Ergodic theory of the stochastic Burgers equation
Ergodic theory of the stochastic Burgers equation
The stochastic Burgers equation is one of the basic evolutionaryÌýSPDEs of Hamilton-Jacobi type related to fluidÌýdynamics and KPZ, among otherÌýthings. The ergodic properties of the system in the compact space case wereÌýunderstood in 2000's. With my coauthors, Eric Cator, Kostya Khanin,ÌýLiyingÌýLi, I have been studying theÌýnoncompact case. The one force - one solutionÌýprinciple has been proved for positive and zero viscosity. TheÌýanalysis isÌýbased on long-term properties of random Lagrangian action minimizers andÌýpolymer measures. TheÌýlatest addition to the program is the convergence ofÌýinfinite volume polymer measures to Lagrangian one-sidedÌýminimizers in theÌýlimit of vanishing viscosity (or, temperature) which results in theÌýconvergence of theÌýassociated global solutions and invariant measures.