Why does the correct scaled dimension for SPDEs count time as two dimensions?

Question

In this video, Felix Otto says that the correct way to count dimensions for parabolic equations is $2+\text{number of space dimensions}$. He said nothing about this. In the accompanying notes it is stated, again with no explanation.

When I searched the literature the only place I could find any reference is again more SPDE/regularity structures people. Here it is stated with little explanation. (Page 10)

Other PDE people usually say that time counts as one dimension. Why in SPDE/regularity structures does time "count for" 2 dimensions?

Does it have something to do with complex spacetime and Wick rotation? If we consider time complex it is 2D. However in all discussed SPDEs, time is strictly real.

Edit: I thought about it some more. Does it have to do with parabolic scaling? I.e. we scale $u(t,x_1,...x_n)$ as $u(c^2 t, c x_1,...,c x_n)$. The $t$ term picks up $c^2$ so it "counts twice"?

Answer 1

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.