For every $\beta>0$, we establish multi-time soft-edge moment convergence for a class of Jack processes, including the Jack--Plancherel process, a discrete $\beta$-analogue of Dyson Brownian motion. The limiting moments admit an absolutely convergent expansion in terms of nonnegative Brownian bridges decorated by pairs of equal and opposite jumps. At equal times, we identify this limit with the joint Laplace statistics of the Airy$_\beta$ process for $\beta\geq1$. The same Brownian expansion yields an asymptotic $\beta$-topological expansion of the marginal $b$-conjecture and, at $\beta=2$, an explicit nonnegative formula for the Witten--Kontsevich intersection numbers.
We initiate a geometric theory of $p$-Brownian motion, the nonlinear Markov process associated with the $p$-Laplacian introduced by Barbu-Rehmeier-R\"ockner. More precisely, we analyze its radial processes thoroughly, relative to an arbitrary center and in every dimension. On the one hand, we show explicit Tanaka--Meye...
Let $\mathcal{C}_\varepsilon$ denote the time it takes a Brownian motion to come within distance $\varepsilon$ of every point in the two dimensional torus $\mathbb{T}^2$ of unit area. We prove that for an explicit non-random centering $m_\varepsilon$, the variables $\{\sqrt{\mathcal{C}_{\varepsilon}} - m_\varepsilon \}...
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 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...
The goal of the present paper is to investigate the exact decay rate of the tail probability $\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assu...
We evaluate moment constants for the times during which Brownian motion admits uniform approximation by an unrestricted affine function or by a line through the origin. The first four moments are rational linear combinations of $1$ and $\zeta(2j+1)/\pi^{2j}$, $1\le j\le4$; the means are $40\zeta(3)/\pi^2-4/3$ and $21\z...
Yueh-Tung Chen· 1 citation· ⚡1
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