This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field's supremum is derived using a generic chaining argument; it implies similar results for the expected supremum, and for the field's modulus of continuity. We also utilize a sharp and convenient condition using iterated Malliavin derivatives, to arrive at similar conclusions for suprema, via a different proof, which does not require full knowledge of the covariance structure.
We introduce a boundedness condition on the Malliavin derivative of a random variable to study subGaussian and other non-Gaussian properties of functionals of random …elds, with particular attention to the estimation of suprema. We relate the boundedness of nth Malliavin derivatives to a new class of "sub-nth Gaussian chaos" processes. An expected supremum estimation, extending the DudleyFernique theorem, is proved for such processes. Sub-nth Gaussian chaos concentration inequalities for the supremum are obtained, using Malliavin derivative conditions; for n = 1, this generalizes the BorellSudakov inequality to a class of sub-Gaussian processes, with a particularly simple and e¢ cient proof; for n = 2 a natural extension to sub-2nd Gaussian chaos processes is established; for n 3 a slightly less e¢ cient Malliavin derivative condition is needed.
The stochastic Anderson model in discrete or continuous space is defined for a class of non-Gaussian spacetime potentials W as solutions u to the multiplicative stochastic heat equationwith diffusivity κ and inverse-temperature β. The relation with the corresponding polymer model in a random environment is given. The large time exponential behavior of u is studied via its almost sure Lyapunov exponent λ = limt→∞ t −1 log u(t, x), which is proved to exist, and is estimated as a function of β and κ for β 2 κ −1 bounded below: positivity and nontrivial upper bounds are established, generalizing and improving existing results. In discrete space λ is of order β 2 / log(β 2 /κ) and in continuous space it is between β 2 (κ/β 2 ) H/(H+1) and β 2 (κ/β 2 ) H/(1+3H) .
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