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_{T\ge t} \Big\{t {\mathbb E}\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big]\Big\} \\ &\le \limsup_{t\to \infty} \sup_{T\ge t} \Big\{ t \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] \Big\}\le \sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2}\end{align*} holds for some constant $c\in (0,1]$ with $c=1$ when $\partial M$ is convex, where $μ_0:= ϕ_0^2μ$ for the first Dirichet eigenfunction $ϕ_0$ of $L$, $\{λ_m\}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in the increasing order counting multiplicities, and the upper bound is finite if and only if $d\le 3$. When $d=4$, $\sup_{T\ge t} \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] $ decays in the order $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-\frac 2 {d-2}}$, as $t\to\infty$.
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.
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{\{…
The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.
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.
Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $μ(\d x):=\e^{V(x)}\d x$ is a probability measure, where $\d x$ is the volume measure, and let $L=Δ+\nabla V$. The exact convergence rate in Wasserstein distance is derived for empirical measures of subordinations for the (reflecting) diffusion process generated by $L$.
We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order p $\in$ [1, $\infty$) between the empirical measure of independent and identically distributed R d-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order rp for some r > 1, and we discuss the optimality of the bounds. Mathematics subject classification. 60B10, 60F10, 60F15, 60E15.
The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than [arXiv:2107.11568], generalize the recent ones in [Stoch. Proc. Appl. 144(2022), 271--287] and are shown to be sharp by a typical …
In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.
We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $α$-dependent sequences. We prove some moments inequalities of order p for any p $\ge$ 1, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well known von Bahr-Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identic…
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 $X_t$ be the (reflecting) diffusion process generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $μ(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s\d s$ under the Wasserstein distance. As a typical example, when $M=\mathbb R^d$ and $V(x)= c_1- c_2 |x|^p$ for some constants $c_1\in \mathbb R, c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when…
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…
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…
We provide some non asymptotic bounds, with explicit constants, that measure the rate of convergence, in expected Wasserstein distance, of the empirical measure associated to an i.i.d. $N$-sample of a given probability distribution on $\mathbb{R}^d$.
We study the Wasserstein distance $W_2$ for Gaussian samples. We establish the exact rate of convergence $\sqrt{\log\log n/n}$ of the expected value of the $W_2$ distance between the empirical and true $c.d.f.$'s for the normal distribution. We also show that the rate of weak convergence is unexpectedly $1/\sqrt{n}$ in the case of two correlated Gaussian samples.
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $\infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slepčev to the case that the true distribution has an unbounded density.
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…
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $μ$ approaches $μ$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radic…
We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phase-transition" in the rates when $d=2+1/H$, akin to the Ajtai-Komlós-Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approxi…
Let $μ_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $μ$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $μ_N$ to $μ$, when measured in the Wasserstein distance of order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds and concentration inequalities, for any values of $p>0$ and $d\geq 1$. We extend also the non asymptotic $L^p$-bounds to stationary $ρ$-mixing sequences, Markov chains, and to some interacting particle systems.
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 are interested in the approximation in Wasserstein distance with index $ρ\ge 1$ of a probability measure $μ$ on the real line with finite moment of order $ρ$ by the empirical measure of $N$ deterministic points. The minimal error converges to $0$ as $N\to+\infty$ and we try to characterize the order associated with this convergence. In \cite{xuberger}, Xu and Berger show that, apart when $μ$ is a Dirac mass and the error vanishes, the order is not larger than $1$ and give a sufficient condition for the order to be equal to this threshold $1$ in terms of the density of the absolutely continu…
We investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the Circular Law. For general entry distributions, we obtain a nearly optimal rate of convergence in 1-Wasserstein distance of order $n^{-1/2+ε}$ and we prove that the optimal rate $n^{-1/2}$ is attained by Ginibre matrices. This shows that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster compared to that of i.i.d. points, which is known to include a logarithmic factor.
We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $μ$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the $1$-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.
Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and $\mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $\mathbb{P}$ is a convex combination between the uniform distribution supported on the two points $\{0,1\}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic c…
We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a Markov chain are also given, under the assumption that the chain is contractive on the space of Lipschitz functions. Throughout …
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.
Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between one-dimensional projections of the measures. Different choices of how to aggregate these projected distances (averaging, random sampling, maximizing) give rise to different distances, requiring different statistical analyses. We define the \emph{Sliced Wasserstein Process}, a stochastic process defined by the empirical Wasserstein distance between projections of em…
A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug-Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler - model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measu…
