Now showing items 1-10 of 21
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 ...
Fiber Brownian motion and the "hot spots" problem
(Duke University Press, 2000-10)
We show that in some planar domains both extrema of the second Neumann eigenfunction lie strictly inside the domain. The main technical innovation is the use of "fiber Brownian motion," a process which switches between two-dimensional and one-dimensional evolution.
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.
Erratum to The Supremum of Brownian Times on Hölder Curves
For [function] f [maps the set]: [0, 1] [into the set] [Real numbers], we consider L [superscript] f [subscript] t , the local time of spacetime Brownian motion on the curve f. Let S [subscript][sigma] be the class of all functions whose Hölder norm of order [sigma] is less than or equal to 1. We show that the supremum of L ...
The supremum of Brownian local times on Holder curves
For f : [maps the set] [0, 1] [into the set of real numbers] R, we consider L ([to the power of] f [subscript] t), the local time of spacetime Brownian motion on the curve f. Let S [subscript alpha] be the class of all functions whose Holder norm of order [alpha] is less than or equal to 1. We show that the supremum of L ...