Random Tessellations and Gibbsian Solutions of Hamilton–Jacobi Equations

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