KPZ equation with fractional derivatives of white noise

In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order $\gamma >0$ in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer [3] 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 $\gamma <1\mathrm{/}2$. However, we show that the renormalization like “$({\mathrm{\partial }}_{x}h{)}^2-\mathrm{\infty }$” is well-posed only if $\gamma <1\mathrm{/}4$.