Related papers: Brownian motion on Perelman's almost Ricci-flat ma…
We elaborate the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott-Strum-Villani, we define this notion based on the geodesic convexity of the Kullback-Leibler divergence in a…
Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Holder regularity is allowed to vary along the paths. In this paper, two kind of multi-parameter extensions of mBm are studied: one is…
When the Ricci curvature of a Riemannian manifold is not lower bounded by a constant, but lower bounded by a continuous function, we give a new characterization of this lower bound through the convexity of relative entropy on the…
Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference…
Brownian motion near soft surfaces is a situation widely encountered in nanoscale and biological physics. However, a complete theoretical description is lacking to date. Here, we theoretically investigate the dynamics of a two-dimensional…
Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian…
We propose a method for developing the flows of stochastic dynamical systems, posed as Ito's stochastic differential equations, on a Riemannian manifold identified through a suitably constructed metric. The framework used for the stochastic…
The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we…
The purpose of this paper is to construct a Brownian motion $X := (X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small enough ${\mathscr…
We study orientability in spaces with Ricci curvature bounded below. Building on the theory developed by Honda, we establish equivalent characterizations of orientability for Ricci limit and RCD spaces in terms of the orientability of their…
By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for…
After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we…
In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of…
We review recent results on the study of the isoperimetric problem on Riemannian manifolds with Ricci lower bounds. We focus on the validity of sharp second order differential inequalities satisfied by the isoperimetric profile of possibly…
We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random…
We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal transport is the large deviation limit of a…
The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci…
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound.…
This work proposes a method for the two-dimensional simulation of Brownian particles in a fluid with restrictions. The method is based on simple numerical rules between two matrices. One of the matrix represent the identification of all…
Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a…