吸收边界上的Langevin过程

A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains

We consider classical solutions to the kinetic Fokker-Planck equation on a bounded domain $\mathcal O \subset~\mathbb{R}^d$ in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain $D = \mathcal O \times \mathbb{R}^d$. Furthermore, a Harnack inequality, as well as a maximum principle, is provided on $D$ for solutions to this kinetic Fokker-Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin p…

Mixed Eigenvalues of Discrete {\LARGE $\pmb p$\,}-Laplacian

This paper deals with the principal eigenvalue of discrete $p$-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation: the difference form, the single summation form and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for pos…

部分吸收介质中相互作用 Brown 气体的广义 Dean-Kawasaki 方程

A generalized Dean-Kawasaki equation for an interacting Brownian gas in a partially absorbing medium

本文将 Dean-Kawasaki 方程推广到一维部分吸收介质中的相互作用 Brown 气体，结合位置—局部时间联合密度、反射边界平均场理论和基于相遇的吸收阈值模型，并扩展至有限区间与陷阱。

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line

We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line. By taking the large particle limit we establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles. This process can be viewed as the weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We establish results on the $L^2$-regularity properties of this density process, showing that it is well behaved in the interior of the domain but may exhibit singularities on the boundary at a dense se…

近距离接触的粘性类型：吸收边界处的布朗运动

Close encounters of the sticky kind: Brownian motion at absorbing boundaries

本文构建了一维粘性布朗运动接触式模型，计算了粒子位置和占用时间的联合概率密度，以首次跨越随机阈值的占用时间定义吸收事件，并揭示了粘性RTP扩散极限与粘性BM的映射关系。

Diffusion limit for a kinetic equation with a thermostatted interface

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $T$ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $ρ(t,y)$ of a heat equation with the boundary condition $ρ(t,0)\equiv T$.

Some reflections on directed lattice paths

This article analyzes directed lattice paths, when a boundary reflecting or absorbing condition is added to the classical models. The lattice paths are characterized by two time-independent sets of rules (also called steps) which have a privileged direction of increase and are therefore essentially one-dimensional objects. Depending on the spatial coordinate, one of the two sets of rules applies, namely one for altitude 0 and one for altitude bigger than 0. The abscissa y=0 thus acts as a border which either absorbs or reflects steps. The absorption model corresponds to the model analyzed by B…

具有势函数的粘性CIR过程：不变测度与精确采样

Sticky CIR process with potential: invariant measure and exact sampling

本文研究了一维粘性CIR过程，该过程在[0,∞)上扩散，并在原点处具有粘性边界条件，出现在基于Hadamard-Langevin动力学的稀疏贝叶斯推断框架中。对于参数范围δ∈(1,2)，其中原点可到达但不可吸收，证明了过程的良定性和不变测度的唯一性，该测度是零点质量与内部加权伽马型密度的混合。通过构造一个显式格林函数，用以构造零势情况下的精确采样器。对于非平凡势函数G，通过Girsanov测度变换建立倾斜不变测度的存在性和唯一性，并开发了两种采样算法：一种是修正的Metropolis-Hastings采样器，可精确采样目标不变测度；另一种是未调整的Langevin算法（ULA），每步成本较低但引入O(h)的偏差。数值实验验证了预测行为：Metropolis-Hastings采样器在所有步长下都能达到目标不变测度，而ULA表现出预期的O(h)偏差。