221,512 research outputs found
Stochastic Gravity
Gravity is treated as a stochastic phenomenon based on fluctuations of the
metric tensor of general relativity. By using a (3+1) slicing of spacetime, a
Langevin equation for the dynamical conjugate momentum and a Fokker-Planck
equation for its probability distribution are derived. The Raychaudhuri
equation for a congruence of timelike or null geodesics leads to a stochastic
differential equation for the expansion parameter in terms of the
proper time . For sufficiently strong metric fluctuations, it is shown that
caustic singularities in spacetime can be avoided for converging geodesics. The
formalism is applied to the gravitational collapse of a star and the
Friedmann-Robertson-Walker cosmological model. It is found that owing to the
stochastic behavior of the geometry, the singularity in gravitational collapse
and the big-bang have a zero probability of occurring. Moreover, as a star
collapses the probability of a distant observer seeing an infinite red shift at
the Schwarzschild radius of the star is zero. Therefore, there is a vanishing
probability of a Schwarzschild black hole event horizon forming during
gravitational collapse.Comment: Revised version. Eq. (108) has been modified. Additional comments
have been added to text. Revtex 39 page
A survey of random processes with reinforcement
The models surveyed include generalized P\'{o}lya urns, reinforced random
walks, interacting urn models, and continuous reinforced processes. Emphasis is
on methods and results, with sketches provided of some proofs. Applications are
discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Exponential Stabilisation of Continuous-time Periodic Stochastic Systems by Feedback Control Based on Periodic Discrete-time Observations
Since Mao in 2013 discretised the system observations for stabilisation problem of hybrid SDEs (stochastic differential equations with Markovian switching) by feedback control, the study of this topic using a constant observation frequency has been further developed. However, time-varying observation frequencies have not been considered. Particularly, an observational more efficient way is to consider the time-varying property of the system and observe a periodic SDE system at the periodic time-varying frequencies. This study investigates how to stabilise a periodic hybrid SDE by a periodic feedback control, based on periodic discrete-time observations. This study provides sufficient conditions under which the controlled system can achieve pth moment exponential stability for p > 1 and almost sure exponential stability. Lyapunov's method and inequalities are main tools for derivation and analysis. The existence of observation interval sequences is verified and one way of its calculation is provided. Finally, an example is given for illustration. Their new techniques not only reduce observational cost by reducing observation frequency dramatically but also offer flexibility on system observation settings. This study allows readers to set observation frequencies according to their needs to some extent
Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample
We provide an example of a Hamilton-Jacobi equation in which stochastic
homogenization does not occur. The Hamiltonian involved in this example
satisfies the standard assumptions of the literature, except that it is not
convex
Spurious memory in non-equilibrium stochastic models of imitative behavior
The origin of the long-range memory in the non-equilibrium systems is still
an open problem as the phenomenon can be reproduced using models based on
Markov processes. In these cases a notion of spurious memory is introduced. A
good example of Markov processes with spurious memory is stochastic process
driven by a non-linear stochastic differential equation (SDE). This example is
at odds with models built using fractional Brownian motion (fBm). We analyze
differences between these two cases seeking to establish possible empirical
tests of the origin of the observed long-range memory. We investigate
probability density functions (PDFs) of burst and inter-burst duration in
numerically obtained time series and compare with the results of fBm. Our
analysis confirms that the characteristic feature of the processes described by
a one-dimensional SDE is the power-law exponent of the burst or
inter-burst duration PDF. This property of stochastic processes might be used
to detect spurious memory in various non-equilibrium systems, where observed
macroscopic behavior can be derived from the imitative interactions of agents.Comment: 11 pages, 5 figure
Stability Analysis of Continuous Waves in Nonlocal Random Nonlinear Media
On the basis of the competing cubic-quintic nonlinearity model, stability
(instability) of continuous waves in nonlocal random non-Kerr nonlinear media
is studied analytically and numerically. Fluctuating media parameters are
modeled by the Gaussian white noise. It is shown that for different response
functions of a medium nonlocality suppresses, as a rule, both the growth rate
peak and bandwidth of instability caused by random parameters. At the same
time, for a special form of the response functions there can be an
''anomalous'' subjection of nonlocality to the instability development which
leads to further increase of the growth rate. Along with the second-order
moments of the modulational amplitude, higher-order moments are taken into
account.Comment: This is a contribution to the Proc. of the Seventh Inter. Conference
''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv,
Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods
and Applications) at http://www.emis.de/journals/SIGMA
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