The Hausdorff dimension of elliptic and elliptic-caloric measure in $\textbf {R}^ N,\;N\geq 3$

Abstract:

The existence of an L-caloric measure with parabolic Hausdorff dimension $4 - \varepsilon$ in ${{\mathbf {R}}^2} \times {{\mathbf {R}}^1}$ is demonstrated. The method is to use a specially constructed quasi-disk Q whose boundary has Hausdorff $\dim = 2 - \varepsilon$. There is an elliptic measure supported on the entire boundary of Q. Then the L-caloric measure on ${\partial _p}Q \times [0,T]$ is compared with the corresponding elliptic measure. The same method gives the existence of an elliptic measure in ${{\mathbf {R}}^n}$ whose support has Hausdorff $\dim n - \varepsilon$ for $n \geq 3$, and an L-caloric measure in ${{\mathbf {R}}^n} \times {{\mathbf {R}}^1}$ supported on a set of parabolic Hausdorff dimension $n + 2 - \varepsilon$.