442 research outputs found
On second order elliptic equations with a small parameter
The Neumann problem with a small parameter
is
considered in this paper. The operators and are self-adjoint second
order operators. We assume that has a non-negative characteristic form
and is strictly elliptic. The reflection is with respect to inward
co-normal unit vector . The behavior of
is effectively described via
the solution of an ordinary differential equation on a tree. We calculate the
differential operators inside the edges of this tree and the gluing condition
at the root. Our approach is based on an analysis of the corresponding
diffusion processes.Comment: 28 pages, 1 figure, revised versio
Gauge Group and Topology Change
The purpose of this study is to examine the effect of topology change in the
initial universe. In this study, the concept of -cobordism is introduced to
argue about the topology change of the manifold on which a transformation group
acts. This -manifold has a fiber bundle structure if the group action is
free and is related to the spacetime in Kaluza-Klein theory or
Einstein-Yang-Mills system. Our results revealed that fundamental processes of
compactification in -manifolds. In these processes, the initial high
symmetry and multidimensional universe changes to present universe by the
mechanism which lowers the dimensions and symmetries.Comment: 8 page
Heat conduction induced by non-Gaussian athermal fluctuations
We study the properties of heat conduction induced by non-Gaussian noises
from athermal environments. We find that new terms should be added to the
conventional Fourier law and the fluctuation theorem for the heat current,
where its average and fluctuation are determined not only by the noise
intensities but also by the non-Gaussian nature of the noises. Our results
explicitly show the absence of the zeroth law of thermodynamics in athermal
systems.Comment: 15 pages, 4 figures, PRE in pres
On stochasticity in nearly-elastic systems
Nearly-elastic model systems with one or two degrees of freedom are
considered: the system is undergoing a small loss of energy in each collision
with the "wall". We show that instabilities in this purely deterministic system
lead to stochasticity of its long-time behavior. Various ways to give a
rigorous meaning to the last statement are considered. All of them, if
applicable, lead to the same stochasticity which is described explicitly. So
that the stochasticity of the long-time behavior is an intrinsic property of
the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic
Multiplicative decompositions and frequency of vanishing of nonnegative submartingales
In this paper, we establish a multiplicative decomposition formula for
nonnegative local martingales and use it to characterize the set of continuous
local submartingales Y of the form Y=N+A, where the measure dA is carried by
the set of zeros of Y. In particular, we shall see that in the set of all local
submartingales with the same martingale part in the multiplicative
decomposition, these submartingales are the smallest ones. We also study some
integrability questions in the multiplicative decomposition and interpret the
notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio
Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
Different initial and boundary value problems for the equation of vibrations
of rods (also called Fresnel equation) are solved by exploiting the connection
with Brownian motion and the heat equation. The analysis of the fractional
version (of order ) of the Fresnel equation is also performed and, in
detail, some specific cases, like , 1/3, 2/3, are analyzed. By means
of the fundamental solution of the Fresnel equation, a pseudo-process ,
with real sign-varying density is constructed and some of its properties
examined. The equation of vibrations of plates is considered and the case of
circular vibrating disks is investigated by applying the methods of
planar orthogonally reflecting Brownian motion within . The composition of
F with reflecting Brownian motion yields the law of biquadratic heat
equation while the composition of with the first passage time of
produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure
Prescription-induced jump distributions in multiplicative Poisson processes
Generalized Langevin equations (GLE) with multiplicative white Poisson noise
pose the usual prescription dilemma leading to different evolution equations
(master equations) for the probability distribution. Contrary to the case of
multiplicative gaussian white noise, the Stratonovich prescription does not
correspond to the well known mid-point (or any other intermediate)
prescription. By introducing an inertial term in the GLE we show that the Ito
and Stratonovich prescriptions naturally arise depending on two time scales,
the one induced by the inertial term and the other determined by the jump
event. We also show that when the multiplicative noise is linear in the random
variable one prescription can be made equivalent to the other by a suitable
transformation in the jump probability distribution. We apply these results to
a recently proposed stochastic model describing the dynamics of primary soil
salinization, in which the salt mass balance within the soil root zone requires
the analysis of different prescriptions arising from the resulting stochastic
differential equation forced by multiplicative white Poisson noise whose
features are tailored to the characters of the daily precipitation. A method is
finally suggested to infer the most appropriate prescription from the data
On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity
This paper develops necessary and sufficient conditions for the preservation
of asymptotic convergence rates of deterministically and stochastically
perturbed ordinary differential equations with regularly varying nonlinearity
close to their equilibrium. Sharp conditions are also established which
preserve the asymptotic behaviour of the derivative of the underlying
unperturbed equation. Finally, necessary and sufficient conditions are
established which enable finite difference approximations to the derivative in
the stochastic equation to preserve the asymptotic behaviour of the derivative
of the unperturbed equation, even though the solution of the stochastic
equation is nowhere differentiable, almost surely
On -transforms of one-dimensional diffusions stopped upon hitting zero
For a one-dimensional diffusion on an interval for which 0 is the
regular-reflecting left boundary, three kinds of conditionings to avoid zero
are studied. The limit processes are -transforms of the process stopped
upon hitting zero, where 's are the ground state, the scale function, and
the renormalized zero-resolvent. Several properties of the -transforms are
investigated
Multiple G-It\^{o} integral in the G-expectation space
In this paper, motivated by mathematic finance we introduce the multiple
G-It\^{o} integral in the G-expectation space, then investigate how to
calculate. We get the the relationship between Hermite polynomials and multiple
G-It\^{o} integrals which is a natural extension of the classical result
obtained by It\^{o} in 1951.Comment: 9 page
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