Meerson Baruch, Oshanin Gleb
Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.
Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), 4 Place Jussieu, 75252 Paris Cedex 05, France.
Phys Rev E. 2022 Jun;105(6-1):064137. doi: 10.1103/PhysRevE.105.064137.
It has been shown recently that the optimal fluctuation method-essentially geometrical optics-provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, -∞<t<∞, fractional Brownian motion (fBm) on the line, which is "pushed" to a large deviation regime by imposed constraints. We test the formalism on three examples where exact solutions are available: the two- and three-point probability distributions of the fBm and the distribution of the area under the fBm on a specified time interval. Then we apply the formalism to several previously unsolved problems by evaluating large-deviation tails of the following distributions: (i) of the first-passage time, (ii) of the maximum of, and (iii) of the area under, fractional Brownian bridge and fractional Brownian excursion, and (iv) of the first-passage area distribution of the fBm. An intrinsic part of a geometrical optics calculation is determination of the optimal path-the most likely realization of the process which dominates the probability distribution of the conditioned process. Due to the non-Markovian nature of the fBm, the optimal paths of a fBm, subject to constraints on a finite interval 0<t≤T, involve both the past -∞<t<0 and the future T<t<∞.
最近的研究表明，最优涨落方法——本质上是几何光学——为布朗运动的大偏差提供了有价值的见解。在此，我们将几何光学形式主义扩展到直线上的双边（-∞<t<∞）分数布朗运动（fBm），它通过施加约束被“推向”大偏差 regime。我们在三个有精确解的例子上检验该形式主义：fBm的两点和三点概率分布以及fBm在指定时间间隔下的面积分布。然后，我们通过评估以下分布的大偏差尾部，将该形式主义应用于几个先前未解决的问题：（i）首次通过时间的分布，（ii）分数布朗桥和分数布朗游程的最大值的分布，（iii）分数布朗桥和分数布朗游程下方的面积的分布，以及（iv）fBm的首次通过面积分布。几何光学计算中一个内在的部分是确定最优路径——过程最可能的实现，它主导了条件过程的概率分布。由于fBm的非马尔可夫性质，在有限区间0<t≤T上受约束的fBm的最优路径涉及过去（-∞<t<0）和未来（T<t<∞）。
