Let λ denote the almost sure Lyapunov exponent obtained by linearizing the stochastic Duffing-van der Pol oscillator ẍ = −ωx + βẋ−Ax −Bxẋ + σxẆt at the origin x = ẋ = 0 in phase space. If λ > 0 then the process {(xt, ẋt) : t ≥ 0} is positive recurrent on R \ {(0, 0)} with stationary probability measure μ, say. For λ > 0 let λ̃ denote the almost sure Lyapunov exponent obtained by linearizing the ...

Let L contain only the equality symbol and let L+ be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L+ structures with ‘edge probability”’ n−α. By Tα, the almost sure theory of random L+-structures we mean the collection of L+-sentences which have limit probability 1. Tα denotes the theory of the generic structures for Kα, (the collec...

We study random perturbations of quasi-periodic uniformly discrete sets in the d-dimensional euclidean space. By means Diffraction Theory, we find conditions under which a set X can be almost surely recovered from its perturbations. This extends recent periodic case result Yakir (Int Math Res Notices 1–19, 2020).

Two counterexamples, addressing questions raised in Adamczak (2019) and Poly Zheng (2019), are provided. Both counterexamples related to chaoses. Let F n = Y + Z , where the random variables belong different chaoses of uniformly bounded degree. It may be that ? a . s 0 L 2 ? E { sup | } < ? for some > yet fails converge a.s.