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Atomic decomposition of predictable martingale Hardy space with variable exponents

Zhiwei Hao (2015)

Czechoslovak Mathematical Journal

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let ( Ω , , ) be a probability space and p ( · ) : Ω ( 0 , ) be a -measurable function such that 0 < inf x Ω p ( x ) sup x Ω p ( x ) < . It is proved that a predictable martingale Hardy space 𝒫 p ( · ) has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with...

Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík (2021)

Czechoslovak Mathematical Journal

We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space d driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted L 2 -space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.

Average convergence rate of the first return time

Geon Choe, Dong Kim (2000)

Colloquium Mathematicae

The convergence rate of the expectation of the logarithm of the first return time R n , after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have l o g [ R n ( x ) P n ( x ) ] = o ( n β ) a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have - ( 1 + ε ) l o g n l o g [ R n ( x ) P n ( x ) ] l o g l o g n eventually, a.s., where P n ( x ) is the probability of the initial n-block in x. In this paper we prove that E [ l o g R ( L , S ) - ( L - 1 ) h ] converges to a constant depending only on the process where R ( L , S ) is the modified first return time with...

Averaged large deviations for random walk in a random environment

Atilla Yilmaz (2010)

Annales de l'I.H.P. Probabilités et statistiques

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman’s transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set , and that the true velocity of the particle is an element (resp. in the boundary) of when the walk is non-nestling...

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