# A unifying approach to first-passage time distributions in diffusing diffusivity and switching diffusion models

## TL;DR

This paper introduces a unified theoretical framework to analyze first-passage time distributions in stochastic processes with evolving diffusivity, covering both continuous and discrete diffusivity models, with explicit formulas and solutions.

## Contribution

It provides a comprehensive approach unifying diffusing diffusivity and switching diffusion models for first-passage time analysis, including general formulas and explicit solutions.

## Key findings

- Derived moment-generating functions for integrated diffusivity
- Provided explicit solutions for specific cases
- Quantified diffusivity dynamics impact on first-passage times

## Abstract

We propose a unifying theoretical framework for the analysis of first-passage time distributions in two important classes of stochastic processes in which the diffusivity of a particle evolves randomly in time. In the first class of "diffusing diffusivity" models, the diffusivity changes continuously via a prescribed stochastic equation. In turn, the diffusivity switches randomly between discrete values in the second class of "switching diffusion" models. For both cases, we quantify the impact of the diffusivity dynamics onto the first-passage time distribution of a particle via the moment-generating function of the integrated diffusivity. We provide general formulas and some explicit solutions for some particular cases of practical interest.