HYPERBOLIC ASYMPTOTICS IN BURGERS TURBULENCE AND EXTREMAL PROCESSES

Large time asymptotics of statistical solution u(t,x) (1.2) of the Bur gers' equation (1.1) is considered, where xi(x) = xi(L)(x) is a statio nary zero mean Gaussian process depending on a large parameter L > 0 s o that xi(L)(x) similar to sigma(L) eta(x/L) (L --> infinity), where s igma(L) = L(2)(2 log L)(1/2) and eta(x) is a given standardized statio nary Gaussian process. We prove that as L --> infinity the hyperbolicl y scaled random fields u(L(2)t,L(2)x) converge in distribution to a ra ndom field with ''saw-tooth'' trajectories, defined by means of a Pois son process on the plane related to high fluctuations of xi(x), which corresponds to the zero viscosity solutions. At the physical level of rigor, such asymptotics was considered before by Gurbatov, Malakhov an d Saichev (1991).