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 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 $\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $ρ\in\mathbb{R}^2$. Each measure $ν^f\in\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,ρ^-,ρ^+)$, which represents the rate at which a line separating two cells associated with marks $ρ^-$ and $ρ^+$ 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 $\textit{conjecture}$ 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 $ν^{f}\in \widehat{\mathcal{M}_G}$, then at a later time the law of $u_x(\cdot, t)$ is given by a measure $ν^{Θ_t(f)}\in\widehat{\mathcal{M}_G}$, for a suitable kernel $Θ_t(f)$. As we vary $t$, the kernel $Θ_t(f)$ 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 $\mathbb{R}^3$.

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