First-passage area distribution and optimal fluctuations of fractional Brownian motion

Abstract

We study the probability distribution $P(A)$ of the area $A=\int_0^T x(t) dt$ swept under fractional Brownian motion (fB\ m) $x(t)$ until its first passage time $T$ to the origin. The process starts at $t=0$ from a specified point $x=L$. We show that $P(A)$ obeys exact scaling relation $$ P(A) = \frac{D^\frac{1}{2H}}{L^{1+\frac{1}{H}}}\,Φ_H\left(\frac{D^\frac{1}{2H} A}{L^{1+\frac{1}{H}}}\right)\,, $$ where $0<H<1$ is the Hurst exponent characterizing the fBm, $D$ is the coefficient of fractional diffusion, and $Φ_H(z)$ is a scaling function. The small-$A$ tail of $P(A)$ has been recently predicted by Meerson and Oshanin [Phys. Rev. E 105, 064137 (2022)], who showed that it has an essential singularity at $A=0$, the character of which depends on $H$. Here we determine the large-$A$ tail of $P(A)$. It is a fat tail, in particular such that the average value of the first-passage area $A$ diverges for all $H$. We also verify the predictions for both tails by performing simple-sampling as well as large-deviation Monte Carlo simulations. The verification includes measurements of $P(A)$ up to probability densities as small as $10^{-190}$. We also perform direct observations of paths conditioned to the area $A$. For the steep small-$A$ tail of $P(A)$ the "optimal paths", i.e. the most probable trajectories of the fBm, dominate the statistics. Finally, we discuss extensions of theory to a more general first-passage functional of the fBm.

Figures (8)

• Figure 1: Example of a random process, starting at $x(0)=L$ and hitting $x=0$ for the first time at time $T$. The area under the curve until time $T$ is denoted by $A$, and its distribution for the fBm is the focus of this work.
• Figure 2: Simulated first-passage area distribution $P(A)$ for the standard random walk versus exact theoretical prediction (\ref{['PAexact']}) for the standard Brownian motion ($H=1/2$) with $L=70$.
• Figure 3: Simulated first-passage area distributions $P(A)$ for the fractional random walk with $H=3/4$ for $L=20$, $L=50$ and $L=100$. The axes are rescaled according to our prediction (\ref{['exactscaling']}) for the fBm. Also shown are two predicted asymptotics for the fBm: Eqs. (\ref{['smallA']}) and (\ref{['largeA']}).
• Figure 4: Simulated first-passage area distributions $P(A)$ for the fractional random walk with $H=1/4$ for $L=5$, $L=10$ and $L=50$. The axes are rescaled according to our prediction (\ref{['exactscaling']}) for the fBm. Also shown is the large-$A$ tail prediction (\ref{['largeA']}).
• Figure 5: The small-$A$ tail of the simulated first-passage area distribution $P(A)$ as a function of $1/A$ for the fractional random walk with $H=1/4$ (symbols). Solid lines: theoretical prediction (\ref{['smallA']}) for the fBm.