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In 1962, Dyson showed that the spectrum of a random Hermitian matrix, whose
entries (real and imaginary) diffuse according to independent Ornstein-Uhlenbeck
processes, evolves as non-colliding Brownian particles held together by a drift term.
When , the largest eigenvalue, with time and space properly
rescaled, tends to the so-called Airy process, which is a non-markovian continuous
stationary process. Similarly the eigenvalues in the bulk, with a different time and
space rescaling, tend...
Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is , and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure....
We prove the existence and smoothness of density for the solution
of a hyperbolic SPDE with free term coefficients depending on
time, under hypoelliptic non degeneracy conditions. The result
extends those proved in Cattiaux and Mesnager, PTRF123 (2002) 453-483 to an infinite dimensional
setting.
In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of -Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the Brownian...
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