Sample Hölder continuity of stochastic processes and majorizing measures

Constructions of Majorizing Measures, Bernoulli Processes and Cotype

We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of well separated ultrametric subspaces. The first scheme of construction provides a simple unified proof of the Majorizing Measure Theorem for Gaussian processes...

A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field’s canonical metric; the full force of Fernique’s zero-one laws and Talagrand’s theory of majorizing ...

Hölder continuity of random processes

For a Young function ϕ and a Borel probability measure m on a compact metric space (T, d) the minorizing metric is defined by τ m,ϕ (s, t) := max{ d(s,t) 0 ϕ −1 (1 m(B(s, ε)))dε, d(s,t) 0 ϕ −1 (1 m(B(t, ε)))dε}. In the paper we extend the result of Kwapien and Rosinski [2] relaxing the conditions on ϕ under which there exists a constant K such that E sup s,t∈T ϕ(|X(s) − X(t)| Kτ m,ϕ (s, t)) 1, ...

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

This paper studies the stochastic heat equation with multiplicative noises: ∂u ∂t = 1 2 ∆u+uẆ , where Ẇ is a mean zero Gaussian noise and uẆ is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutio...