Browsing EPrint Collection  Mathematics by Author "Bass, Richard F."
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The boundary Harnack principle for nondivergence form elliptic operators
Burdzy, Krzysztof; Bass, Richard F. (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 ... 
A boundary Harnack principle in twisted Hölder domains
Burdzy, Krzysztof; Bass, Richard F. (Annals of Mathematics, 199109)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 ... 
Conditioned Brownian motion in planar domains
Burdzy, Krzysztof; Bass, Richard F. (SpringerVerlag GmbH, 199504)We give an upper bound for the Green functions of conditioned Brownian motion in planar domains. A corollary is the conditional gauge theorem in bounded planar domains. 
A critical case for Brownian slow points
Burdzy, Krzysztof; Bass, Richard F. (SpringerVerlag GmbH, 199601)Let X [subscript] t be a Brownian motion and let S(c) be the set of reals r [is greather than or equal to] 0 such that X ([subscript] r+t) − X [subscript] r [is less than or equal to] c [square root of] t, 0 [is less ... 
Cutting Brownian Paths
Burdzy, Krzysztof; Bass, Richard F. (American Mathematical Society, 199901)Let Z [subscript] t be twodimensional 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 ... 
Eigenvalue expansions for Brownian motion with an application to occupation times
Burdzy, Krzysztof; Bass, Richard F. (Institute of Mathematical Statistics, 19960131)Let B be a Borel subset of R [to the power of] d with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B. Let A [subscript] 1 be the time spent by Brownian ... 
Erratum to The Supremum of Brownian Times on Hölder Curves
Bass, Richard F.; Burdzy, Krzysztof (Birkhauser, 20020521)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 ... 
Fiber Brownian motion and the "hot spots" problem
Burdzy, Krzysztof; Bass, Richard F. (Duke University Press, 200010)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 ... 
Hölder domains and the boundary Harnack principle
Banuelos, Rodrigo; Bass, Richard F.; Burdzy, Krzysztof (Duke University Press, 199110)A version of the boundary Harnack principle is proven. 
Intersection local time for points of infinite multiplicity
Burdzy, Krzysztof; Bass, Richard F.; Khoshnevisan, Davar (Institute of Mathematical Statistics, 199404)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 twodimensional Brownian motion spends a units of local time. The measure ... 
Lifetimes of conditioned diffusions
Burdzy, Krzysztof; Bass, Richard F. (SpringerVerlag GmbH, 1992)We investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and nondivergence form can be found. The critical value of the parameter is found for ... 
The Martin boundary in nonLipschitz domains
Burdzy, Krzysztof; Bass, Richard F. (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) = ... 
On domain monotonicity of the Neumann heat kernel
Burdzy, Krzysztof; Bass, Richard F. (Academic Press (Elsevier), 19930815)Some examples are given of convex domains for which domain monotonicity of the Neumann heat kernel does not hold. 
On the Robin problem in fractal domains
Bass, Richard F.; Burdzy, Krzysztof; Chen, ZhenQing (2005)We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum is greater than 0, and some other families of domains where it is ... 
Positivity of Brownian transition densities
Burdzy, Krzysztof; Barlow, Martin T.; Bass, Richard F. (Electronic Journal of Probability, 19970924)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 ... 
A probabilistic proof of the boundary Harnack principle
Burdzy, Krzysztof; Bass, Richard F. (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 ... 
A representation of local time for Lipschitz surfaces
Banuelos, Rodrigo; Bass, Richard F.; Burdzy, Krzysztof (SpringerVerlag 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 ... 
Stochastic bifurcation models
Burdzy, Krzysztof; Bass, Richard F. (Institute of Mathematical Statistics, 199901)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 RayKnight theorems), and on time and ... 
Stochastic differential equations driven by stable processes for which pathwise uniqueness fails
Burdzy, Krzysztof; Bass, Richard F.; Chen, ZhenQing (NorthHolland (Elsevier), 200405)Let Z [subscript] t be a onedimensional symmetric stable process of order [alpha] with [alpha is an element of the set] (0, 2) and consider the stochastic differential equation dX [subscript] t = [omega] (X [subscript] ... 
The supremum of Brownian local times on Holder curves
Burdzy, Krzysztof; Bass, Richard F. (Elsevier, 200111)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 ...