Fractional Stokes-Boussinesq-Langevin equationand Mittag-Leffler correlation decay

his paper presents some stationary processes which are solutions of the fractional Stokes-Boussinesq-Langevin equation. These processes have reflection positivity and their correlation functions, which may exhibit the Alder-Wainwright effect or long-range dependence, are expressed in terms of the Mittag-Leffler functions. These properties are established rigorously via the theory of KMO-Langevin equation and a combination of Mittag-Leffler functions and fractional derivatives. A~relationship to fractional Riesz-Bessel motion is also investigated. This relationship permits to study the effects of long-range dependence and second-order intermittency simultaneously

The Stochastic Wave Equation with Fractional Noise: a random field approach

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in \cite{dalang99}, when the noise is white in time. Under this condition, we show that the solution is $L^2(\Omega)$-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is {\em different} (and more general) than the one obtained for the wave equation

Space and time inversions of stochastic processes and Kelvin transform

Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion

Brownian Manifolds, Negative Type and Geo-temporal Covariances

We survey Brownian manifolds -- manifolds that can parametrise Brownian motion -- and those that cannot. We consider covariances of space-time processes, particularly those when space is the sphere -- geo-temporal processes. There are connections with functions of negative type.Comment: 12 pages; corrected typos and acknowledgemen

Solutions and approximations of some Lévy-driven stochastic (partial) differential equations

In this work we look at solutions to stochastic partial differential equations (SPDEs) with noise induced by a Lévy process in the context of Marcus integrals. The canonical Marcus integral is known from the study of SDEs with Lévy noise. We recapture the fundamental results on the existence of solution flows to the Marcus SDE and the convergence of Wong-Zakai approximations. We also prove a generalized Itô formula for said solutions and use this result to establish equations for the inverse flow. We are then looking at extensions of Marcus integrals to the case of SPDEs and find solutions for these equations. Our focus mainly lies on multi-dimensional first-order transport equations driven by Lévy noise. Existence and uniqueness results for the Marcus SPDE are established using a method of characteristics. For second-order equations we prove the existence and uniqueness of mild solutions for equations driven by pure jump Lévy processes, also in terms of Marcus SPDEs. Finally, we study a one-dimensional second-order advection-diffusion equation on the half-line, with Lévy noise at the boundary. Both Dirichlet and Neumann boundary conditions are considered, and the closed form formulae for mild solutions are determined. We also define Wong-Zakai type approximations of the solution by classical solutions and show convergence in the setting of the M1-topology in the Skorokhod space