91,334 research outputs found
The stability of conditional Markov processes and Markov chains in random environments
We consider a discrete time hidden Markov model where the signal is a
stationary Markov chain. When conditioned on the observations, the signal is a
Markov chain in a random environment under the conditional measure. It is shown
that this conditional signal is weakly ergodic when the signal is ergodic and
the observations are nondegenerate. This permits a delicate exchange of the
intersection and supremum of -fields, which is key for the stability of
the nonlinear filter and partially resolves a long-standing gap in the proof of
a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar result
is obtained also in the continuous time setting. The proofs are based on an
ergodic theorem for Markov chains in random environments in a general state
space.Comment: Published in at http://dx.doi.org/10.1214/08-AOP448 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Conditional ergodicity in infinite dimension
The goal of this paper is to develop a general method to establish
conditional ergodicity of infinite-dimensional Markov chains. Given a Markov
chain in a product space, we aim to understand the ergodic properties of its
conditional distributions given one of the components. Such questions play a
fundamental role in the ergodic theory of nonlinear filters. In the setting of
Harris chains, conditional ergodicity has been established under general
nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional
state spaces are rarely amenable to the classical theory of Harris chains due
to the singularity of their transition probabilities, while topological and
functional methods that have been developed in the ergodic theory of
infinite-dimensional Markov chains are not well suited to the investigation of
conditional distributions. We must therefore develop new measure-theoretic
tools in the ergodic theory of Markov chains that enable the investigation of
conditional ergodicity for infinite dimensional or weak-* ergodic processes. To
this end, we first develop local counterparts of zero-two laws that arise in
the theory of Harris chains. These results give rise to ergodic theorems for
Markov chains that admit asymptotic couplings or that are locally mixing in the
sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary
absolutely regular sequences. We proceed to show that local ergodicity is
inherited by conditioning on a nondegenerate observation process. This is used
to prove stability and unique ergodicity of the nonlinear filter. Finally, we
show that our abstract results can be applied to infinite-dimensional Markov
processes that arise in several settings, including dissipative stochastic
partial differential equations, stochastic spin systems and stochastic
differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP879 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization
We introduce a class of densely defined, unbounded, 2-Hochschild cocycles
([PT]) on finite von Neumann algebras . Our cocycles admit a coboundary,
determined by an unbounded operator on the standard Hilbert space associated to
the von Neumann algebra . For the cocycles associated to the
-equivariant deformation ([Ra]) of the upper halfplane
, the "imaginary" part of the coboundary operator is
a cohomological obstruction - in the sense that it can not be removed by a
"large class" of closable derivations, with non-trivial real part, that have a
joint core domain, with the given coboundary.Comment: This is the print versio
On the V-states for the generalized quasi-geostrophic equations
We prove the existence of the V-states for the generalized inviscid SQG
equations with These structures are special rotating simply
connected patches with fold symmetry bifurcating from the trivial solution
at some explicit values of the angular velocity. This produces, inter alia, an
infinite family of non stationary global solutions with uniqueness.Comment: 54 page
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