277 research outputs found
Rattling and freezing in a 1-D transport model
We consider a heat conduction model introduced in \cite{Collet-Eckmann 2009}.
This is an open system in which particles exchange momentum with a row of
(fixed) scatterers. We assume simplified bath conditions throughout, and give a
qualitative description of the dynamics extrapolating from the case of a single
particle for which we have a fairly clear understanding. The main phenomenon
discussed is {\it freezing}, or the slowing down of particles with time. As
particle number is conserved, this means fewer collisions per unit time, and
less contact with the baths; in other words, the conductor becomes less
effective. Careful numerical documentation of freezing is provided, and a
theoretical explanation is proposed. Freezing being an extremely slow process,
however, the system behaves as though it is in a steady state for long
durations. Quantities such as energy and fluxes are studied, and are found to
have curious relationships with particle density
Resonances and Twist in Volume-Preserving Mappings
The phase space of an integrable, volume-preserving map with one action and
angles is foliated by a one-parameter family of -dimensional invariant
tori. Perturbations of such a system may lead to chaotic dynamics and
transport. We show that near a rank-one, resonant torus these mappings can be
reduced to volume-preserving "standard maps." These have twist only when the
image of the frequency map crosses the resonance curve transversely. We show
that these maps can be approximated---using averaging theory---by the usual
area-preserving twist or nontwist standard maps. The twist condition
appropriate for the volume-preserving setting is shown to be distinct from the
nondegeneracy condition used in (volume-preserving) KAM theory.Comment: Many typos fixed and notation simplified. New order
averaging theorem and volume-preserving variant. Numerical comparison with
averaging adde
Differentially Private Distributed Optimization
In distributed optimization and iterative consensus literature, a standard
problem is for agents to minimize a function over a subset of Euclidean
space, where the cost function is expressed as a sum . In this paper,
we study the private distributed optimization (PDOP) problem with the
additional requirement that the cost function of the individual agents should
remain differentially private. The adversary attempts to infer information
about the private cost functions from the messages that the agents exchange.
Achieving differential privacy requires that any change of an individual's cost
function only results in unsubstantial changes in the statistics of the
messages. We propose a class of iterative algorithms for solving PDOP, which
achieves differential privacy and convergence to the optimal value. Our
analysis reveals the dependence of the achieved accuracy and the privacy levels
on the the parameters of the algorithm. We observe that to achieve
-differential privacy the accuracy of the algorithm has the order of
Pricing and hedging game options in currency models with proportional transaction costs
The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option
Infinitely Many Stochastically Stable Attractors
Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume that each
attractor supports a stochastically stable probability measure and that the
union of the basins of attraction of each attractor covers Lebesgue almost all
points of M. We prove that the time averages of almost all orbits under random
perturbations are given by a finite number of probability measures. Moreover
these probability measures are close to the probability measures supported by
the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure
Representation of Markov chains by random maps: existence and regularity conditions
We systematically investigate the problem of representing Markov chains by
families of random maps, and which regularity of these maps can be achieved
depending on the properties of the probability measures. Our key idea is to use
techniques from optimal transport to select optimal such maps. Optimal
transport theory also tells us how convexity properties of the supports of the
measures translate into regularity properties of the maps via Legendre
transforms. Thus, from this scheme, we cannot only deduce the representation by
measurable random maps, but we can also obtain conditions for the
representation by continuous random maps. Finally, we present conditions for
the representation of Markov chain by random diffeomorphisms.Comment: 22 pages, several changes from the previous version including
extended discussion of many detail
Stochastic stability at the boundary of expanding maps
We consider endomorphisms of a compact manifold which are expanding except
for a finite number of points and prove the existence and uniqueness of a
physical measure and its stochastical stability. We also characterize the
zero-noise limit measures for a model of the intermittent map and obtain
stochastic stability for some values of the parameter. The physical measures
are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy
Formula
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