Joshua Abramson | Searcharxiv

arXiv2012

Structure of shocks in Burgers turbulence with Lévy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by Lévy noise, or equivalently when the initial potential is a two-sided Lévy process $ψ_0$. When $ψ_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} ψ_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $ψ_0$ is abrupt we show that the shock structure is discrete. When $ψ_0$ is eroded we show that there are no rarefaction intervals.

math.PR↗

arXiv2012

Lipschitz minorants of Brownian Motion and Levy processes

For $α> 0$, the $α$-Lipschitz minorant of a function $f: \mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to \mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le α|s-t|$ for all $s,t \in \mathbb{R}$, should such a function exist. If $X=(X_t)_{t \in \mathbb{R}}$ is a real-valued Lévy process that is not pure linear drift with slope $\pm α$, then the sample paths of $X$ have an $α$-Lipschitz minorant almost surely if and only if $| \mathbb{E}[X_1] | < α$. Denoting the minorant by $M$, we investigate properties of the random closed set $\mathcal{Z} := {t \in \mathbb{R} : M_t = X_t \wedge X_{t-}}$, which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator "made stationary" in a suitable sense. We give conditions for the contact set $\mathcal{Z}$ to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the Lévy measure of the associated subordinator. We study the limit of $\mathcal{Z}$ as $α\to \infty$ and find for the so-called abrupt Lévy processes introduced by Vigon that this limit is the set of local infima of $X$. When $X$ is a Brownian motion with drift $β$ such that $|β| < α$, we calculate explicitly the densities of various random variables related to the minorant.

math.PR↗