Hydrodynamic limits for TASEP with space-time discontinuities

Abstract

We develop a hydrodynamic theory for a height-dependent version of the totally asymmetric simple exclusion process in which the jump rate at a growth site is sampled from a macroscopic two-dimensional speed function evaluated at the spatial coordinate and the current height level. The speed function is allowed to have discontinuities along locally finitely many curves. Through the TASEP height-function representation, the process is coupled to an inhomogeneous directed last-passage percolation model whose exponential rates vary discontinuously in the two macroscopic LPP coordinates. Combining the law of large numbers for this last-passage model with an extension of the variational coupling method, we prove a hydrodynamic limit for the height function and for the associated particle density. The limiting current is characterised by a Lax-Oleinik type variational formula built from the discontinuous last-passage shape function. We then identify the first-order PDE structure selected by the microscopic dynamics. At points of differentiability of the limiting current and continuity of the sampled coefficient, the current solves a Hamilton-Jacobi equation whose Hamiltonian depends discontinuously on the spatial variable and on the value of the solution itself. At discontinuities, the variational formula leads to a natural envelope-based discontinuous viscosity formulation, and we prove that the limiting current satisfies this formulation. Finally, when the coefficient has only spatial discontinuities, we prove uniqueness of the Hamilton-Jacobi solution in the natural class of nondecreasing Lipschitz currents, and identify its spatial derivative as the maximal-current weak solution of the associated scalar conservation law with discontinuous flux.