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Quantitative stability analysis of stochastic generalized equations
We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints
Persistence and survival in equilibrium step fluctuations
Results of analytic and numerical investigations of first-passage properties
of equilibrium fluctuations of monatomic steps on a vicinal surface are
reviewed. Both temporal and spatial persistence and survival probabilities, as
well as the probability of persistent large deviations are considered. Results
of experiments in which dynamical scanning tunneling microscopy is used to
evaluate these first-passage properties for steps with different microscopic
mechanisms of mass transport are also presented and interpreted in terms of
theoretical predictions for appropriate models. Effects of discrete sampling,
finite system size and finite observation time, which are important in
understanding the results of experiments and simulations, are discussed.Comment: 30 pages, 12 figures, review paper for a special issue of JSTA
Canonical active Brownian motion
Active Brownian motion is the complex motion of active Brownian particles.
They are active in the sense that they can transform their internal energy into
energy of motion and thus create complex motion patterns. Theories of active
Brownian motion so far imposed couplings between the internal energy and the
kinetic energy of the system. We investigate how this idea can be naturally
taken further to include also couplings to the potential energy, which finally
leads to a general theory of canonical dissipative systems. Explicit analytical
and numerical studies are done for the motion of one particle in harmonic
external potentials. Apart from stationary solutions, we study non-equilibrium
dynamics and show the existence of various bifurcation phenomena.Comment: 11 pages, 6 figures, a few remarks and references adde
Stochastic Turing patterns in the Brusselator model
A stochastic version of the Brusselator model is proposed and studied via the
system size expansion. The mean-field equations are derived and shown to yield
to organized Turing patterns within a specific parameters region. When
determining the Turing condition for instability, we pay particular attention
to the role of cross diffusive terms, often neglected in the heuristic
derivation of reaction diffusion schemes. Stochastic fluctuations are shown to
give rise to spatially ordered solutions, sharing the same quantitative
characteristic of the mean-field based Turing scenario, in term of excited
wavelengths. Interestingly, the region of parameter yielding to the stochastic
self-organization is wider than that determined via the conventional Turing
approach, suggesting that the condition for spatial order to appear can be less
stringent than customarily believed.Comment: modified version submitted to Phys Rev. E. 5. 3 Figures (5 panels)
adde
Linear Stochastic Models of Nonlinear Dynamical Systems
We investigate in this work the validity of linear stochastic models for
nonlinear dynamical systems. We exploit as our basic tool a previously proposed
Rayleigh-Ritz approximation for the effective action of nonlinear dynamical
systems started from random initial conditions. The present paper discusses
only the case where the PDF-Ansatz employed in the variational calculation is
``Markovian'', i.e. is determined completely by the present values of the
moment-averages. In this case we show that the Rayleigh-Ritz effective action
of the complete set of moment-functions that are employed in the closure has a
quadratic part which is always formally an Onsager-Machlup action. Thus,
subject to satisfaction of the requisite realizability conditions on the noise
covariance, a linear Langevin model will exist which reproduces exactly the
joint 2-time correlations of the moment-functions. We compare our method with
the closely related formalism of principal oscillation patterns (POP), which,
in the approach of C. Penland, is a method to derive such a linear Langevin
model empirically from time-series data for the moment-functions. The
predictive capability of the POP analysis, compared with the Rayleigh-Ritz
result, is limited to the regime of small fluctuations around the most probable
future pattern. Finally, we shall discuss a thermodynamics of statistical
moments which should hold for all dynamical systems with stable invariant
probability measures and which follows within the Rayleigh-Ritz formalism.Comment: 36 pages, 5 figures, seceq.sty for sequential numbering of equations
by sectio
Arbitrary Truncated Levy Flight: Asymmetrical Truncation and High-Order Correlations
The generalized correlation approach, which has been successfully used in
statistical radio physics to describe non-Gaussian random processes, is
proposed to describe stochastic financial processes. The generalized
correlation approach has been used to describe a non-Gaussian random walk with
independent, identically distributed increments in the general case, and
high-order correlations have been investigated. The cumulants of an
asymmetrically truncated Levy distribution have been found. The behaviors of
asymmetrically truncated Levy flight, as a particular case of a random walk,
are considered. It is shown that, in the Levy regime, high-order correlations
between values of asymmetrically truncated Levy flight exist. The source of
high-order correlations is the non-Gaussianity of the increments: the increment
skewness generates threefold correlation, and the increment kurtosis generates
fourfold correlation.Comment: 19 pages, 1 figure, To be submitted to Physica
Fractal-cluster theory and thermodynamic principles of the control and analysis for the self-organizing systems
The theory of resource distribution in self-organizing systems on the basis
of the fractal-cluster method has been presented. This theory consists of two
parts: determined and probable. The first part includes the static and dynamic
criteria, the fractal-cluster dynamic equations which are based on the
fractal-cluster correlations and Fibonacci's range characteristics. The second
part of the one includes the foundations of the probable characteristics of the
fractal-cluster system. This part includes the dynamic equations of the
probable evolution of these systems. By using the numerical researches of these
equations for the stationary case the random state field of the one in the
phase space of the , , criteria have been obtained. For the
socio-economical and biological systems this theory has been tested.Comment: 37 pages, 20 figures, 4 table
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