布朗运动与Goldstein--Kac电报过程之间Wasserstein距离的定量控制

长期时间区间内平面对称Markov随机飞行的边缘分布表现得像Goldstein-Kac电报过程

Marginals of the planar symmetric Markov random flight on long time intervals behave like the Goldstein-Kac telegraph process

本文研究平面对称Markov随机飞行在坐标轴上的投影，证明当时间趋于无穷大时，其边缘分布密度渐近等价于经典一维Goldstein-Kac电报过程，并通过数值计算证实了该结论。

The explicit probability distribution of the sum of two telegraph processes

We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$, both developing with finite constant speed $c>0$, that, at the initial time instant $t=0$, simultaneously start from the origin $0\in\Bbb R$ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate $λ>0$. Closed-form expressions for the transition density $p(x,t)$ and the probability distribution function $Φ(x,t)=\text{Pr} \{ S(t)<x \}, \; x\in\Bbb R, \; t>0,$ of the sum $S(t)=X_1(t)+X_2(t)$ of these processes at arbitrary time instant $t>0…

电报过程与布朗运动之间的Wasserstein误差估计

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。

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$. The processes $X_k(t), \; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \; c_2>0,$ that start at the initial time instant $t=0$ from the origin of the real line $\Bbb R$ and are controlled by two independent homogeneous Poisson processes of rates $λ_1>0, \; λ_2>0$, respectively. Closed-form expression for the probability distribution function of the Euclidean distance $$ρ(t) = |X_1(t) - X_2(t) |, \qquad t>0,$$ between these processes at arbitrary time …

基于Kac流的电报生成模型

Telegrapher's Generative Model via Kac Flows

提出一种基于阻尼波动方程（电报方程）的新型生成模型，利用Kac过程与电报方程的Feynman-Kac关系，通过流匹配训练神经网络近似速度场，实现样本生成，并展示其相对于扩散模型的优势。

Some results on the telegraph process confined by two non-standard boundaries

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and $H>0$. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold $H$) it is either absorbed, with probability $α$, or reflected upwards (downwards), with probability $1-α$, for $0<α<1$. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through lin…