We want to sketch the support of a probability measure on Euclidean space from samples that have been drawn from the measure. This problem is closely related to certain manifold learning problems, where one assumes that the sample points are drawn from a manifold that is embedded in Euclidean space. Here we propose to sketch the support of the probability measure (that does not need to be a man...

Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to e...

We propose the model, which allows us to approximate fractional Levy noise and fractional Levy motion. Our model is based (i) on the Gnedenko limit theorem for an attraction basin of stable probability law, and (ii) on regarding fractional noise as the result of fractional integration/differentiation of a white Levy noise. We investigate self-affine properties of the approximation and conclude ...

Abstract. Fractional Lévy motion (fLm) is the natural generalization of fractional Brownian motion in the context of self-similar stochastic processes and stable probability distributions. In this paper we give an explicit derivation of the propagator of fLm by using path integral methods. The propagators of Brownian motion and fractional Brownian motion are recovered as particular cases. The f...

Fundamental solutions of space-time fractional diffusion equations can be interpret as probability density functions. This fact creates a strong link with stochastic processes. Recasting probability density functions in terms of subordination laws has emerged to be important to built up stochastic processes. In particular, for diffusion processes, subordination can be understood as a diffusive ...

Abstract. We study the long-time asymptotics of the probability Pt that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [−L, L] up to time t. We show that for any H ∈]0, 1], for both subdiffusion and superdiffusion regimes, this probability obeys ln(Pt) ∼ −t2H/L2, i.e. may decay slower than exponential (subdiffusion) or faster than expon...

We study the existence of mild solutions for semilinear fractional differential equations with nonlocal initial conditions in $L^p([0,1],E)$, where $E$ is a separable Banach space. The main ingredients used in the proof of our results are measure of noncompactness, Darbo and Schauder fixed point theorems. Finally, an application is proved to illustrate the results of this work. 

Consider a Markov chain {X n } n≥0 with an ergodic probability measure π. Let Ψ a function on the state space of the chain, with α-tails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α N n Ψ(X n) to a α-stable law. A " martingale approximation " approach and " coupling " approach give two different sets of condition...