In Ricci flow, the pseudolocality theorem says roughly that regularity in some region implies that as time goes on, there is some regularity in a smaller region. The first version is due to Perelman. I threw a more precise statement in at the end of this post.
This sort of thing is not true for the heat equation. For example we can create initial data which is zero in the ball of radius one and horrible in the annulus $\{2 < |x| < 3\}$. For small times the situation in the annulus affects a small ball (arbitrarily much and arbitrarily quickly if the annulus is arbitrarily horrible). This can be seen from the kernel formula for the solution.
My question is: which parabolic PDE defined in $\mathbb{R^n}$ exhibit pseudolocality while still having infinite speed of propagation? I expect we should see this sort of theorem outside of geometric flows. My motivation is to understand what makes pseudolocality possible. There are two things I see which may block the counterexample about the heat equation from being carried over to the Ricci flow.
The simpler possibility is that if we model the curvature evolution under Ricci flow by a reaction-diffusion equation, e.g. $$\partial_t u = \Delta u + u^2$$ then perhaps (in a quantifiable way) if we make the data in the annulus large at time zero, then maybe we cap the existence time $T$. Then if we try the trick of making the initial data in the annulus arbitrarily large, we start running into the $\min$ in the statement of the theorem. The bad data in the annulus is not able to affect the small ball around the origin before the solution stops being smooth.
The more complicated possibility is that the nature of Riemannian manifolds puts a limit on the nature of the initial data. Comparison theorems tell us that we can't assign arbitrarily large positive curvature to fixed-size regions in a manifold.
The term "pseudolocality" seems to be localized to geometric flows. Pseudolocality also exists for mean curvature flow (Bing-Long Chen and Le Yin). I have seen parabolic PDE with a finite speed of propagation but I have seen no mention of such a control for PDE with infinite speed of propagation.
Theorem (Pseudolocality for Ricci flow): There are $\epsilon$ and $\delta$ depending on the dimension $n$ with the following property. Suppose $(g(t), M)$ is a Ricci flow for $t \in [0, T)$. If $x_0 \in M$ satisfies, at time $t=0$,
- For all $x \in B_{g(0)}(x_0, R)$ we have $|Rm|(x) < (r_0)^{-2}$
- $Vol(B_R(x_0)) > (1 - \delta)\omega_n r_0^n$
Then for all $t < \min(T, (\epsilon r_0)^2)$ and $x \in B_{g(t)}(x_0, \epsilon R)$ we have
- $|Rm|(x) < (\epsilon r_0)^{-2}$
