概率论

We consider a transient excited random walk on $Z$ and study the asymptotic behavior of the occupation time of a currently most visited site. In particular, our results imply that, in contrast to the random walks in random environment, a…

For the independent reference model with popularity vector $p\in\Delta_N^\circ$, let $H_C(p)$ denote the exact stationary hit rate of an LRU cache of capacity $C$. We prove that, for every $1\le C<N$, the uniform popularity vector is the…

The Einstein relation describes the response of a diffusing particle to a small constant external force. It states that, as the force tends to zero, the ratio of the limiting velocity to the force magnitude converges to the diffusivity…

We give a complete classification of the eternal solutions for the KPZ fixed point. Each of these is a (possibly infinite) patching together of the known eternal solutions, called Busemann functions. The resulting evolution of the KPZ fixed…

We present a numerical framework for approximating the $\mu$-domain in the planar Skorokhod embedding problem PSEP, recently introduced in \cite{gross2019}. We show that under weak convergence of a sequence of probability measures…

The present paper is devoted to a systematic study of the $p$-Brownian convergence introduced in \cite{boudabra2026stability} (in press) to study the stability of the planar Skorokhod embedding problem \cite{gross2019,Boudabra2020}. The…

We study weighted Helmholtz--Hodge decompositions of drift vector fields associated with second-order diffusion operators on $\mathbb{R}^d$, $d\ge 2$. Given a decomposition of the form \[ \mathbf{G}=A\nabla\Phi+\mathbf{B}, \] we relate the…

Fastest arrival events, where the first among many diffusing particles reaches a target, are central in triggering signal initiation in molecular stochastic systems. Classical approaches to simulate such events rely on full trajectory…

Understanding the statistics of level crossings in stochastic processes is crucial across many scientific disciplines. The traditional Kac-Rice formula gives the mean rate of level crossings and has found broad use. However, that mean rate…

We extend Stein's method to include dependence with respect to an auxiliary random variable, for conditional laws for which Stein's characterizations do exist.

In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a…

We consider immigration processes with binomial catastrophes and random survival parameters. Two sources of randomness are analyzed. In the first model, the survival parameter is independently resampled at each catastrophe. In the second…

Let $M_n$ denote the largest interpoint distance among independent random points $X_1,\dots,X_n$ uniformly distributed in a compact set in $\mathbb{R}^d$. Weak limit laws for $M_n$ are known in several geometric settings, in particular for…

We establish Chernoff-type bounds for the largest eigenvalue of sums of Hermitian random matrices generated by a time-inhomogeneous Markov chain. Our primary regime assumes a compact state space and contractivity of each Markov kernel in…

We study the stability of compensated jump integrals under convergence of quadratic variation alone. Let \(X\) and \(\{X^n\}_{n\ge1}\) be c\`adl\`ag processes with jump measures \(\mu,\mu_n\) and predictable compensators \(\nu,\nu_n\).…

We discuss a probabilistic approximation framework for the three-dimensional attractive point interaction on a finite time horizon. By iterating the Doob transforms of the explicit heat kernel associated with the singular Schr\"odinger…

We study a two-component stochastic Klein--Gordon system on \(\mathbb T^3\) with fixed distinct speeds and pure cross interaction \(u_1u_2\). The mixed paracontrolled operators \[ I_i(w<\Psi_j)\circ \Psi_k \] are organized by color--phase…

We study downward deviations of the maximum local time of the discrete-time simple random walk on $\mathbb{Z}^d$, $d\ge 3$. In our previous paper \cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching…

Consider $n$ real/complex, independent/dependent random variables with respective tail bounds and $g$ a measurable function of the r.v.'s. Consider $f$ the "sharpest" tail bound of $g$ (sharpest in the sense that if $f$ were any less, then…

In this paper, we study the problem of testing whether or not a given probability measure $\mu$ on $\mathbb{R}^{d}$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We…