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

We study the probability distribution P(A) of the area A = integral(T)(0) 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 exact scaling relation P(A) = D-1/2H/L-1+ 1/H Phi(H) (D-1/2H A/L-1+ 1/H), where 0 < H < 1 is the Hurst exponent characterizing the fBm, D is the coefficient of fractional diffusion, and Phi(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 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.