561 research outputs found
Motion in a Random Force Field
We consider the motion of a particle in a random isotropic force field.
Assuming that the force field arises from a Poisson field in , , and the initial velocity of the particle is sufficiently large, we
describe the asymptotic behavior of the particle
Area Distribution of Elastic Brownian Motion
We calculate the excursion and meander area distributions of the elastic
Brownian motion by using the self adjoint extension of the Hamiltonian of the
free quantum particle on the half line. We also give some comments on the area
of the Brownian motion bridge on the real line with the origin removed. We will
stress on the power of self adjoint extension to investigate different possible
boundary conditions for the stochastic processes.Comment: 18 pages, published versio
Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
We have derived a fractional Fokker-Planck equation for subdiffusion in a
general space-and- time-dependent force field from power law waiting time
continuous time random walks biased by Boltzmann weights. The governing
equation is derived from a generalized master equation and is shown to be
equivalent to a subordinated stochastic Langevin equation.Comment: 5 page
Stochastic integration based on simple, symmetric random walks
A new approach to stochastic integration is described, which is based on an
a.s. pathwise approximation of the integrator by simple, symmetric random
walks. Hopefully, this method is didactically more advantageous, more
transparent, and technically less demanding than other existing ones. In a
large part of the theory one has a.s. uniform convergence on compacts. In
particular, it gives a.s. convergence for the stochastic integral of a finite
variation function of the integrator, which is not c\`adl\`ag in general.Comment: 16 pages, some typos correcte
On the distribution of estimators of diffusion constants for Brownian motion
We discuss the distribution of various estimators for extracting the
diffusion constant of single Brownian trajectories obtained by fitting the
squared displacement of the trajectory. The analysis of the problem can be
framed in terms of quadratic functionals of Brownian motion that correspond to
the Euclidean path integral for simple Harmonic oscillators with time dependent
frequencies. Explicit analytical results are given for the distribution of the
diffusion constant estimator in a number of cases and our results are confirmed
by numerical simulations.Comment: 14 pages, 5 figure
Classical motion in force fields with short range correlations
We study the long time motion of fast particles moving through time-dependent
random force fields with correlations that decay rapidly in space, but not
necessarily in time. The time dependence of the averaged kinetic energy and
mean-squared displacement is shown to exhibit a large degree of universality;
it depends only on whether the force is, or is not, a gradient vector field.
When it is, p^{2}(t) ~ t^{2/5} independently of the details of the potential
and of the space dimension. Motion is then superballistic in one dimension,
with q^{2}(t) ~ t^{12/5}, and ballistic in higher dimensions, with q^{2}(t) ~
t^{2}. These predictions are supported by numerical results in one and two
dimensions. For force fields not obtained from a potential field, the power
laws are different: p^{2}(t) ~ t^{2/3} and q^{2}(t) ~ t^{8/3} in all dimensions
d\geq 1
Quantum noise and stochastic reduction
In standard nonrelativistic quantum mechanics the expectation of the energy
is a conserved quantity. It is possible to extend the dynamical law associated
with the evolution of a quantum state consistently to include a nonlinear
stochastic component, while respecting the conservation law. According to the
dynamics thus obtained, referred to as the energy-based stochastic Schrodinger
equation, an arbitrary initial state collapses spontaneously to one of the
energy eigenstates, thus describing the phenomenon of quantum state reduction.
In this article, two such models are investigated: one that achieves state
reduction in infinite time, and the other in finite time. The properties of the
associated energy expectation process and the energy variance process are
worked out in detail. By use of a novel application of a nonlinear filtering
method, closed-form solutions--algebraic in character and involving no
integration--are obtained for both these models. In each case, the solution is
expressed in terms of a random variable representing the terminal energy of the
system, and an independent noise process. With these solutions at hand it is
possible to simulate explicitly the dynamics of the quantum states of
complicated physical systems.Comment: 50 page
Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed semimartingale
is a time-changed stochastic integral driven by the original semimartingale. As
a direct consequence, a specialized form of the Ito formula is derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with drift
extend to a larger class of stochastic differential equations involving a
time-change with continuous paths. A form of the general solution of linear
equations in this new class is established, followed by consideration of some
examples analogous to the classical equations. Through these examples, each
coefficient of the stochastic differential equations in the new class is given
meaning. The new feature is the coexistence of a usual drift term along with a
term related to the time-change.Comment: 27 pages; typos correcte
Some results and problems for anisotropic random walks on the plane
This is an expository paper on the asymptotic results concerning path
behaviour of the anisotropic random walk on the two-dimensional square lattice
Z^2. In recent years Mikl\'os and the authors of the present paper investigated
the properties of this random walk concerning strong approximations, local
times and range. We give a survey of these results together with some further
problems.Comment: 20 page
Random walk generated by random permutations of {1,2,3, ..., n+1}
We study properties of a non-Markovian random walk , , evolving in discrete time on a one-dimensional lattice of
integers, whose moves to the right or to the left are prescribed by the
\text{rise-and-descent} sequences characterizing random permutations of
. We determine exactly the probability of finding
the end-point of the trajectory of such a
permutation-generated random walk (PGRW) at site , and show that in the
limit it converges to a normal distribution with a smaller,
compared to the conventional P\'olya random walk, diffusion coefficient. We
formulate, as well, an auxiliary stochastic process whose distribution is
identic to the distribution of the intermediate points , ,
which enables us to obtain the probability measure of different excursions and
to define the asymptotic distribution of the number of "turns" of the PGRW
trajectories.Comment: text shortened, new results added, appearing in J. Phys.
- âŠ