On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field

Transition densities of spectrally positive Lévy processes

We prove asymptotic behaviour of transition density for a large class of spectrally one-sided Lévy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.

Conditional Law and Occupation Times of Two-Sided Sticky Brownian Motion

Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.

Values of Brownian intersection exponents III: Two-sided exponents

This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent $ξ(3,3)$ describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is $(73-2 \sqrt {73}) / 12$. More generally, the values of $ξ(w_1, >..., w_k)$ and $\tx (w_1', ..., w_k')$ are determined for all $ k \ge 2$, $w_1, w_2\ge 1$, $w_3, ...,w_k\in[0,\infty)$ and all $w_1',...,w_k'\in[0,\infty)$. The …

Dirichlet Heat Kernel Estimates for Rotationally Symmetric Lévy processes

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by Ψ(ξ)= ϕ(|ξ|^2) with ϕbeing a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are expl…

Lévy 过程的双边长时间优化奇异控制问题与 Dynkin 博弈

Two sided long-time optimization singular control problems for Lévy processes and Dynkin's games

本文建立了 Lévy 过程双边奇异控制与 Dynkin 博弈的联系，证明最优控制可在反射屏障策略中求得，并给出三个示例。

非自治随机微分方程Milstein格式的李雅普诺夫指数的双边估计

Two-sided estimates of Lyapunov exponents for Milstein schemes of non-autonomous stochastic differential equations

本文针对非自治随机微分方程的Milstein格式，给出李雅普诺夫指数的精确上下估计，证明连续场景的稳定效应可保留至离散场景，核心是利用Milstein格式的随机噪声二阶项完成泰勒展开估计。

Law of two-sided exit by a spectrally positive strictly stable process

For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper end point, and ii) the joint distribution of the time, the undershoot, and the jump of the process when it makes first exit the other way around. For i), the density of the time of first exit is expressed as an infinite sum of functions, each the product of a polynomial and an exponential function, with all coefficients determined by the roots of a Mittag-Lef…

Moments of Gaussian chaoses in Banach spaces

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including L_q spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in L_q spaces based on exponential random variables.

Two-sided Green function estimates for killed subordinate Brownian motions

A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-ϕ(-Δ)$, where $ϕ$ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with $ϕ$ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of indepe…

Formulas for ASEP with Two-Sided Bernoulli Initial Condition

For the asymmetric simple exclusion process on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the probability that site x is occupied at time t; (2) a correlation function, the probability that site 0 is occupied at time 0 and site x is occupied at time t; (3) the distribution function for the total flux across 0 at time t and its exponential generating function.

Heavy-tailed random walks on complexes of half-lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution $μ_k$. If $χ_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and $α_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all $α_k χ_k \in (0,1)$ is determined by the sign…

Strict inequality for the chemical distance exponent in two-dimensional critical percolation

We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain …

Sharp two-sided Green function estimates for Dirichlet forms degenerate at the boundary

In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-α}$ and killing potentials $κ(x)=cx_d^{-α}$, $0<α<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $β_1, β_2$, $β_3$ and $β_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $α$, ${\mathcal B}$ and a parameter $p\in ((α-1)_+, α+β_1)…

Dirichlet heat kernel for the Laplacian in a ball

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates.

On Chemical Distance and Local Uniqueness of a Sufficiently Supercritical Finitary Random Interlacement

In this paper, we study geometric properties of the unique infinite cluster $Γ$ in a sufficiently supercritical Finitary Random Interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d, \ d\ge 3$. We prove that the chemical distance in $Γ$ is, with stretched exponentially high probability, of the same order as the Euclidean distance in $\mathbb{Z}^d$. This also implies a shape theorem parallel to those for Bernoulli percolation and random interlacements. We also prove local uniqueness of $\mathcal{FI}^{u,T}$, which says any two large clusters in $\mathcal{FI}^{u,T}$ "close to each other" will with…

单侧弹道沉积中轴线附近的GUE涨落

GUE Fluctuations Near the Axis in One-Sided Ballistic Deposition

本文研究单侧弹道沉积模型，通过有向最后通过渗流的图形化构造证明高度函数在t轴附近的涨落服从GUE系综及Tracy-Widom分布，并确定测地线横向涨落的游走指数为2/3。

双侧热核估计 for √(8/3)-Liouville Brownian motion

Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion

本文研究了当γ=√(8/3)时，Liouville Brownian motion的热核上界和下界，基于√(8/3)-LQG度量，结果在指数上至多相差多项对数因子。

A two-sided estimate for the Gaussian noise stability deficit

The Gaussian noise-stability of a set A in R^n is defined by S_rho(A) = P (X in A and Y in A) where X and Y are standard Gaussian vectors whose correlation is rho. Borell's inequality states that for all 0 < rho < 1, among all sets A with a given Gaussian measure, the quantity S_rho(A) is maximized when A is a half-space. We give a novel short proof of this fact, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set A and its corresponding half-spac…