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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"?

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  • $\begingroup$ Yes, scaling is the key word. Consider the equation $(\partial_t {-}\Delta) u = 0.$ Then $u^{\lambda} = u(\lambda^2t, \lambda x)$ solves the equation. The dimension they refer to can be considered the "dimension of scaling" (to see why it's a dimension, try to compute the $L^1$ norm of the rescaled solution). Maybe to probabilists this is well known: in time $t$ Brownian motion usually reaches a box of size $\sqrt{t}$. But I think in PDE theory in general scaling plays a crucial role. $\endgroup$ – Kore-N Sep 8 '18 at 15:10

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