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

Alexander K. Hartmann, Baruch Meerson

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We study the probability distribution P(A) of the area A=∫0Tx(t)dt swept under fractional Brownian motion (fBm) 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)=D12HL1+1HφHD12HAL1+1H, where 0<H<1 is the Hurst exponent characterizing the fBm, D is the coefficient of fractional diffusion, and φH(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)2470-004510.1103/PhysRevE.105.064137], 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 on 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.

Original languageAmerican English
Article number014146
JournalPhysical Review E
Issue number1
StatePublished - Jan 2024

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