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DY: Fachverband Dynamik und Statistische Physik
DY 59: Brownian Motion and Anomalous Diffusion
DY 59.6: Talk
Friday, March 22, 2024, 10:45–11:00, BH-N 334
First-passage area distribution and optimal fluctuations of fractional Brownian motion — •Alexander K. Hartmann1 and Baruch Meerson2 — 1University of Oldenburg, Germany — 2Hebrew University of Jerusalem, Israel
We study the probability distribution P(A) of the area A=∫0T x(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 the exact scaling relation P(A) = D1/2H/L1+1/H ΦH(D1/2H A/L1+1/H) , 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 [1] as having an essential singularity at A=0. Here [2] we determine the large-A tail of P(A). It is a fat tail, with the average value A diverging for all H. We also verify the predictions for both tails by performing simple-sampling as well as large-deviation Monte Carlo [3] 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.
[1] B. Meerson and G. Oshanin, Phys. Rev. E 105, 064137 (2022).
[2] A.K. Hartmann and B. Meerson, preprint arXiv:2310.14003 (2023).
[3] A.K. Hartmann, Phys. Rev. E 89, 052103 (2014).
Keywords: large-deviation algorithm; correlations; simulations; area; first-passage time