On isoperimetric sets of radially symmetric measures

径向对称分支随机游走在$\mathbb{R}^d$中的最大位移

The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$

本文研究径向对称分支随机游走的最大位移，证明在生存条件下其增长与分支布朗运动一致，主要工具为带移动障碍的投票定理。

Entropy Jumps for Radially Symmetric Random Vectors

We establish a quantitative bound on the entropy jump associated to the sum of independent, identically distributed (IID) radially symmetric random vectors having dimension greater than one. Following the usual approach, we first consider the analogous problem of Fisher information dissipation, and then integrate along the Ornstein-Uhlenbeck semigroup to obtain an entropic inequality. In a departure from previous work, we appeal to a result by Desvillettes and Villani on entropy production associated to the Landau equation. This obviates strong regularity assumptions, such as presence of a spe…

Optimal Brownian Stopping between radially symmetric marginals in general dimensions

Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stopped at $τ$-- equal to $B_τ\sim ν$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in "subharmoic order" into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely t…

Quantitative Isoperimetric Inequalities on the Real Line

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures μon the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.

Isoperimetric Inequalities for Non-Local Dirichlet Forms

Let $(E,\F,μ)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,μ(\d x)\,μ(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d x,\d y)$$ in $L^2(μ)$. We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with $\E$. In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\R^n$, which include the existing fractional isoperimetric inequality as a special …

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the $O(1)$-terms as the number $N$ of particles increases. Notably, our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions prop…

径向对称环境中鞅的最优控制

Optimal control of martingales in a radially symmetric environment

针对固定二次变差的连续多维鞅，本文在径向对称环境下显式求解随机控制问题，给出按半径在纯径向与纯切向策略间切换的最优策略，并以 HJB 粘性解证明其最优性。

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ μ(λA + (1-λ)B)^{1/n} \geq λμ(A)^{1/n} + (1-λ)μ(B)^{1/n} \] holds true for an unconditional product measure $μ$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $μ$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measu…

带径向对称势和硬壁的二维库仑气体的配分函数

Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

该研究针对带径向对称势的二维库仑气体可解模型，探究液滴处硬壁位置对配分函数大$N$渐近展开的影响，发现不同硬壁位置下$\text{log}\thinspace N$阶项系数有差异，还得到了$\text{sqrt}(N)$阶及常数项的普适数值常数。

Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures

We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov's original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov's panc…