Monotonicity of time-dependent transportation costs and coupling by reflection

Optimal transportation and monotonic quantities on evolving manifolds

In this note we will adapt Topping's $\mathcal{L}$-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold $(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M$. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List's (and hence also of Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.

关于具有白噪声的均值反射SPDEs的运输成本不等式

Transportation cost inequalities for mean reflection SPDEs with white noise

本文研究了均值反射随机偏微分方程的解在统一范数下的二次运输成本不等式，提出了一种新的方程类型，其中补偿反射部分不依赖于路径，而是依赖于解的分布。

A note on Harnack and Transportation inequalities For Stochastic Differential Equations with reflections

We establish transportation cost inequalities, with respect to the uniform and $L_2$-metric, on the path space of continuous functions, for laws of solutions of stochastic differential equations with reflections. We also consider the case of stochastic differential equations involving local times. Harnack inequalities for the associated semigroups are also established.

A Note on Transportation Cost Inequalities for Diffusions with Reflections

We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusions with suitable drifts and diffusions. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.

Site-monotonicity properties for reflection positive measures with applications to quantum spin systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and $XY$ model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site…

Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

By using the mirror coupling for solutions of SDEs driven by pure jump Lévy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we…