We consider beta-Dyson Brownian motion, with $\beta>= 1$, started from a deterministic configuration with uniformly bounded support. Let $\mu_t$ be the semicircular free-convolution flow issued from the initial empirical measure, and set $S_t = supp(\mu_t)$. For every fixed $T, \epsilon>0$, with probability at least $1 - C \exp(-(log n)^2)$, every particle remains within an epsilon-neighborhood of $S_t$ for all $0<= t<= T$. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an $o((n \eta)^(-1))$ comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.
We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional $Z_T=\int_0^T W_s^2ds$, where $(W_t)_{t\geq 0}$ is a Brownian motion with drift $\mu$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le \theta T$, as $T\to\in...
Frank Aurzada, Yuvraj Dutta, Max Wiegand· 0 citations
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up...
Joseph Najnudel, Thammadol Tansrivorarat· 0 citations
We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For $\beta=2$, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal c...
Let $B$ be a standard Brownian motion, $x\ge 0,\, \nu<1$, and $b:[0,\infty)\to\mathbb R$ is a continuous function locally of finite variation starting from 0. We define the perturbed Brownian motion reflected at the boundary $b$ by establishing strong existence and pathwise uniqueness of a solution to the equation \[ W...
We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The proce...
We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}\Delta X_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove...
Leonid Mytnik, Johanna Weinberger· 0 citations
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