We consider a collection of independent standard Brownian particles (or random walks), starting from a configuration where at least one particle is positive, and study the first time they all become negative. This is clearly equivalent to studying the first hitting time from the negative orthant or the first exit time from the complement of the negative orthant. While it turns out to be possible to compute the distribution of these hitting times for one and two particles, the distribution (and even its tail asymptotics) is not known in closed form for three or more particles. In this paper we study the tail asymptotics of the distribution as the number of particles tends to infinity. Our main techniques come from spectral geometry: we prove new asymptotic estimates for the principal eigenvalue of the complement of a high-dimensional orthant, which we believe are of independent interest.
The exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case
对于满足Cramér矩条件且漂移向量至少有一个负分量的多元Lévy过程，我们得到了沿正分量的固定向量平移到无穷远的正正交点的概率的精确渐近性.这个问题的动机是在F中引入的多变量破产问题。Avram et al.（2008）在二维情况下。我们的解决方案依赖于Y. Pan和K. Borovkov（2017）用于多变量随机游动和适当的时间离散化。
For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having a mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, whic…
We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$ variables attains a relative maximum on a vertex of the $n$-dimensional simplex tends to one as the dimension $n$ grows. The bounds we obtain for the orthant probabilities are tight up to $\log(n)$ factors.
It is shown by the author in 2017 that for the union of N orthants in the n-dimensional space there exists an efficient and systematic way to find the exact value, using a suitable partial order relation construction. In this paper our events are hyperrectangles (or n-orthotopes), the Cartesian product of intervals: another important sets (or events) in both theory and practice. We have discovered a new efficient algorithm for the union of such events. With other important Boolean functions we present optimization problem formulations for both hyperreectangles and orthants.
We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional $α$-stable processes $X_t$, where $α\in(1,2)$. We also find a spectral-type integral formula for the transition operators $P_0^t$ of $X_t$ killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for $P_0^t$.
We consider the rates of convergence of the quenched central limit theorem for hitting times of one-dimensional random walks in a random environment. Previous results had identified polynomial upper bounds for the rates of decay which are sometimes slower than $n^{-1/2}$ (the optimal rate in the classical Berry-Esseen estimates). Here we prove that the previous upper bounds are in fact the best possible polynomial rates for the quenched CLT.
Weak weak quenched limits for the path-valued processes of hitting times and positions of a transient, one-dimensional random walk in a random environment
The purpose of the paper is to find the joint distribution of the hitting time and place of two-dimensional Brownian motion hitting the negative horizontal axis. We provide various formulas for Green functions as well as for the conditional distributions. In the end we treat an analogous problem for the case when vertical component has a negative drift.
We consider the superposition of symmetric simple exclusion dynamics speeded-up in time, with spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We show that the mixing time has an exponential lower bound in the system size if the potential of the hydrodynamic equation has more than two local minima. We also apply our estimates to show that the normalized hitting times of rare events converge to a mean one exponential random variable if the potential has a unique minimum.
We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $γ$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.
On the hitting times of continuous-state branching processes with immigration
研究了具有移民的连续态分支过程及其在此时刻停止的原态的常能级的首中时的二维联合分布。我们给出了它的拉普拉斯变换的显式表达式。利用这个公式，我们研究了零的极性，并给出了一个瞬时性或常返性的充分必要判据。我们遵循的方法志贺，T。（1990）[A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type.概率论and Related Fields，85（4），425-447]，通过找到生成器的一些$λ$不变函数。
Fixation probabilities and hitting times under low levels of frequency-dependent selection
在群体遗传学中，单位区间上的扩散经常被用来模拟等位基因的频率路径。在这种情况下，我们得到近似的固定概率，预期的首中时和预期的网站下的频谱依赖于频率的选择。具体地说，我们重新推导和扩展了进化对策论的三分之一规则（Nowak et al.，2004）和随机减速的影响（Altrock和Traulsen，2009）。由于类似的效果在其他应用领域的兴趣，我们制定我们的结果一般一维扩散。
We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.
A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, …
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.
Consider the linear stochastic biharmonic heat equation on a $d$-dimensional torus ($d=1,2,3$), driven by a space-time white noise and with periodic boundary conditions: \begin{equation} \label{0} \left(\frac{\partial}{\partial t}+(-Δ)^2\right) v(t,x)= σ\dot W(t,x),\ (t,x)\in(0,T]\times \mathbb{T}^d,\ v(0,x)=v_0(x). \end{equation} We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for $d=2$, they include a $z(\log \tfrac{c}{z})^{1/2}$ term. Consider $D$ independent copies of the ran…
