1,359 research outputs found
On distributions of functionals of anomalous diffusion paths
Functionals of Brownian motion have diverse applications in physics,
mathematics, and other fields. The probability density function (PDF) of
Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger
equation in imaginary time. In recent years there is a growing interest in
particular functionals of non-Brownian motion, or anomalous diffusion, but no
equation existed for their PDF. Here, we derive a fractional generalization of
the Feynman-Kac equation for functionals of anomalous paths based on
sub-diffusive continuous-time random walk. We also derive a backward equation
and a generalization to Levy flights. Solutions are presented for a wide number
of applications including the occupation time in half space and in an interval,
the first passage time, the maximal displacement, and the hitting probability.
We briefly discuss other fractional Schrodinger equations that recently
appeared in the literature.Comment: 25 pages, 4 figure
Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
We calculate analytically the probability density of the time
at which a continuous-time Brownian motion (with and without drift) attains its
maximum before passing through the origin for the first time. We also compute
the joint probability density of the maximum and . In the
driftless case, we find that has power-law tails: for large and for small . In
presence of a drift towards the origin, decays exponentially for large
. The results from numerical simulations are in excellent agreement with
our analytical predictions.Comment: 13 pages, 5 figures. Published in Journal of Statistical Mechanics:
Theory and Experiment (J. Stat. Mech. (2007) P10008,
doi:10.1088/1742-5468/2007/10/P10008
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde
Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction
We consider three one-dimensional continuous-time Markov processes on a
lattice, each of which models the conduction of heat: the family of Brownian
Energy Processes with parameter , a Generalized Brownian Energy Process, and
the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these
three processes is a parabolic equation, the linear heat equation in the case
of the BEP and the KMP, and a nonlinear heat equation for the GBEP().
We prove the hydrodynamic limit rigorously for the BEP, and give a formal
derivation for the GBEP().
We then formally derive the pathwise large-deviation rate functional for the
empirical measure of the three processes. These rate functionals imply
gradient-flow structures for the limiting linear and nonlinear heat equations.
We contrast these gradient-flow structures with those for processes describing
the diffusion of mass, most importantly the class of Wasserstein gradient-flow
systems. The linear and nonlinear heat-equation gradient-flow structures are
each driven by entropy terms of the form ; they involve dissipation
or mobility terms of order for the linear heat equation, and a
nonlinear function of for the nonlinear heat equation.Comment: 29 page
Convex hull of n planar Brownian paths: an exact formula for the average number of edges
We establish an exact formula for the average number of edges appearing on
the boundary of the global convex hull of n independent Brownian paths in the
plane. This requires the introduction of a counting criterion which amounts to
"cutting off" edges that are, in a specific sense, small. The main argument
consists in a mapping between planar Brownian convex hulls and configurations
of constrained, independent linear Brownian motions. This new formula is
confirmed by retrieving an existing exact result on the average perimeter of
the boundary of Brownian convex hulls in the plane.Comment: 14 pages, 8 figures, submitted to JPA. (Typo corrected in equation
(14).
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