On ergodicity of some Markov processes

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluation of Hashing Methods Performance on Binary Feature Descriptors

In this paper we evaluate performance of data-dependent hashing methods on binary data. The goal is to find a hashing method that can effectively produce lower dimensional binary representation of 512-bit FREAK descriptors. A representative sample of recent unsupervised, semi-supervised and supervised hashing methods was experimentally evaluated on large datasets of labelled binary FREAK feature descriptors

Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian.Comment: 19 page

On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment

We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.Comment: Published at http://dx.doi.org/10.1214/105051604000000945 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org