In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order γ > 0 in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer (Invent Math 198:269–504, 2014) as an application of the theory of regularity structures. Our aim is to see to what extent his theory works if noises become rougher. We can expect that his theory works if and only if γ < 1/2. However, we show that the renormalization like “(∂ x h) 2 − ∞” is well-posed only if γ < 1/4.
|Number of pages||64|
|Journal||Stochastics and Partial Differential Equations: Analysis and Computations|
|Publication status||Published - Jan 1 2016|
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics