1,003 research outputs found
Gradient flows of the entropy for finite Markov chains
Let K be an irreducible and reversible Markov kernel on a finite set X. We
construct a metric W on the set of probability measures on X and show that with
respect to this metric, the law of the continuous time Markov chain evolves as
the gradient flow of the entropy. This result is a discrete counterpart of the
Wasserstein gradient flow interpretation of the heat flow in R^n by Jordan,
Kinderlehrer, and Otto (1998). The metric W is similar to, but different from,
the L^2-Wasserstein metric, and is defined via a discrete variant of the
Benamou-Brenier formula.Comment: An error in Example 2.6 has been corrected and several changes have
been made accordingly. To appear in J. Funct. Ana
A spatial version of the It\^{o}-Stratonovich correction
We consider a class of stochastic PDEs of Burgers type in spatial dimension
1, driven by space-time white noise. Even though it is well known that these
equations are well posed, it turns out that if one performs a spatial
discretization of the nonlinearity in the "wrong" way, then the sequence of
approximate equations does converge to a limit, but this limit exhibits an
additional correction term. This correction term is proportional to the local
quadratic cross-variation (in space) of the gradient of the conserved quantity
with the solution itself. This can be understood as a consequence of the fact
that for any fixed time, the law of the solution is locally equivalent to
Wiener measure, where space plays the role of time. In this sense, the
correction term is similar to the usual It\^{o}-Stratonovich correction term
that arises when one considers different temporal discretizations of stochastic
ODEs.Comment: Published in at http://dx.doi.org/10.1214/11-AOP662 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
We consider a non-standard finite-volume discretization of a strongly
non-linear fourth order diffusion equation on the -dimensional cube, for
arbitrary . The scheme preserves two important structural properties
of the equation: the first is the interpretation as a gradient flow in a mass
transportation metric, and the second is an intimate relation to a linear
Fokker-Planck equation. Thanks to these structural properties, the scheme
possesses two discrete Lyapunov functionals. These functionals approximate the
entropy and the Fisher information, respectively, and their dissipation rates
converge to the optimal ones in the discrete-to-continuous limit. Using the
dissipation, we derive estimates on the long-time asymptotics of the discrete
solutions. Finally, we present results from numerical experiments which
indicate that our discretization is able to capture significant features of the
complex original dynamics, even with a rather coarse spatial resolution.Comment: 27 pages, minor change
Entropic Ricci curvature bounds for discrete interacting systems
We develop a new and systematic method for proving entropic Ricci curvature
lower bounds for Markov chains on discrete sets. Using different methods, such
bounds have recently been obtained in several examples (e.g., 1-dimensional
birth and death chains, product chains, Bernoulli-Laplace models, and random
transposition models). However, a general method to obtain discrete Ricci
bounds had been lacking. Our method covers all of the examples above. In
addition, we obtain new Ricci curvature bounds for zero-range processes on the
complete graph. The method is inspired by recent work of Caputo, Dai Pra and
Posta on discrete functional inequalities.Comment: Published at http://dx.doi.org/10.1214/15-AAP1133 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Poisson stochastic integration in Banach spaces
We prove new upper and lower bounds for Banach space-valued stochastic
integrals with respect to a compensated Poisson random measure. Our estimates
apply to Banach spaces with non-trivial martingale (co)type and extend various
results in the literature. We also develop a Malliavin framework to interpret
Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a
Clark-Ocone representation formula.Comment: 26 page
Non-tangential maximal functions and conical square functions with respect to the Gaussian measure
We study, in with respect to the gaussian measure,
non-tangential maximal functions and conical square functions associated with
the Ornstein-Uhlenbeck operator by developing a set of techniques which allow
us, to some extent, to compensate for the non-doubling character of the
gaussian measure. The main result asserts that conical square functions can be
controlled in -norm by non-tangential maximal functions. Along the way we
prove a change of aperture result for the latter. This complements recent
results on gaussian Hardy spaces due to Mauceri and Meda.Comment: 21 pages, revised version with various arguments simplified and
generalise
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