An example on highly singular parabolic measure (Q803796)

We consider solutions of parabolic equations. \(Lu=a(x,t)u_{xx}-u_ t=0\) in the region \(\{t>0\}\), where \[ (0.1)\;0<\Lambda_ 1\leq a(x,t)\leq \Lambda_ 2<+\infty \] and a is of class \(C^ 2\) in the region; so all solutions are classical. Because \(a(x,t)\leq \Lambda_ 2\) the Dirichlet problem (with bounded, continuous data on the line \(t=0)\) has solutions and these satisfy the maximum principle. The value of a solution at \((x,t)\) can be treated as a linear functional of the boundary values; the value is therefore represented by a probability measure \(\omega^{(x,t)}\) on the boundary - the parabolic measure. Theorem. For each \(\delta>0\) there is some coefficient \(a(x,t)\), \(C^ 2\) in \(\{t>0\}\), satisfying (0.1), for which all parabolic measures are concentrated on a single boundary set of Hausdorff dimension \(<\delta\).