F eb 1 99 8 Burgers ’ Flows as Markovian Diffusion Processes

We analyze the unforced and deterministically forced Burgers equation in the framework of the (diffusive) interpolating dynamics that solves the so-called Schrödinger boundary data problem for the random matter transport. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In general, the usage of a continuity equation ∂ t ρ = −∇(vρ), where v = v(x, t) stands for the Burgers field and ρ is the density of transported matter, is at variance with the explicit diffusion scenario. Under these circumstances, we give a complete characterisation of the diffusive transport that is governed by Burgers velocity fields. The result extends both to the approximate description of the transport driven by an incompressible fluid and to motions in an infinitely compressible medium. Also, in conjunction with the Born statistical postulate in quantum theory, it pertains to the probabilis-tic (diffusive) counterpart of the Schrödinger picture quantum dynamics. We give a generalisation of this dynamical problem to cases governed by non-conservative force fields when it appears indispensable to relax the gradient velocity field assumption. The Hopf-Cole procedure has been appropriately generalised to yield solutions in that case.