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In this talk I show how one can construct a family of Gibbs like measures on the set of piecewise differentiable functions that are invariant with respect to the flow of Hamilton-Jacobi PDE (HJE).

In higher dimensions, when the Hamiltonian function is independent of the spatial variable, the set of piecewise linear convex functions is invariant. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and the Laguerre tessellations. Each measure in our family is uniquely characterized by a kernel, which represents the rate at which a line separating two cells associated with slopes $\rho^-$ 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.