Speaker: John Dever (Gatech)
Time: Tuesday, October 10, 2018 at 8:30 am
Place: MONT 214
Title: Local space and time scaling exponents for diffusion on a compact metric space.

Abstract:
In this talk we define and investigate a local space scaling exponent $$\alpha$$ and a local time scaling exponent $$\beta.$$
The exponent $$\alpha$$ is the local Hausdorff dimension. We provide several examples of spaces with continuously variable $$\alpha.$$ We also investigate a local Hausdorff measure and establish uniqueness up to a strong equivalence of measures satisfying the Ahlfors regularity property $$\mu(B_r(x)) \asymp r^{\alpha(x)}.$$
The exponent $$\beta$$ is roughly defined as a critical exponent of the expected number of steps needed for a discrete time random walk on an approximation of the space at scale $$\epsilon$$ to leave a ball multiplied by a power $$\epsilon^\beta$$ as the scale $$\epsilon$$ goes to $$0.$$ This exponent can then be localized. Next, we use $$\beta$$ to re-normalize the time scale by introducing local exponential waiting times with mean at site $$x$$ of [/latex]\epsilon^{\beta(x)}.[/latex] This gives a continuous time walk. We then examine the exit time regularity condition that if $$\mathscr{T}(B)$$ is the supremum of the exit times from a ball $$B$$ then $$\mathscr{T}(B_r(x))\asymp r^{\beta(x)}.$$
We show that this condition has a number of useful consequences, one of which is a Faber-Krahn type inequality $$\lambda_{1,\epsilon}(B)\geq \frac{c}{R^{\beta(x_0)}}$$ where $$B=B_R(x_0)$$, $$\lambda_{1,\epsilon}(B)$$ is the bottom of the spectrum of the generator of the time-renormalized walk killed outside of $$B$$ at stage $$\epsilon$$ and $$c$$ is a constant independent of $$\epsilon, x_0,$$ and $$R.$$ Lastly, we investigate convergence of approximating forms using $$\Gamma-$$convergence techniques.