## Abstract

We study the probability distribution P(A) of the area A =

R 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

ΦH

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 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)], 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.

R 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

ΦH

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 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)], 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.

Original language | American English |
---|---|

Publisher | arXiv |

Pages | 1-9 |

Number of pages | 9 |

Volume | 2310.14003 |

DOIs | |

State | Published - 2023 |