Density large deviations for multidimensional stochastic hyperbolic conservation laws

Julien Barré (1, 2) , Cedric Bernardin (3) , Raphaël Chetrite (3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IUF - Institut universitaire de France
3 LJAD - Laboratoire Jean Alexandre Dieudonné

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law conservation law. When the conductivity and dif-fusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the problem has been solved in [4]. When this proportionality does not hold, we compute explicitly the large deviation function for a step-like density profile, and we show that the associated optimal current has a non trivial structure. We also derive a lower bound for the large deviation function, valid for a general weak solution, and leave the general large deviation function upper bound as a conjecture.

Mots clés: Large Deviations Principle; Stochastic conservation laws; Kawasaki dynamics; Active particles

Domaines: Mécanique statistique [cond-mat.stat-mech]; Probabilités [math.PR]

Julien Barré, Cedric Bernardin, Raphaël Chetrite. Density large deviations for multidimensional stochastic hyperbolic conservation laws. 2017. ⟨hal-01465293v1⟩