53 research outputs found
Large Deviations Principle for a Large Class of One-Dimensional Markov Processes
We study the large deviations principle for one dimensional, continuous,
homogeneous, strong Markov processes that do not necessarily behave locally as
a Wiener process. Any strong Markov process in that is
continuous with probability one, under some minimal regularity conditions, is
governed by a generalized elliptic operator , where and are
two strictly increasing functions, is right continuous and is
continuous. In this paper, we study large deviations principle for Markov
processes whose infinitesimal generator is where
. This result generalizes the classical large deviations
results for a large class of one dimensional "classical" stochastic processes.
Moreover, we consider reaction-diffusion equations governed by a generalized
operator . We apply our results to the problem of wave front
propagation for these type of reaction-diffusion equations.Comment: 23 page
Langevin dynamics with a tilted periodic potential
We study a Langevin equation for a particle moving in a periodic potential in
the presence of viscosity and subject to a further external field
. For a suitable choice of the parameters and the
related deterministic dynamics yields heteroclinic orbits. In such a regime, in
absence of stochastic noise both confined and unbounded orbits coexist. We
prove that, with the inclusion of an arbitrarly small noise only the confined
orbits survive in a sub-exponential time scale.Comment: 38 pages, 6 figure
Heat bounds and the blowtorch theorem
We study driven systems with possible population inversion and we give
optimal bounds on the relative occupations in terms of released heat. A precise
meaning to Landauer's blowtorch theorem (1975) is obtained stating that
nonequilibrium occupations are essentially modified by kinetic effects. Towards
very low temperatures we apply a Freidlin-Wentzel type analysis for continuous
time Markov jump processes. It leads to a definition of dominant states in
terms of both heat and escape rates.Comment: 11 pages; v2: minor changes, 1 reference adde
Dobrushin states in the \phi^4_1 model
We consider the van der Waals free energy functional in a bounded interval
with inhomogeneous Dirichlet boundary conditions imposing the two stable phases
at the endpoints. We compute the asymptotic free energy cost, as the length of
the interval diverges, of shifting the interface from the midpoint. We then
discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure
with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit
in a suitable scaling in which the length of the interval diverges and the
temperature vanishes. The limiting state is not translation invariant and
describes a localized interface. This result can be seen as the probabilistic
counterpart of the variational convergence of the associated excess free
energy.Comment: 34 page
Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
The Freidlin-Wentzell large deviation principle is established for the
distributions of stochastic evolution equations with general monotone drift and
small multiplicative noise. As examples, the main results are applied to derive
the large deviation principle for different types of SPDE such as stochastic
reaction-diffusion equations, stochastic porous media equations and fast
diffusion equations, and the stochastic p-Laplace equation in Hilbert space.
The weak convergence approach is employed in the proof to establish the Laplace
principle, which is equivalent to the large deviation principle in our
framework.Comment: 31 pages, published in Appl. Math. Opti
Time-averaging for weakly nonlinear CGL equations with arbitrary potentials
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the
form: under the periodic boundary conditions, where and
is a smooth function. Let be
the -basis formed by eigenfunctions of the operator . For a
complex function , write it as and
set . Then for any solution of the linear
equation we have . In this work it is
proved that if equation with a sufficiently smooth real potential
is well posed on time-intervals , then for any its
solution , the limiting behavior of the curve
on time intervals of order , as
, can be uniquely characterized by a solution of a certain
well-posed effective equation:
where is a resonant averaging of the nonlinearity . We
also prove a similar results for the stochastically perturbed equation, when a
white in time and smooth in random force of order is added
to the right-hand side of the equation.
The approach of this work is rather general. In particular, it applies to
equations in bounded domains in under Dirichlet boundary conditions
The geometry of spontaneous spiking in neuronal networks
The mathematical theory of pattern formation in electrically coupled networks
of excitable neurons forced by small noise is presented in this work. Using the
Freidlin-Wentzell large deviation theory for randomly perturbed dynamical
systems and the elements of the algebraic graph theory, we identify and analyze
the main regimes in the network dynamics in terms of the key control
parameters: excitability, coupling strength, and network topology. The analysis
reveals the geometry of spontaneous dynamics in electrically coupled network.
Specifically, we show that the location of the minima of a certain continuous
function on the surface of the unit n-cube encodes the most likely activity
patterns generated by the network. By studying how the minima of this function
evolve under the variation of the coupling strength, we describe the principal
transformations in the network dynamics. The minimization problem is also used
for the quantitative description of the main dynamical regimes and transitions
between them. In particular, for the weak and strong coupling regimes, we
present asymptotic formulas for the network activity rate as a function of the
coupling strength and the degree of the network. The variational analysis is
complemented by the stability analysis of the synchronous state in the strong
coupling regime. The stability estimates reveal the contribution of the network
connectivity and the properties of the cycle subspace associated with the graph
of the network to its synchronization properties. This work is motivated by the
experimental and modeling studies of the ensemble of neurons in the Locus
Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive
performance and behavior
Weak noise approach to the logistic map
Using a nonperturbative weak noise approach we investigate the interference
of noise and chaos in simple 1D maps. We replace the noise-driven 1D map by an
area-preserving 2D map modelling the Poincare sections of a conserved dynamical
system with unbounded energy manifolds. We analyze the properties of the 2D map
and draw conclusions concerning the interference of noise on the nonlinear time
evolution. We apply this technique to the standard period-doubling sequence in
the logistic map. From the 2D area-preserving analogue we, in addition to the
usual period-doubling sequence, obtain a series of period doubled cycles which
are elliptic in nature. These cycles are spinning off the real axis at
parameters values corresponding to the standard period doubling events.Comment: 22 pages in revtex and 8 figures in ep
Abrupt Convergence and Escape Behavior for Birth and Death Chains
We link two phenomena concerning the asymptotical behavior of stochastic
processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape
behavior usually associated to exit from metastability. The former is
characterized by convergence at asymptotically deterministic times, while the
convergence times for the latter are exponentially distributed. We compare and
study both phenomena for discrete-time birth-and-death chains on Z with drift
towards zero. In particular, this includes energy-driven evolutions with energy
functions in the form of a single well. Under suitable drift hypotheses, we
show that there is both an abrupt convergence towards zero and escape behavior
in the other direction. Furthermore, as the evolutions are reversible, the law
of the final escape trajectory coincides with the time reverse of the law of
cut-off paths. Thus, for evolutions defined by one-dimensional energy wells
with sufficiently steep walls, cut-off and escape behavior are related by time
inversion.Comment: 2 figure
Harmonic Systems With Bulk Noises
We consider a harmonic chain in contact with thermal reservoirs at different
temperatures and subject to bulk noises of different types: velocity flips or
self-consistent reservoirs. While both systems have the same covariances in the
nonequilibrium stationary state (NESS) the measures are very different. We
study hydrodynamical scaling, large deviations, fluctuations, and long range
correlations in both systems. Some of our results extend to higher dimensions
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