Tangent measures and absolute continuity of harmonic measure

Absolute Continuity between the Surface Measure and Harmonic Measure Implies Rectifiability

In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS

A Criterion for the Absolute Continuity of the Harmonic Measure Associated with an Elliptic Operator

for some l > 0 and for all x E 0 and C; E R" . We also assume a;j = aj;. For such operators, the Dirichlet problem is solvable in n if and only if it is solvable for the Laplace operator, according to a theorem of Littman, Stampacchia and Weinberger [I]. This means that if n ~ R" is a sufficiently nice bounded region (the unit ball, B, is an example) and J is a given continuous function on the ...

Absolute Continuity and Singularity of Probability Measures in Functional Spaces

Harmonic measure and subanalytically tame measures

We introduce the notion of analytically and subanalytically tame measures. These are measures which behave well in the globally subanalytic context; they preserve tameness: integrals of globally subanalytic functions with parameters resp. analytic functions with parameters restricted to globally subanalytic compact sets are definable in an o-minimal structure. We consider the harmonic measure f...

Absolute Continuity between the Wiener and Stationary Gaussian Measures

It is known that the entropy distance between two Gaussian measures is finite if, and only if, they are absolutely continuous with respect to one another. Shepp [5] characterized the correlations corresponding to stationary Gaussian measures that are absolutely continuous with respect to the Wiener measure. By analyzing the entropy distance, we show that one of his conditions, involving the spe...