Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ(dx):=e^{V(x)} d x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=Δ+\nabla V$ with $τ:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the conditional empirical measure
$μ_t^ν:= \mathbb E^ν\big(\frac 1 t \int_0^t δ_{X_s}d s\big|t<τ\big)$ for the diffusion process with initial distribution $ν$ such that $ν(\partial M)<1$. Then $$\lim_{t\to\infty} \big\{t\mathbb W_2(μ_t^ν,μ_0)\big\}^2 = \frac 1 {\{μ(ϕ_0)ν(ϕ_0)\}^2} \sum_{m=1}^\infty \frac{\{ν(ϕ_0)μ(ϕ_m)+ μ(ϕ_0) ν(ϕ_m)\}^2}{(λ_m-λ_0)^3},$$ where $ν(f):=\int_Mf {d} ν$ for a measure $ν$ and $f\in L^1(ν)$, $μ_0:=ϕ_0^2μ$, $\{ϕ_m\}_{m\ge 0}$ is the eigenbasis of $-L$ in $L^2(μ)$ with the Dirichlet boundary,
$\{λ_m\}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and $\mathbb W_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.
We investigate long-time behaviors of empirical measures associated with subordinated Dirichlet diffusion processes on a compact Riemannian manifold $M$ with boundary $\partial M$ to some reference measure, under the quadratic Wasserstein distance. For any initial distribution not concentrated on $\partial M$, we obtain the rate of convergence and even the precise limit for the conditional expectation of the quadratic Wasserstein distance conditioned on the process killed upon exiting $M\setminus\partial M$. In particular, the results coincide with the recent ones proved by F.-Y. Wang in \cite…
The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the rate of convergence of conditional empirical measures associated to subordinated Dirichlet diffusion processes on a connected compact Riemannian manifold with absorbing boundary. We give the sharp rate of convergence for any initial distribution and prove the precise limit for a large class of initial distributions. We follow the basic idea of Wang, but all…
We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein Diffusion, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya.
Von Renesse and the author (Ann. Prob. '09) developed a second order calculus on the Wasserstein space P([0,1]) of probability measures on the unit interval. The basic objects of interest had been Dirichlet form, semigroup and continuous Markov process, called Wasserstein diffusion. The goal of this paper is to derive approximations of these objects on the infinite dimensional space P([0, 1]) in terms of appropriate objects on finite dimensional spaces. In particular, we will approximate the Wasserstein diffusion in terms of interacting systems of Brownian motions.
We introduce a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction on the real line as mathematically rigorous description of colloidal motion of fluids. The associated measure-valued process provides a weak solution to a corrected Dean-Kawasaki equation for supercooled liquids without dissipation. Our construction is based on the introduction and analysis of a fundamentally new family of equilibrium measures for the associated dynamics and their Dirichlet forms. We identify the intrinsic metric as the quadrat…
Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ({\rm d} x):={\rm e}^{V(x)}{\rm d} x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=Δ+\nabla V$ with $τ:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s}{\rm d} s$ under the condition $t<τ$ for the diffusion process. If $d\le 3$, then for any initial distribution not fully supported on $\partial M$, \begin{align*} &c\sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2} \le \liminf_{t\to \infty} \inf_…
We prove tightness of a family of path measures $ν_{\varepsilon}$ on tubes $L(\varepsilon)$ of small diameters around a closed and connected submanifold $L$ of another Riemannian manifold $M$. Together with a convergence result for Dirichlet semigroups on tubular neighbourhoods, that implies weak convergence of the measures as the tube radius $\varepsilon$ tends to zero to a measure supported by the path space of the submanifold. As a consequence, we obtain weak convergence of the measures obtained by conditioning Brownian motion to stay within the tubes $L(\varepsilon)$ up to a finite time $T…
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 particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context o…
We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension $d\ge 2$. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet-Ferguson measure, and is the counterpart on multi-dimensional base spaces to the Modified Massive Arratia Flow over the unit interval described in V. Konarovskyi, M.-K. von Renesse, Comm. Pure Appl. Math., 72, 0764-0800 (2019). Together with two different constructions of the process, we discuss its ergodicity, invar…
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universitat de Barcelona（巴塞罗那大学）
(Conditional) Generative Adversarial Networks (GANs) have found great success in recent years, due to their ability to approximate (conditional) distributions over extremely high dimensional spaces. However, they are highly unstable and computationally expensive to train, especially in the time series setting. Recently, it has been proposed the use of a key object in rough path theory, called the signature of a path, which is able to convert the min-max formulation given by the (conditional) GAN framework into a classical minimization problem. However, this method is extremely expensive in ter…
We investigate yet another approach to understand the limit behaviour of Brownian motion conditioned to stay within a tubular neighbourhood around a closed and connected submanifold of a Riemannian manifold. In this context, we identify a second order generator subject to Dirichlet conditions on the boundary of the tube and study its associated semigroups. After a suitable rescaling and renormalization procedure, we obtain convergence of these semigroups, both in $L^2$ and in Sobolev spaces of arbitrarily large index, to a limit semigroup, as the tube diameter tends to zero. As a byproduct, we…
We prove that the linear heat flow in a RCD(K,\infty) metric measure space (X,d,m) satisfies a contraction property with respect to every L^p-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a precise estimate for the optimal W_\infty-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the RCD(K,\infty) lower Ricci bound and the corresponding Bakry-Émery condition for the canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension to the non-smooth metric measur…
We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The…
We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.
For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the law of the Markov process itself. In the last section we want to give an application to the Wasserstein diffusion.
By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $μ_t$ of non-symmetric subordinated diffusion processes in an abstract framework. The main results are applied to the subordinations of several typical models, which include the (reflecting) diffusion processes on compact manifolds, the conditional diffusion processes, the Wright-Fisher diffusion process, and hypoelliptic diffusion processes on {\bf SU}(2). In particular, for the (reflecting) diffusion processes on…
This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both sta…
This paper studies posterior concentration behavior of the base probability measure of a Dirichlet measure, given observations associated with the sampled Dirichlet processes, as the number of observations tends to infinity. The base measure itself is endowed with another Dirichlet prior, a construction known as the hierarchical Dirichlet processes (Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566-1581]). Convergence rates are established in transportation distances (i.e., Wasserstein metrics) under various conditions on the geometry of the support of the true base measure. As a consequenc…
