Walter Schachermayer, Uwe Schmock, Josef Teichmann
We give an easy counter-example to Problem 7.20 from C. Villani's book on mass transport: in general, the quadratic Wasserstein distance between $n$-fold normalized convolutions of two given measures fails to decrease monotonically.
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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.
Quadratic Wasserstein distance between Gaussian laws revisited with correlation
本文提出一种简单推导方法，基于正交矩阵使两高斯分布协方差矩阵共享同一相关矩阵，重新推导了两个高斯分布间二次Wasserstein距离的公式。
Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
本文研究了凸序下Wasserstein投影的连续性与正则性，并在二次Gaussian情形下证明了投影的Gaussian性质，刻画了协方差矩阵及非Gaussian投影的存在条件。
In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^1$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied …
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…
In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $W^2_2(μ,ν)$ between two probability measures $μ$ and $ν$ with finite second order moments on $\mathbb{R}^d$ is the composition of a martingale coupling with an optimal transport map ${\mathcal T}$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between $μ$ and ${\mathcal T}\#μ$. Next, we give a direct proof that $σ\mapsto W_2^2(σ,ν)$ is differentiable at $μ$ in the Lions sense iff there is a unique optimal coupling between $μ$ and $ν$ and this coupling i…
Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of fully non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.
Empirical measures and random walks on compact spaces in the quadratic Wasserstein metric
本文扩展了2维流形上的结果，证明了d维紧Riemann流形上经验测度在二次Wasserstein度量下的精确收敛速率上界，并将i.i.d.条件放宽至混合平稳样本及紧Lie群上的随机游走，发现半单群上无需谱隙假设即可达近最优速率。
We investigate here the optimal transportation problem on configuration space for the quadratic cost. It is shown that, as usual, provided that the corresponding Wasserstein is finite, there exists one unique optimal measure and that this measure is supported by the graph of the derivative (in the sense of the Malliavin calculus) of a ``concave'' (in a sense to be defined below) function. For finite point processes, we give a necessary and sufficient condition for the Wasserstein distance to be finite.
This thesis consists of three parts. In the first part, we study $\mathbb{L}^p$ solutions of a large class of BSDEs. Existence, comparison theorem, uniqueness and a stability result are proved. In the second part, we establish the solvability of quadratic semimartingale BSDEs. In contrast to current literature, we use Lipschitz-quadratic regularization and obtain the existence and uniqueness results with minimal assumptions. The third part is a brief summary of quadratic semimartingales and the monotone stability result. This provides an alternative proof of monotone stability result for quadr…
This is a follow up of our previous paper - Trybuła and Zawisza \cite{TryZaw}, where we considered a modification of a monotone mean-variance functional in continuous time in stochastic factor model. In this article we address the problem of optimizing the mentioned functional in a market with a stochastic interest rate. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to derive the optimal investment strategy and the value function.
A higher-order Otto calculus approach to the Gaussian completely monotone conjecture
本文基于Otto微积分和Wasserstein梯度流，证明了对数凹初始测度下高斯完全单调猜想至五阶成立，并推导了Wasserstein空间上的Faa di Bruno公式及内能泛函的高阶微分。
This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance $W_2$. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving an elliptic boundary value problem involving the Witten Laplacian,…
The existence and uniqueness of stationary distributions and the exponential convergence in $L^p$-Wasserstein distance are derived for distribution dependent SDEs from associated decoupled equations. To establish the exponential convergence, we introduce a twinned Talagrand inequality of the original SDE and the associated decoupled equation, and explicit convergence rate is obtained. Our results can be applied to SDEs without uniformly dissipative drift and distribution dependent diffusion term, which cover the Curie-Weiss model and the granular media model in double-well landscape with quadr…
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…
Exact rate of convergence for the empirical measure of a subordinated process in $p$-Wasserstein distance
本文研究紧 Riemann 流形上从属扩散的非对称跳过程经验测度，利用 Bernstein 型不等式与 PDE 方法得到 $p$-Wasserstein 距离精确收敛速度及二次情形重整化极限。
Copulas farthest from independence in quadratic Wasserstein distance
本文证明了Catalano和Lavenant关于二次Wasserstein距离下最远离独立性的copulas的猜想，即M和W达到最大值1/10，并刻画了等号条件，同时给出最优Monge映射及新的相依性度量。
We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arb…
The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent…
Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration
本文研究半离散二次Wasserstein能量的性质，构建Langevin温度的熵正则化松弛控制，证明相关动力学的适定性与遍历性，为该领域提供理论支撑。
Second-order Wasserstein response for Lévy laws
本文研究 Lévy 律在弱扰动下的 Wasserstein 距离二阶展开，通过方差加权特征微分获得一致展开，并给出二次网格速率、度量速度及响应线性规划等结果。
In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding $N$-player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equival…
Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process
该研究利用带二次平均代价的Wasserstein距离实现布朗运动与Goldstein--Kac电报过程的非渐近过程层面控制，推导了对应时间平均p阶矩的非渐近估计，证明采用多种耦合技术。
It is shown that under suitable regularity conditions, differential entropy is a Lipschitz functional on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be Lipschitz continuous in distributions over the product space with respect to Ornstein's $\bar d$-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interfer…
Wasserstein error estimates between telegraph processes and Brownian motion
本文在路径Wasserstein距离下提供了电报过程与布朗运动泛函间的非渐近误差界，证明其在Kac机制下趋于零且在翻转率较小时有效，并通过数值实验验证了误差界的尖锐性。
The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked by the curse of dimensionality, whereby the number of data points needed to accurately estimate them grows exponentially with dimension. Gaussian smoothing was recently introduced as a means to alleviate the curse of dimensionality, giving rise to a parametric convergence rate in any dimension, while preserving the Wasserstein metric and topological structure…
A notion of BSDE on the Wasserstein space and its applications to control problems and PDEs
本文引入了Wasserstein空间上的BSDE新概念，将其推广至平均场设定，并证明其与平均场控制问题和PDEs的对应关系，建立了比较原理与存在性结果。
Local convergence rates for Wasserstein gradient flows and McKean-Vlasov equations with multiple stationary solutions
针对自由能存在非全局最小值临界点导致log-Sobolev不等式全局失效的问题，提出局部不等式方法，为颗粒介质方程及Vlasov-Fokker-Planck方程提供了局部指数收敛速率，并证明了粒子系统自由能的快速衰减。
We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear …
Couplings Farthest from the Independent Gaussian
研究联合分布与边际乘积的最大2-Wasserstein距离，证明高斯情形下单调耦合最远，并刻画一般测度的最远分布。
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling in…
We study the Wasserstein metric $W_p$, a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance $W_1$ between the distribution of quadratic residues in a finite field $\mathbb{F}_p$ and uniform distribution by $\lesssim p^{-1/2}$ (the Polya-Vinogradov inequality implies $\lesssim p^{-1/2} \log{p}$). We also show for continuous $f:\mathbb{T} \rightarrow \mathbb{R}_{}$ with mean value 0 $$ (\mbox{number of roots of}~f) \cdot \left( \sum_{k=1}^{\infty}{ \frac{ |\hat{f}(k)|^2}{k^2…
Empirical Convergence of Even-Order Gromov-Wasserstein Functionals
本文研究了偶阶Gromov-Wasserstein函数的样本复杂度，证明了在特定条件下误差率与样本量呈反比关系，并扩展了欧几里得空间中的二次上界结果。
Geometry and analytic properties of the sliced Wasserstein space
本文研究切片Wasserstein空间的几何与度量性质，证明在良好测度下SW度量与负Sobolev范数可比，在接近离散测度时与Wasserstein度量可比，并刻画其切空间、曲线速度及SW长度，进而揭示对梯度流的影响。
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