Preservation of log-concavity on summation

对数凹性的保持与Wasserstein-Fisher-Rao梯度流的收敛性

Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows

本文证明WFR梯度流在强对数凹目标分布下保持对数凹性，并导出无需热启动的对称KL散度非渐近收敛速率，其可分解为Wasserstein与Fisher-Rao贡献，证实相关猜想。

通过 Hamilton-Jacobi 方程沿广义热流传播弱对数凹性

Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations

本文提出弱对数凹性概念，证明其可沿广义热半群传播，并基于 HJB 方程的随机控制与反射耦合分析，推导出非凸势下 Schrödinger 算子基态的对数半凹性及抛物方程基本解的双侧对数 Hessian 估计。

On the framework of $L_{p}$ summations for functions

We develop the framework of $L_p$ operations for functions by introducing two primary new types $L_{p,s}$ summations for $p>0$: the $L_{p,s}$ convolution sum and the $L_{p,s}$ Asplund sum for functions. The first type is defined as the linear summations of functions in terms of the $L_p$ coefficients ($C_{p,λ,t}$, $D_{p,λ,t}$), the so-called the $L_{p,s}$ supremal-convolution when $p\geq1$ and the $L_{p,s}$ inf-sup-convolution when $0<p<1$, respectively. The second type $L_{p,s}$ summation is created by the $L_p$ averages of bases for $s$-concave functions. We show that they are equivalent in …