On global solutions of the random Hamilton-Jacobi equations and the KPZ problem

Abstract

In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton--Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1+1. In the case of general viscous Hamilton--Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

33 pages

References in corpus (5)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• An exact solution for the KPZ equation with flat initial conditions
• The one-dimensional KPZ equation and its universality class
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Second class particles and cube root asymptotics for Hammersley's process

Cited by in corpus (12)

• Stationary solutions to the stochastic Burgers equation on the line
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• A PDE hierarchy for directed polymers in random environments
• Viscous shock solutions to the stochastic Burgers equation
• Invariant measures for stochastic conservation laws on the line
• Dynamic polymers: invariant measures and ordering by noise
• Jointly invariant measures for the Kardar-Parisi-Zhang Equation
• Localization length of the $1+1$ continuum directed random polymer
• Intertwining the Busemann process of the directed polymer model
• Localization of directed polymers in continuous space
• Simultaneous global inviscid Burgers flows with periodic Poisson forcing
• Point Fields of Last Passage Percolation and Coalescing Fractional Brownian Motions