Partial Survival and Crossing Statistics for a Diffusing Particle in a Transverse Shear Flow

TL;DR

This paper analyzes the survival probability and crossing statistics of a non-Gaussian stochastic process involving a diffusing particle in a shear flow, deriving explicit expressions for survival exponents and crossing densities.

Contribution

It provides an explicit formula for the survival exponent (p) in a non-Gaussian shear flow model, linking it to model parameters and crossing statistics.

Findings

Survival probability decays as a power law with exponent (p).

Derived explicit expression for (p) in terms of  and flow parameters.

Calculated mean and variance of crossing density for the process.

Abstract

We consider a non-Gaussian stochastic process where a particle diffuses in the $y$-direction, $dy/dt=\eta(t)$, subject to a transverse shear flow in the $x$-direction, $dx/dt=f(y)$. Absorption with probability $p$ occurs at each crossing of the line $x=0$. We treat the class of models defined by $f(y) = \pm v_{\pm}(\pm y)^\alpha$ where the upper (lower) sign refers to $y>0$ ($y<0$). We show that the particle survives up to time $t$ with probability $Q(t) \sim t^{-\theta(p)}$ and we derive an explicit expression for $\theta(p)$ in terms of $\alpha$ and the ratio $v_+/v_-$. From $\theta(p)$ we deduce the mean and variance of the density of crossings of the line $x=0$ for this class of non-Gaussian processes.