Crossover from the Brownian Castle to Edwards-Wilkinson

Cannizzaro Giuseppe, University of Warwick

In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang (KPZ) and the Edwards-Wilkinson (EW). Notoriously, the KPZ equation is known to interpolate between them in that its small-scale statistics are those of EW while its large-scale fluctuations are those of KPZ. In a recent work with M. Hairer, we showed that the universality picture outlined above is not exhaustive and identified a new universality class together with the universal process at its core, the Brownian Castle (BC). After reviewing the origin, construction and characterising properties of BC, the talk will be devoted to show that there exist a huge family of processes that play a role similar to that of the KPZ equation, connecting though the BC and EW universality classes. We called these processes $\nu$-Brownian Castle, for $\nu$ a probability measure on $[0,1]$, and are linked to the Brownian Net and the stochastic flows of kernels of Schertzer, Sun and Swart. Time allowing, we will show that (one of) these processes naturally arise as the limit, under a suitable scaling, of a microscopic model given by a stochastic PDE. This is joint (ongoing) work with M. Hairer, T. Rosati and R. Sun.

Area: IS22 - Universality, Stochastic PDEs and Random Growth (Giuseppe Cannizzaro)

Keywords: Brownian Castle

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