Local Solution to the Multilayer KPZ Equation
Abstract
In this article we prove local wellposedness of the system of equations partial _t h_{i}= \sum _{j=1}^{i}partial _x^2 h_{j}+ (partial _x h_{i})^2 + ξ on the circle where 1≤ i≤ N and ξ is a spacetime white noise. We attempt to generalize the renormalization procedure which gives the HopfCole solution for the single layer equation and our h_1 (solution to the first layer) coincides with this solution. However, we observe that cancellation of logarithmic divergences that occurs at the first layer does not hold at higher layers and develop explicit combinatorial formulae for them.
 Publication:

Journal of Statistical Physics
 Pub Date:
 June 2019
 DOI:
 10.1007/s10955019022784
 arXiv:
 arXiv:1901.00882
 Bibcode:
 2019JSP...175.1080C
 Keywords:

 Renormalization;
 Regularity structures;
 Stochastic partial differential equations;
 Mathematics  Probability
 EPrint:
 29 pages