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Register a new userThe walk on moving spheres: a new tool for simulating Brownian motions exit time from a domain
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In this paper we introduce a new method for the simulation of the exit time and position of a $delta$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $delta$-dimensional Bessel process and the $delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with existing methods are performed.
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Let $D$ be an unbounded domain in $RR^d$ with $dgeq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $overline D$ is transient. Next assume that RBM $X$ on $overline D$ is transient and let $Y$ be its time change by Revuz measure ${bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $overline D$. We further show that if there is some $r>0$ so that $Dsetminus overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
We consider random walk on dynamical percolation on the discrete torus $mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
We prove the existence of the intersection local time for two independent, d -dimensional fractional Brownian motions with the same Hurst parameter H. Assume d greater or equal to 2, then the intersection local time exists if and only if Hd<2.
We consider the scaling behavior of the range and $p$-multiple range, that is the number of points visited and the number of points visited exactly $pgeq 1$ times, of simple random walk on ${mathbb Z}^d$, for dimensions $dgeq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $Dsubset {mathbb R}^d$, as $Nuparrowinfty$. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case $D$ is a cube in $dgeq 3$, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in $dgeq 2$, both weakly converge to proportional exit times of Brownian motion from $D$, and that the corresponding limit moments are `polyharmonic, solving a hierarchy of Poisson equations.
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