Local solution to the multi-layer KPZ equation

摘要: In this article we prove local well-posedness of the system equations \(\partial _t h_{i}= \sum _{j=1}^{i}\partial _x^2 h_{j}+ (\partial _x h_{i})^2 + \xi \) on circle where \(1\le i\le N\) and \(\xi is a space-time white noise. We attempt to generalize renormalization procedure which gives Hopf-Cole solution for single layer equation our \(h_1\) (solution first layer) coincides with solution. However, observe that cancellation logarithmic divergences occurs at does not hold higher layers develop explicit combinatorial formulae them.