Related papers: Narrow Escape, Part II: The circular disk
In nanoscale space and pico- to nanoseconds enormous physical, chemical and biological processes take place, while the motions of involved particles/molecules under thermal fluctuations are usually analyzed using the conventional theory of…
We study the problem of particles undergoing Brownian motion in an expanding sphere whose surface is an absorbing boundary for the particles. The problem is akin to that of the diffusion of impurities in a grain of polycrystalline material…
How long a stochastic process survives before leaving a domain depends not only on its intrinsic dynamics but also on how it is observed. Classical first-passage theory assumes continuous monitoring with absorbing boundaries…
We consider the first exit time $\tau = \min \{n\ge 1 : S_n\le 0\}$ from the positive halfline of a random walk $S_n = \sum_1^n \xi_i, n\ge 1$ with i.d.d. summands having a negative drift ${\mathbb E} \xi = -a< 0$. Let $\xi^+ = \max (0,…
The paper addresses Brownian motion in the logarithmic potential with time-dependent strength, $U(x,t) = g(t) \log(x)$, subject to the absorbing boundary at the origin of coordinates. Such model can represent kinetics of…
For drifted Brownian motion $X(t)= x - \mu t + B_t \ (\mu >0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $\tau(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time…
Cells have evolved efficient strategies to probe their surroundings and navigate through complex environments. From metastatic spread in the body to swimming cells in porous materials, escape through narrow constrictions - a key component…
We show in detail some results, outlined in a previous paper regarding the case of Brownian motion (BM), about the distribution of the $n$th-passage time of a one-dimensional diffusion obtained by a space or time transformation of BM,…
We obtain bounds for the expected loss of torsional rigidity of a cylinder $\Omega_L=(-L/2,L/2) \times \Omega\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $\Omega_L,$ and runs until the first time…
We investigate a moving boundary problem for a Brownian particle on the semi-infinite line in which the boundary moves by a distance proportional to the time between successive collisions of the particle and the boundary. Phenomenologically…
A possible mechanism leading to anomalous diffusion is the presence of long-range correlations in time between the displacements of the particles. Fractional Brownian motion, a non-Markovian self-similar Gaussian process with stationary…
We study here the extreme statistics of Brownian particles escaping from a cusp funnel: the fastest Brownian particles among $n$ follow an ensemble of optimal trajectories located near the shortest path from the source to the target. For…
We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time…
We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the…
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization…
We study a Brownian motion with drift in a wedge of angle $\beta$ which is obliquely reflected on each edge along angles $\varepsilon$ and $\delta$. We assume that the classical parameter $\alpha=\frac{\delta+\varepsilon - \pi}{\beta}$ is…
Suppose that an infinite lattice gas of constant density $n_0$, whose dynamics are described by the symmetric simple exclusion process, is brought in contact with a spherical absorber of radius $R$. Employing the macroscopic fluctuation…
We investigate the Kramers escape from a potential well of a test particle driven by fractional Gaussian noise with Hurst exponent 0<H<1. From a numerical analysis we demonstrate the exponential distribution of escape times from the well…
Consider the all-time maximum of a Brownian motion with negative drift. Assume that this process is sampled at certain points in time, where the time between two consecutive points is rendered by an Erlang distribution with mean $1/\omega$.…
We study the second-order asymptotics around the superdiffusive strong law~\cite{MMW} of a multidimensional driftless diffusion with oblique reflection from the boundary in a generalised parabolic domain. In the unbounded direction we prove…