Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Boundary behavior of harmonic functions andBrownian motion.

M. O’Neill1

1Department of MathematicsClaremont McKenna College

November 8, 2009

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Harmonic Measure and Brownian Motion

Harmonic measure is the exit distribution of Brownianmotion started from a given point z0 in a given domain.For domains with smooth boundary, it is also ∂g

∂n |ds| whereg is Green’s function with pole at z0.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

McMillan’s twist point theorem

For simply connected domains in the plain, and with respect tothe harmonic measure, almost every boundary point is either atwist point or a cone point.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Cone Points

A cone point in a planar domain is any boundary point which isthe vertex of a triangle contained in the interior of the domain.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Twist Points

A twist point is a boundary point ξ of a planar domain such that

− lim inf arg (z − ξ) = lim sup arg (z − ξ) = +∞

where the limes are taken as z approaches ξ along any curvecontained in the interior of the domain and arg denotes a fixedsingle valued branch of the argument.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

In two dimensions

For any planar domain, the harmonic measure issupported on a set of σ- finite one dimensional Hausdorffmeasure. (Makarov, Jones-Wolff, Wolff)In the infinitely connected case, the dimension can be lessthan 1. (Carleson)

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

In two dimensions

For simply connected domains:

The restriction of the harmonic measure to the cone pointsis in the same measure class as the one dimensionalHausdorff measure λ1.On the set of twist points, the harmonic measure and λ1are mutually singular

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Harmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

In three or more dimensions

For domains in Rn with n ≥ 3:

There is τ(n) > 0 such that the dimension of the support ofthe harmonic measure is always less than n − τ(n).(Bourgain)There is a domain Ω in R3 and there is ε > 0 such that nosubset of ∂Ω with Hausdorff dimension less than 2 + ε canhave full harmonic measure.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Using Ito’s formula

A single valued, continuous and harmonic branch of theargument of the gradient of Green’s function can bedefined in the domain (minus a slit, or one may conditionand then think locally).u(Xt ) is a martingale and a time change of a Brownianmotion, perhaps stopped at a stopping time.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Using Ito’s formula

e.g. lim inf u(Xt ) < +∞ a.s., (in fact, LIL for this martingaleand its conjugate can be used to derive the Makarovdimension estimates.)geometric arguments connect the almost sure exitbehavior of Brownian motion to the behavior of thetrajectories of the Green function

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Lipschitz Subdomains and Stopping Times

To finish (one half of) the proof of the twist point theorem, onemust show that the set of boundary points which are not conepoints but for which lim sup U(Xt ) < +∞ has harmonicmeasure zero.There are two main ingredients:

Fatou’s theorem for convergence of bounded harmonicfunctions along Green lines.Construction of Lipschitz sub-domains and use of stoppingtime arguments with the strong Markov property.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

Twist point theoremFatou’s theorem

Why reprove Fatou’s theorem?

To try to make arguments in higher dimesnions relating exitof Brownian motion to the behavior at the boundary of theGreen lines, one will need the same sort of Fatou theoremalong the Green trajectories.The same question is posed by R. Bass in the bookProbabilistic Techniques in Analysis (chapter 3, pg. 213)

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Definition of NTA domains

A Non Tangentially Accessible (NTA) domain is one in which:

each boundary point has both an interior and exteriornon-tangential ball at every scale andnon-tangential balls of the same scale for differentboundary points can be connected by a chain ofnon-tangential balls whose length is comparable to theEuclidean distance between the balls.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Fatou on Green lines

Adjustments to the reasoning for the two dimensional proofgive a Fatou theorem for Green lines in NTA domains.The main ingredients are again the Strong Markov propertyand the construction of stopping time subdomains.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

A dimension 3 analog for NTA domains

Let η be a Brownian path and let τ(η) denote its first exit timefrom a given NTA domain Ω.Let

E(η) =⋂ε>0

Xt (η)− Xτ (η)

|Xt (η)− Xτ (η)|: τ − ε < t < τ

Then E(η) is almost surely either a half sphere or a sphere, andthe half sphere exits are almost surely at cone points.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Outline

1 Harmonic MeasureHarmonic measure and Brownian motionMcMillan’s twist point theoremDimension estimates.

2 Probabilistic ApproachesTwist point theoremFatou’s theorem

3 Extensions to NTA domainsA Fatou theoremA twist point theoremDimension estimates?

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Cone points

A theorem of Dahlberg shows that in Lipshitz domains,harmonic measure and surface measure are in the samemeasure class.It is not hard to see that harmonic measure restricted tocone points is in the same measure class as the (n-1)dimensional Hausdorff measure.

Boundary behavior

Harmonic MeasureProbabilistic Approaches

Extensions to NTA domains

A Fatou theoremA twist point theoremDimension estimates?

Twist points

So the interesting part of the harmonic measure is again at thetwist points.

If one can find an explicit geometric charecterization of thetwist points which support harmoic measure in, say, theVon koch snowflake perhaps that would help tocharacterize the corresponding points in the Wolff snowballexample.

What more can be said about the process ∇g(Xt )|∇g(Xt )|?

Boundary behavior