We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale \varepsilon \ll 1 and multiplied by \varepsilon^{3/4} then, as \varepsilon \to 0, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise.
In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter H < 1/4, smoothened out at scale \varepsilon \ll 1 and multiplied by \varepsilon^{1/4-H} then, as \varepsilon \to 0, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially.
