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On the exponential Orlicz norms of stopped Brownian motion

Goran Peškir (1996)

Studia Mathematica

Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ψ p ( x ) = e x p ( | x | p ) - 1 with 0 < p ≤ 2) of m a x 0 ≤ t ≤ τ | B t | or | B τ | to be finite, where B = ( B t ) t ≥ 0 is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that ∥ m a x 0 ≤ t ≤ τ | B t | ∥ ψ 1 < ∞ as soon as E ( τ k ) = O ( C k k k ) for some constant C > 0 as k → ∞ (or equivalently ∥ τ ∥ ψ 1 < ∞ ). In particular, if τ ∼ Exp(λ) or | N ( 0 , σ 2 ) | then the last condition is satisfied, and we obtain ∥ m a x 0 ≤ t ≤ τ | B t | ∥ ψ 1 ≤ K √ E ( τ ) with some universal constant K > 0....

On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model

Franco Flandoli, Massimiliano Gubinelli, Francesco Russo (2009)

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

We study the pathwise regularity of the map φ↦I(φ)=∫0T〈φ(Xt), dXt〉, where φ is a vector function on ℝd belonging to some Banach space V, X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture...

On the relation between elliptic and parabolic Harnack inequalities

Waldemar Hebisch, Laurent Saloff-Coste (2001)

Annales de l’institut Fourier

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suffices to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suffices to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for Δ on M , (i.e., for ∂ t + Δ ) and elliptic Harnack inequality for - ∂ t 2 + Δ on ℝ × M .

Currently displaying 41 – 60 of 66