Motion in a Random Force Field

We consider the motion of a particle in a random isotropic force field. Assuming that the force field arises from a Poisson field in $\mathbb{R}^d$, $d \geq 4$, and the initial velocity of the particle is sufficiently large, we describe the asymptotic behavior of the particle

Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces

We prove a Miyadera-Voigt type perturbation theorem for strong Feller semigroups. Using this result, we prove well-posedness of the semilinear stochastic equation dX(t) = [AX(t) + F(X(t))]dt + GdW_H(t) on a separable Banach space E, assuming that F is bounded and measurable and that the associated linear equation, i.e. the equation with F = 0, is well-posed and its transition semigroup is strongly Feller and satisfies an appropriate gradient estimate. We also study existence and uniqueness of invariant measures for the associated transition semigroup.Comment: Revision based on the referee's comment

Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces

We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-and- time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.Comment: 5 page

Loop-Erasure of Plane Brownian Motion

We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.Comment: 10 page

Area Distribution of Elastic Brownian Motion

We calculate the excursion and meander area distributions of the elastic Brownian motion by using the self adjoint extension of the Hamiltonian of the free quantum particle on the half line. We also give some comments on the area of the Brownian motion bridge on the real line with the origin removed. We will stress on the power of self adjoint extension to investigate different possible boundary conditions for the stochastic processes.Comment: 18 pages, published versio

Feller property and infinitesimal generator of the exploration process

We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale

Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure

Many transport processes in nature exhibit anomalous diffusive properties with non-trivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multi-point structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.Comment: Accepted in Phys. Rev. Let

On exact time-averages of a massive Poisson particle

In this work we study, under the Stratonovich definition, the problem of the damped oscillatory massive particle subject to a heterogeneous Poisson noise characterised by a rate of events, \lambda (t), and a magnitude, \Phi, following an exponential distribution. We tackle the problem by performing exact time-averages over the noise in a similar way to previous works analysing the problem of the Brownian particle. From this procedure we obtain the long-term equilibrium distributions of position and velocity as well as analytical asymptotic expressions for the injection and dissipation of energy terms. Considerations on the emergence of stochastic resonance in this type of system are also set forth.Comment: 21 pages, 5 figures. To be published in Journal of Statistical Mechanics: Theory and Experimen

Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators $\Gamma_i$ (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times

Multiplicative decompositions and frequency of vanishing of nonnegative submartingales

In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y=N+A, where the measure dA is carried by the set of zeros of Y. In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio