First Passage in Infinite Paraboloidal Domains

TL;DR

This paper analyzes the first-passage properties of diffusing particles in and around generalized paraboloids, revealing dimension-dependent survival probabilities and decay behaviors, with implications for understanding diffusion in complex geometries.

Contribution

It provides a comprehensive analysis of first-passage times for particles in paraboloidal domains, highlighting the effects of geometry and dimension on survival probabilities and decay laws.

Findings

Inside paraboloids, survival probability decays as a stretched exponential independent of dimension.

Outside paraboloids, decay depends on dimension, with finite survival in higher dimensions.

Estimated the distance to the closest surviving particle in a filled paraboloid interior.

Abstract

We study first-passage properties for a particle that diffuses either inside or outside of generalized paraboloids, defined by y=a(x_1^2+...+x_{d-1}^2)^{p/2} where p>1, with absorbing boundaries. When the particle is inside the paraboloid, the survival probability S(t) generically decays as a stretched exponential, ln(S) ~ -t^{(p-1)/(p+1)}, independent of the spatial dimensional. For a particle outside the paraboloid, the dimensionality governs the asymptotic decay, while the exponent p specifying the paraboloid is irrelevant. In two and three dimensions, S ~ t^{-1/4} and S ~(ln t)^{-1}, respectively, while in higher dimensions the particle survives with a finite probability. We also investigate the situation where the interior of a paraboloid is uniformly filled with non-interacting diffusing particles and estimate the distance between the closest surviving particle and the apex of the paraboloid.