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Lecturer: Fraydoun Rezakhanlou (University of California)
Abstract: Hamilton–Jacobi equation (HJE) is one of the most popular and studied PDE which enjoys vast applications in numerous areas of science. Originally HJEs were formulated in connection with the completely integrable Hamiltonian ODEs of celestial mechanics. They have also been used to study the evolution of the value functions in control and differential game theory. HJE associated with space-time stationary Hamiltonian functions are used to study turbulence in hydrodynamics. Several growth models in physics and biology are described by such HJEs and their viscous variants. In these models, a random interface separates regions associated with different phases and the interface can be locally approximated by the graph of a solution to a HJE. Naturally we would like to understand how the randomness affects the solutions and how the statistics of solutions are propagated with time. Lagrangian techniques in Aubry-Mather theory for action-minimizing trajectories, PDE techniques of weak KAM theory, and probabilistic methods related to first/last passage percolation problems have been employed to study long-time behavior of solutions. Most notably, a unique invariant measure has been constructed for any prescribed average velocity for some important examples of Hamiltonian functions. In these lectures I will give an overview of some of the existing results for the statistics of random solutions to HJEs. In particular, I will discuss a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such Gibbsian solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.
References: Updated notes of the lecture will be made available at https://math.berkeley.edu/~rezakhan/KHJE_l'Aquila.pdf
Information. This minicourse is part of the school "Scaling Limits and Generalized Hydrodynamics", see https://indico.gssi.it/e/ScalingLimits.