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.