First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

TL;DR

This paper derives exact distributions for certain functionals of Brownian motion in logarithmic potentials and extends these results to heterogeneous diffusion processes, highlighting the impact of different Langevin interpretations.

Contribution

It provides explicit formulas for the distribution of first-passage functionals in logarithmic potentials and extends the analysis to heterogeneous diffusion models with different stochastic interpretations.

Findings

Explicit PDF for $eta=0$ case

Laplace transform for $eta eq0$

Extension to heterogeneous diffusion processes

Abstract

We study the statistics of random functionals $\mathcal{Z}=\int_{0}^{\mathcal{T}}[x(t)]^{\gamma-2}dt$, where $x(t)$ is the trajectory of a one-dimensional Brownian motion with diffusion constant $D$ under the effect of a logarithmic potential $V(x)=V_0\ln(x)$. The trajectory starts from a point $x_0$ inside an interval entirely contained in the positive real axis, and the motion is evolved up to the first-exit time $\mathcal{T}$ from the interval. We compute explicitly the PDF of $\mathcal{Z}$ for $\gamma=0$, and its Laplace transform for $\gamma\neq0$, which can be inverted for particular combinations of $\gamma$ and $V_0$. Then we consider the dynamics in $(0,\infty)$ up to the first-passage time to the origin, and obtain the exact distribution for $\gamma>0$ and $V_0>-D$. By using a mapping between Brownian motion in logarithmic potentials and heterogeneous diffusion, we extend this result to functionals measured over trajectories generated by $\dot{x}(t)=\sqrt{2D}[x(t)]^{\theta}\eta(t)$, where $\theta<1$ and $\eta(t)$ is a Gaussian white noise. We also emphasize how the different interpretations that can be given to the Langevin equation affect the results. Our findings are illustrated by numerical simulations, with good agreement between data and theory.