Student Probability/PDE Seminar: Lipschitz Minorants of Lévy Processes

Seminar | April 5 | 2:10-3:30 p.m. | 891 Evans Hall

 Mehdi Ouaki, UC Berkeley

 Department of Mathematics

For $\alpha >0$, the $\alpha$-Lipschitz minorant of a càdlàg function f is the greatest $\alpha$-Lipschitz function that is dominated by f. We study the joint law of any two-sided Lévy process $(X_t)_{t \in \mathbb R}$ and its $\alpha$-Lipschitz minorant $(M_t)_{t \in \mathbb R}$. In particular, we consider $\mathcal Z$ to be the set of points where $X$ meets $M$, and prove that $((X_t),\mathcal Z)$ is a stationary and regenerative space-time system. Under some determined conditions, when the set $\mathcal Z$ is almost surely discrete, we have an i.i.d sequence of excursions of X above M. In the special case, when X is a Brownian motion with drift, we give explicit path decompositions of those excursions. This $\alpha$-Lipschitz minorant appears as the solutions of the Hamilton-Jacobi PDE when the initial condition is a Lévy noise and the corresponding Lagrangian is of the form $L(v)=\alpha \vert v \vert$. This talk is based on a joint work with Steven N. Evans.