Makiko Sasada

KdV- and Toda-type discrete locally-defined dynamics and generalized Pitman’s transform

The Korteweg-de Vries equation (KdV equation) and the Toda lattice are typical and well-known classical integrable systems. For the KdV equation, the (almost-sure) well-posedness of a solution starting from a general ergodic random field on the line is still an open problem, though the invariance, as well as the well-posedness of a solution, of the white noise was proved recently by Killip, Murphy and Visan recently. In this talk, I will consider discretized versions of KdV equation and Toda lattice on the infinite one-dimensional lattice. These systems are understood as "deterministic vertex model”, which are discretely indexed in space and time, and their deterministic dynamics is defined locally via lattice equations. They have another formulation via the generalized Pitman’s transform, which is a new and crucial observation for our result. We show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Also, a detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. This talk is based on a joint work with David Croydon and Satoshi Tsujimoto.