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Solving the KPZ equation. (English) Zbl 1281.60060

This (monograph) paper proposes an approach to obtaining the solution of the Kardar-Parisi-Zhang equation (KPZ-equation) which has been proposed as a model for surface growth. The KPZ-equation involves a structural parameter \(\lambda\) and the idea is to perform a Wild expansion of the solution in powers of \(\lambda\) but, instead of working with the infinite series as the whole, which may be untracktable, one truncates the series at the fourth order, and then one uses a completely different technique to manage the remainder. This approach allows us to consider this equation with new points of view, and moreover, to obtain new practical results on the solution of the Fokker-Planck equation associated to a particle diffusing in a rough space-time dependent potential. The key ingredient of the construction so performed is a family of processes indexed by binary trees where the trivial tree consists of only its root.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K59 Quasilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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