Sala Seminari (Dip. Matematica).
In my ongoing work with J. Quastel we consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite jump ranges and general jump rates. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. In contrast to the celebrated result by L. Bertini and G. Giacomin (in the case of the nearest neighbour interaction) and its extension by A. Dembo and L.-C. Tsai (for jumps of sizes at most three) we do not use the Hopf-Cole transform and work with the KPZ equation using regularity structures. The price which we have to pay for this approach is a non-trivial renormalization which has not been observed before for equations with stationary noises. In my talk i will give a general review of the Hopf-Cole solution to the KPZ equation and the results of the aforementioned authors. After that I will introduce a general weakly asymmetric exclusion process and explain the difficulty with renormalization.