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A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov…
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…
This paper is devoted to the analysis of random motions on the line and in the space R^d (d > 1) performed at finite velocity and governed by a non-homogeneous Poisson process with rate \lambda(t). The explicit distributions p(x,t) of the…
Probabilistic dependency graphs (PDGs) are a flexible class of probabilistic graphical models, subsuming Bayesian Networks and Factor Graphs. They can also capture inconsistent beliefs, and provide a way of measuring the degree of this…
Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
Gaussian processes (GPs) are the most common formalism for defining probability distributions over spaces of functions. While applications of GPs are myriad, a comprehensive understanding of GP sample paths, i.e. the function spaces over…
We extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to…
This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order…
Principal stratification is a widely used framework for addressing post-randomization complications. After using principal stratification to define causal effects of interest, researchers are increasingly turning to finite mixture models to…
Nonparametric density estimation is considered for a discretely observed stationary continuous-time process. For each of three given time sampling procedures either random or deterministic, we establish that histograms and frequency…
Recent years have witnessed significant progress in developing effective training and fast sampling techniques for diffusion models. A remarkable advancement is the use of stochastic differential equations (SDEs) and their…
We study the problem of estimating a set of $d$ linear queries with respect to some unknown distribution $\mathbf{p}$ over a domain $\mathcal{J}=[J]$ based on a sensitive data set of $n$ individuals under the constraint of local…
Partial differential equations (PDEs) are a central tool for modeling the dynamics of physical, engineering, and materials systems, but high-fidelity simulations are often computationally expensive. At the same time, many scientific…
This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and…
In many sampled-data applications, observers are designed based on approximately discretized models of continuous-time systems, where usually only the discretized system is analyzed in terms of its detectability. In this paper, we show that…
Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space $\HH$, where $L$ is a $\RR$ valued L\'evy process, $A:H\to…
We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the…
This paper reconsiders the problem of testing the equality of two unspecified continuous distributions. The framework, which we propose, allows for readable and insightful data visualisation and helps to understand and quantify how two…
There has recently been much interest in Gaussian fields on linear networks and, more generally, on compact metric graphs. One proposed strategy for defining such fields on a metric graph $\Gamma$ is through a covariance function that is…