Related papers: Theory of Second and Higher Order Stochastic Proce…
In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe…
This paper is centered around the approximation of dynamical systems by means of Gaussian processes. To this end, trajectories of such systems must be collected to be used as training data. The measurements of these trajectories are…
We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian…
We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a…
We consider linear stochastic differential-algebraic equations with constant coefficients and additive white noise. Due to the nature of this class of equations, the solution must be defined as a generalised process (in the sense of Dawson…
This book covers a wide range of problems involving the applications of stochastic processes, stochastic calculus, large deviation theory, group representation theory and quantum statistics to diverse fields in dynamical systems,…
Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles…
This paper addresses the detection of a stochastic process in noise from irregular samples. We consider two hypotheses. The \emph{noise only} hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables…
For many applications with multivariate data, random field models capturing departures from Gaussianity within realisations are appropriate. For this reason, we formulate a new class of multivariate non-Gaussian models based on systems of…
A new class of random partial differential equations of parabolic type is considered, where the stochastic term consists of an irregular noisy drift, not necessarily Gaussian, for which a suitable interpretation is provided. After freezing…
We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic…
The Langevin system subjected to non-Gaussian noise has been discussed, by using the second-order moment approach with two kinds of models for generating the noise. We have derived the effective differential equation (DE) for a variable…
Inferring dynamical models from low-resolution temporal data continues to be a significant challenge in biophysics, especially within transcriptomics, where separating molecular programs from noise remains an important open problem. We…
This brief article gives an overview of quantum mechanics as a {\em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum…
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or…
Time-irreversible stochastic processes are frequently used in natural sciences to explain non-equilibrium phenomena and to design efficient stochastic algorithms. Our main goal in this thesis is to analyse their dynamics by means of large…
It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time…
A scalar Langevin-type process $X(t)$ that is driven by Ornstein-Uhlenbeck noise $\eta(t)$ is non-Markovian. However, the joint dynamics of $X$ and $\eta$ is described by a Markov process in two dimensions. But even though there exists a…