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Résumé |
The optimal fluctuation method — essentially geometrical optics — gives a deep insight into large
deviations of Brownian motion, and it achieves this purpose by simple
means. Here we illustrate these points by telling three short stories about Brownian motions, ``pushed"
into a large-deviation regime by constraints.
Story 1 deals with a long-time survival of a Brownian particle in 1 + 1 dimension against absorption by a
wall which advances according to a power law $x_w (t) \sim
t^{\gamma} $, where $\gamma> 1/2$. We also calculate the large deviation function (LDF) of the particle
position at an earlier time, conditional on the survival by a later
time.
Story 2 addresses a stretched Brownian motion above an absorbing obstacle in the plane. We compute the
short-time LDF of the position of the surviving Brownian
particle at an intermediate point.
In story 3 we compute the short-time LDF of the winding angle of a Brownian particle wandering around a
reflecting disk in the plane.
In all three stories we uncover singularities of the LDFs which can be interpreted as dynamical phase
transitions and which have a simple geometric origin. We also use
the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations.
We argue that, in stories 1 and 2, this is the Ferrari–Spohn
distribution.
The talk is based on a recent paper by B. Meerson and N. R. Smith, J. Phys. A: Math. Theor. 52, 415001
(2019) . |
