Random Tessellations and Gibbsian solutions of Hamilton-Jacobi Equations

Abstract

We pursue two goals in this article. As our first goal, we construct a family ${\mathcal{M}}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:{\mathbb{R}}^2\to \mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\text{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\rho \in {\mathbb{R}}^2$. Each measure ${\nu }^f\in {\mathcal{M}}_G$ in our family is uniquely characterized by a kernel $f(x,{\rho }^{-},{\rho }^{+})$, which represents the rate at which a line separating two cells associated with marks ${\rho }^{-}$ and ${\rho }^{+}$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set ${\mathcal{M}}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\text{conjecture}$ the invariance of a suitable subfamily $\widehat{{\mathcal{M}}_G}$ of ${\mathcal{M}}_G$. More precisely, we expect that if the initial slope $u_x(\cdot ,0)$ is selected according to a measure ${\nu }^f\in \widehat{{\mathcal{M}}_G}$, then at a later time the law of $u_x(\cdot ,t)$ is given by a measure ${\nu }^{{\Theta }_t(f)}\in \widehat{{\mathcal{M}}_G}$, for a suitable kernel ${\Theta }_t(f)$. As we vary $t$, the kernel ${\Theta }_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of ${\mathbb{R}}^3$.

64 pages, 1 figure. Revised version. To appear in Comm. Math. Phys

References in corpus (3)

• Scalar conservation laws with monotone pure-jump Markov initial conditions
• Scalar Conservation Laws with white noise initial data
• The argmin process of random walks, Brownian motion and Lévy processes