the links on the video and notes are not there. But indeed as they explain in "Stochastic PDEs, Regularity Structures, and Interacting Particle Systems" at page 13, the main reason is that the Schauder estimates are improving regularity by +2

The Schauder estimate shows that the use of parabolic scaling of space-time is natural when measuring regularity.

For $f$ in $C^\alpha((0,\infty)\times T^{d})$ define

$$u(s,x) := \int_{\Lambda_s} \,K(s-r,x-y)\; f(r,y) \, d y \; dr \;$$

interpreted in a distributional sense if $\alpha<0$. Then if $\alpha \notin Z$, we have

$$\|u\|_{C^{\alpha+2}((0,t)\times T^{d})} \lesssim \|f \|_{C^{\alpha}((0,t)\times T^{d})}\,.$$

which in turn originates from the scaling of the heat kernel as mentioned in the comments.

So we adjust our norms for computational ease.

the links on the video and notes are not there. But indeed as they explain in "Stochastic PDEs, Regularity Structures, and Interacting Particle Systems" at page 13, the main reason is that the Schauder estimates are improving regularity by +2

The Schauder estimate shows that the use of parabolic scaling of space-time is natural when measuring regularity.

For $f$ in $C^\alpha((0,\infty)\times T^{d})$ define

$$u(s,x) := \int_{\Lambda_s} \,K(s-r,x-y)\; f(r,y) \, d y \; dr \;$$

interpreted in a distributional sense if $\alpha<0$. Then if $\alpha \notin Z$, we have

$$\|u\|_{C^{\alpha+2}((0,t)\times T^{d})} \lesssim \|f \|_{C^{\alpha}((0,t)\times T^{d})}\,.$$

which in turn originates from the scaling of the heat kernel as mentioned in the comments.

In words, parabolic solutions have half smoothness in time compared to space. So we adjust our norms for computational ease.