245 research outputs found
Counting statistics: a Feynman-Kac perspective
By building upon a Feynman-Kac formalism, we assess the distribution of the
number of hits in a given region for a broad class of discrete-time random
walks with scattering and absorption. We derive the evolution equation for the
generating function of the number of hits, and complete our analysis by
examining the moments of the distribution, and their relation to the walker
equilibrium density. Some significant applications are discussed in detail: in
particular, we revisit the gambler's ruin problem and generalize to random
walks with absorption the arcsine law for the number of hits on the half-line.Comment: 10 pages, 6 figure
Universal properties of branching random walks in confined geometries
Characterizing the occupation statistics of a radiation flow through confined
geometries is key to such technological issues as nuclear reactor design and
medical diagnosis. This amounts to assessing the distribution of the travelled
length and the number of collisions performed by the underlying
stochastic transport process, for which remarkably simple Cauchy-like formulas
were established in the case of branching Pearson random walks with
exponentially distributed jumps. In this Letter, we show that such formulas
strikingly carry over to the much broader class of branching processes with
arbitrary jumps, provided that scattering is isotropic and the average jump
size is finite.Comment: 5 pages, 3 figure
First-passage time of a Brownian motion: two unexpected journeys
The distribution of the first-passage time for a Brownian particle with
drift subject to hitting an absorber at a level is well-known and
given by its density , which is normalized only if . In this article,
we show that there are two other families of diffusion processes (the first
with one parameter and the second with two parameters) having the same first
passage-time distribution when . In both cases we establish the
propagators and study in detail these new processes. An immediate consequence
is that the distribution of the first-passage time does not allow us to know if
the process comes from a drifted Brownian motion or from one of these new
processes.Comment: 16 pages, 2 figures; improved mathematical notatio
Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture
Monte Carlo (MC) simulations allowing to describe photons propagation in
statistical mixtures represent an interest that goes way beyond the domain of
optics, and can cover, e.g., nuclear reactor physics, image analysis or life
science just to name a few. MC simulations are considered a ``gold standard''
because they give exact solutions (in the statistical sense), however, in the
case of statistical mixtures they are enormously time consuming and their
implementation is often extremely complex. For this reason, the aim of the
present contribution is to propose a new approach that should allow us in the
future to simplify the MC approach. This is done through an explanatory
example, i.e.; by deriving the `exact' analytical expression for the
probability density function of photons' random steps (single step function,
SSF) propagating in a medium represented as a binary (isotropic-Poisson)
statistical mixture. The use of the SSF reduces the problem to an `equivalent'
homogeneous medium behaving exactly as the original binary statistical mixture.
This will reduce hundreds time-consuming MC simulations to only one equivalent
simple MC simulation. To the best of our knowledge the analytically `exact' SSF
for a binary (isotropic-Poisson) statistical mixture has never been derived
before
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