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In this paper, we prove a sufficient and necessary condition for the transition probability distribution of a general, time-inhomogeneous linear SDE to possess a density function and study the differentiability of the density function and…
In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical…
In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic…
Copulas allow a flexible and simultaneous modeling of complicated dependence structures together with various marginal distributions. Especially if the density function can be represented as the product of the marginal density functions and…
Quantifying uncertainty in future climate projections is hindered by the prohibitive computational cost of running physical climate models, which severely limits the availability of training data. We propose a data-efficient framework for…
This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a particular class of L\'evy processes with finite jump intensity, using neural networks. We propose a framework that…
Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain…
We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the…
We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using cross-fluctuations, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic…
Despite their groundbreaking performance for many generative modeling tasks, diffusion models have fallen short on discrete data domains such as natural language. Crucially, standard diffusion models rely on the well-established theory of…
The dynamics of a one-dimensional stochastic model is studied in presence of an absorbing boundary. The distribution of fluctuations is analytically characterized within the generalized van Kampen expansion, accounting for higher order…
The multivariate extended skew-normal distribution allows for accommodating raw data which are skewed and heavy tailed, and has at least three appealing statistical properties, namely closure under conditioning, affine transformations, and…
Standard regression approaches assume that some finite number of the response distribution characteristics, such as location and scale, change as a (parametric or nonparametric) function of predictors. However, it is not always appropriate…
Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process, including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE, which in certain…
Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE),…
Stochastic differential equations (SDEs) are increasingly used in longitudinal data analysis, compartmental models, growth modelling, and other applications in a number of disciplines. Parameter estimation, however, currently requires…
We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of…
This paper provides a systematic method to build wind speed models based on stochastic differential equations (SDEs). The resulting models produce stochastic processes with a given probability distribution and exponential decaying…
In this article, we consider McKean stochastic differential equations, as well as their corresponding McKean-Vlasov partial differential equations, which admit a unique stationary state, and we study the linearized It\^o diffusion process…
In the present study we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe…