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Now showing items 1-10 of 15

#### The Martin boundary in non-Lipschitz domains

(American Mathematical Society, 1993)

The Martin boundary with respect to the Laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the Euclidean
boundary in C [to the power of gamma] domains, where [gamma](x) = bx log log(1/x)/ log log log(1/x), b small. A counterexample shows that this result is very nearly sharp.

#### Intersection local time for points of infinite multiplicity

(Institute of Mathematical Statistics, 1994-04)

For each a [is an element of the set] (0, 1/2), there exists a random measure [beta] [subscript] a which is supported on the set of points where two-dimensional Brownian motion spends a units of local time. The measure [beta] [subscript] a is carried by a set which has Hausdorff dimension equal to 2−a. A Palm measure ...

#### The boundary Harnack principle for non-divergence form elliptic operators

(Cambridge University Press, 1994)

If L is a uniformly elliptic operator in non–divergence form, the boundary Harnack principle for the ratio of positive L–harmonic functions holds in Hölder domains of order [alpha] if [alpha] > 1/2. A counterexample shows that 1/2 is sharp. For Hölder domains of order [alpha] with [alpha is an element of the set] (0, 1], the ...

#### Hölder domains and the boundary Harnack principle

(Duke University Press, 1991-10)

A version of the boundary Harnack principle is proven.

#### A representation of local time for Lipschitz surfaces

(Springer-Verlag GmbH, 1990)

Suppose that D [is an element of the set of Real numbers to the power of n], n [is greater than or equal to] 2, is a Lipschitz domain and let N[subscript]t(r) be the number of excursions of Brownian motion inside D with diameter greater than r which started before time t. Then rN[subscript]t(r) converges as r --> 0 to a ...

#### A boundary Harnack principle in twisted Hölder domains

(Annals of Mathematics, 1991-09)

The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in twisted Hölder domains of order [alpha] for [alpha is an element of the set](1/2, 1]. For each [alpha is an element of the set] (0, 1/2), there exists a twisted Hölder domain of order [alpha] for which the boundary Harnack principle ...

#### Stochastic bifurcation models

(Institute of Mathematical Statistics, 1999-01)

We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.

#### A probabilistic proof of the boundary Harnack principle

(Birkhauser Boston, Inc., 1990)

The main purpose of this paper is to give a probabilistic proof of Theorem 1.1, one using elementary properties of Brownian motion. We also obtain the fact that the Martin boundary equals the Euclidean boundary as an easy corollary of Theorem 1.1. The boundary Harnack principle may be viewed as a Harnack inequality for ...

#### Positivity of Brownian transition densities

(Electronic Journal of Probability, 1997-09-24)

Let B be a Borel subset of R [to the power of] d and let p(t, x, y) be the transition densities of Brownian motion killed on leaving B. Fix x and y in B. If p(t, x, y) is positive for one t, it is positive for every value of t. Some related results are given.

#### Cutting Brownian Paths

(American Mathematical Society, 1999-01)

Let Z [subscript] t be two-dimensional Brownian motion. We say that a straight line
L is a cut line if there exists a time t [is an element of the set] (0, 1) such that the trace of {Z [subscript] s : 0 [is less than or equal to] s < t} lies on one side of L and the trace of {Z [subscript] s : t < s < 1} lies on the other ...