289 research outputs found
Criterion on stability for Markov processes applied to a model with jumps
We formulate and prove a new criterion for stability of e-processes. In particular we show that any e-process which is averagely bounded and concentrating is asymptotically stable. This general result is applied to a stochastic process with jumps that is a continuous counterpart of the chain considered in Szarek (Ann. Probab. 34:1849-1863, 2006)
Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations
In this paper we prove the law of large numbers and central limit theorem for
trajectories of a particle carried by a two dimensional Eulerian velocity
field. The field is given by a solution of a stochastic Navier--Stokes system
with a non-degenerate noise. The spectral gap property, with respect to
Wasserstein metric, for such a system has been shown in [9]. In the present
paper we show that a similar property holds for the environment process
corresponding to the Lagrangian observations of the velocity. In consequence we
conclude the law of large numbers and the central limit theorem for the tracer.
The proof of the central limit theorem relies on the martingale approximation
of the trajectory process
On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS
We consider measures which are invariant under a measurable iterated function
system with positive, place-dependent probabilities in a separable metric
space. We provide an upper bound of the Hausdorff dimension of such a measure
if it is ergodic. We also prove that it is ergodic iff the related skew product
is.Comment: 16 pages; to appear in Journal of Stat. Phy
On the structure of the body of states with positive partial transpose
We show that the convex set of separable mixed states of the 2 x 2 system is
a body of constant height. This fact is used to prove that the probability to
find a random state to be separable equals 2 times the probability to find a
random boundary state to be separable, provided the random states are generated
uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An
analogous property holds for the set of positive-partial-transpose states for
an arbitrary bipartite system.Comment: 10 pages, 1 figure; ver. 2 - minor changes, new proof of lemma
On a Unique Ergodicity of Some Markov Processes
It is proved that the sufficient condition for the uniqueness of an invariant
measure for Markov processes with the strong asymptotic Feller property formulated
by Hairer and Mattingly (Ann Math 164(3):993–1032, 2006) entails the existence of
at most one invariant measure for e-processes as well. Some application to timehomogeneous
Markov processes associated with a nonlinear heat equation driven by
an impulsive noise is also given
Hastings' additivity counterexample via Dvoretzky's theorem
The goal of this note is to show that Hastings' counterexample to the
additivity of minimal output von Neumann entropy can be readily deduced from a
sharp version of Dvoretzky's theorem on almost spherical sections of convex
bodies.Comment: 12 pages; v.2: added references, Appendix A expanded to make the
paper essentially self-containe
Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
We seek to derive the probability--expressed in terms of the Hilbert-Schmidt
(Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit
system is separable, by implementing the well-known Peres-Horodecki test on the
partial transposes (PT's) of the associated 4 x 4 density matrices). But the
full implementation of the test--requiring that the determinant of the PT be
nonnegative for separability to hold--appears to be, at least presently,
computationally intractable. So, we have previously implemented--using the
auxiliary concept of a diagonal-entry-parameterized separability function
(DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal
minors of the PT. This yielded an exact upper bound on the separability
probability of 1024/{135 pi^2} =0.76854$. Here, we piece together
(reflection-symmetric) results obtained by requiring that each of the four 3 x
3 principal minors of the PT, in turn, be nonnegative, giving an
improved/reduced upper bound of 22/35 = 0.628571. Then, we conclude that a
still further improved upper bound of 1129/2100 = 0.537619 can be found by
similarly piecing together the (reflection-symmetric) results of enforcing the
simultaneous nonnegativity of certain pairs of the four 3 x 3 principal minors.
In deriving our improved upper bounds, we rely repeatedly upon the use of
certain integrals over cubes that arise. Finally, we apply an independence
assumption to a pair of DESF's that comes close to reproducing our numerical
estimate of the true separability function.Comment: 16 pages, 9 figures, a few inadvertent misstatements made near the
end are correcte
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