate result of [Bai08], i.e. the determination of the Poisson boundary of the relativistic Bro…
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 process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and dist…
We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space $\mathbb{R}^{1+1}$ with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.
In this paper exponential stability of nonlinear fractional order stochastic system with Poisson jumps is studied in finite dimensional space. Existence and uniqueness of solution, stability and exponential stability results are established by using boundedness properties of Mittag-Leffler matrix function, fixed point route and local assumptions on nonlinear terms. A numerical example is given to illustrate the efficiency of the obtained results. Finally, conclusion is drawn.
In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator ${\cal L}$ and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel $q(t, x, y)$, which we call the fundamental solution for the time fractional Poisson equation, if the semigrou for ${\cal L}$ has an…
In this note we consider a class of neutral stochastic functional differential equations with finite delay driven simultaneously by a fractional Brownian motion and a Poisson point processes in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle.
The Mittag-Leffler function $E_α$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_α(ϕ) :=E_α(\int (e^{iϕ(x)}-1)dμ(x)) \] which we prove to fulfill all requirements of the Bochner-Minlos theorem. The identity of the support of this new measure with the support of the infinite-dimensional Poisson measure ($α=1$) allows the development of a fractional infinite-dimensional analysis modeled on Poisson analysis through the combinatorial harmonic analysis on configuration spaces. …
The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a Lévy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These …
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 Kolmogorov-Feller equations for the probabilities at time t can be representated by an infinite linear system of ordinary differential equations of first order in a transformed time variable. These new equations resemble a linear version of the discrete coagulation-fragmentation equations, well-known from the non-equilibrium theory of gelation, cluster-dynam…
Generalizations of Bell polynomials, Bell numbers, and Stirling numbers of the second kind have been introduced and their generating functions were evaluated.
