### A critical function for the planar brownian convex hull

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We prove a polynomial growth estimate for random fields satisfying the Kolmogorov continuity test. As an application we are able to estimate the growth of the solution to the Cauchy problem for a stochastic diffusion equation.

Let (Ω, $\mathcal{F}$, (${\mathcal{F}}_{t}$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to ${\mathcal{F}}_{\infty}$ (theσ-algebra generated by (${\mathcal{F}}_{t}$)t≥0) a coherent family of probability measures (${\mathbb{Q}}_{t}$) indexed byt≥0, each of them being defined on ${\mathcal{F}}_{t}$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual augmentation....

Let (Ω, $\mathcal{F}$, (${\mathcal{F}}_{t}$)t≥0, $\mathbb{P}$) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to ${\mathcal{F}}_{\infty}$ (the σ-algebra generated by (${\mathcal{F}}_{t}$)t≥0) a coherent family of probability measures (${\mathbb{Q}}_{t}$) indexed by t≥0, each of them being defined on ${\mathcal{F}}_{t}$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if one takes its usual...

The $\sigma $-finite measure ${\mathcal{P}}_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$-transform processes with respect to these functions are utilized for the construction of ${\mathcal{P}}_{sup}$.

Given a two-dimensional fractional multiplicative process ${\left({F}_{t}\right)}_{t\in [0,1]}$ determined by two Hurst exponents ${H}_{1}$ and ${H}_{2}$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if ${H}_{1}={H}_{2}$.