Related papers: Gradient Bounds for Kolmogorov Type Diffusions
We analyze the Gray-Scott reaction--diffusion system on $\Omega\subset\mathbb{R}^n$ ($n\ge 1$) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of…
We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities.…
This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion on the Euclidean space, which is deeply related with a family of fractional Gagliardo-Nirenberg-Sobolev inequalities. Generically,…
This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in…
The main objective of this paper is to obtain generalization of some Gruss-type inequalities in case of functional bounds by using a generalized Katugampola fractional integral.
Diffusion models have become the go-to method for many generative tasks, particularly for image-to-image generation tasks such as super-resolution and inpainting. Current diffusion-based methods do not provide statistical guarantees…
We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy…
We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly…
We derive a uniform bound for the difference of two contractive semigroups, if the difference of their generators is form-bounded by the Hermitian parts of the generators themselves. We construct a semigroup dynamics for second order…
We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of…
This work discusses the homogenization analysis for diffusion processes on scale-free metric graphs, using weak variational formulations. The oscillations of the diffusion coefficient along the edges of a metric graph induce internal…
This paper deals with the problem of outliers in high frequency observation data from diffusion processes. Robust estimation methods are needed because the inclusion of outliers can lead to incorrect statistical inference even in the…
Using present a unified approach, we establish a Kolmogorov type comparison theorem for the classes of $2\pi$-periodic functions defined by a special class of operators having certain oscillation properties, which includes the classical…
We study the existence of densities for distributions of piecewise deterministic Markov processes. We also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times. In…
This review paper, written for the second edition of the Handbook of Markov Chain Monte Carlo, provides an introduction to the study of convergence analysis for Markov chain Monte Carlo (MCMC), aimed at researchers new to the field. We…
Recently, through a unified gradient flow perspective of Markov chain Monte Carlo (MCMC) and variational inference (VI), particle-based variational inference methods (ParVIs) have been proposed that tend to combine the best of both worlds.…
Grad's method is used on the linearized Boltzmann collision operator to derive the most general expressions for the collision coefficients for a multi-component, multi-temperature plasma up to rank-2. In doing so, the collision coefficients…
We study in some generality intertwinings between $h$-transforms of Karlin-McGregor semigroups associated with one dimensional diffusion processes and those of their Siegmund duals. We obtain couplings so that the corresponding processes…
Diffusion models have emerged as powerful generative models, inspiring extensive research into their underlying mechanisms. One of the key questions in this area is the loss functions these models shall train with. Multiple formulations…
The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward…