An Introduction to Analysis on Wiener Space

Üstünel, Ali Süleyman

doi:10.1007/bfb0096328

Cited by 107 publications

(114 citation statements)

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“…Excellent and complete treatment of these results and many more can be found for instance in the monographs [16] and [22]; both have been a constant source of inspiration for us.…”

Section: Preliminairiesmentioning

confidence: 90%

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Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

Vizcarra

Viens

2007

Progress in Probability

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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.

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“…Excellent and complete treatment of these results and many more can be found for instance in the monographs [16] and [22]; both have been a constant source of inspiration for us.…”

Section: Preliminairiesmentioning

confidence: 90%

“…The latter has applications ranging from other areas of probability theory, to physics, to …nance, to name a few; a very short selection of references might include [2], [5], [6], [7], [12], [13], [14], [15], [16], [17], [22]. We will not attempt to give an overview of such a wide array of areas.…”

Section: Introductionmentioning

confidence: 99%

Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

Vizcarra

Viens

2007

Progress in Probability

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“…Let F be a random variable in the space L 2 (Ω M , F M , P M ) generated by M . Its Malliavin derivative DF with respect to M , when it exists, is a random field on the parameter space R + × R (see [16,21] for more details). Thus, the Malliavin derivative here is defined only in terms of the randomness in M .…”

Section: Almost-sure Convergencementioning

confidence: 99%

Lyapunov Exponents for Stochastic Anderson Models With Non-Gaussian Noise

2008

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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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“…6 r dT, is the λ-capacity of the "ball" ω ∈ B : p(ω) 6 r ; cf.Üstünel [13] and Fukushima et al [7] for details. It turns out that under very general conditions, λ p (r; ) − r has polynomial decay rate; cf.…”

Section: Remark (A)mentioning

confidence: 99%

“…Gross [8] orÜstünel [13] for a modern treatment as well as some of the new developments in this area. In particular, we mention the following important properties:…”

Section: Introductionmentioning

confidence: 99%