Browsing Mathematics, Department of by Author "Burdzy, Krzysztof"
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2D Brownian motion in a system of reflecting barriers: effective diffusivity by a sampling method
Burdzy, Krzysztof; Holyst, Robert; Ingerman, David (Institute of Physics, 19940207)We study twodimensional Brownian motion in an ordered periodic system of linear reflecting barriers using the sampling method and conformal transformations. We calculate the effective diffusivity for the Brownian particle. ... 
2D Brownian motion in a system of traps: Application of conformal transformations
Burdzy, Krzysztof; Holyst, Robert; Salisbury, Thomas S. (Institute of Physics, 1992)We study twodimensional Brownian motion in a periodic system of traps using conformal transformations. The system is periodic in the x and y directions. We calculate the ratio of the drift along the yaxis to the drift ... 
An annihilatingbranching particle model for the heat equation with average temperature zero
Burdzy, Krzysztof; Quastel, Jeremy (2005)We consider two species of particles performing random walks in a domain in [Real numbers] [superscript] d with reflecting boundary conditions, which annihilate on contact. In addition there is a conservation law so that ... 
An asymptotically 4stable process
Burdzy, Krzysztof; Madrecki, Andrzej (CRC Press, 1995)An asymptotically 4stable process is constructed. The model identifies the 4stable process with a sequence of processes converging in a very weak sense. It is proved that the 4th variation of the process is a linear ... 
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 ... 
Brownian motion in a Brownian crack
Burdzy, Krzysztof; Khoshnevisan, Davar (Institute of Mathematical Statistics, 199808)Let D be the Wiener sausage of width [epsilon] around twosided Brownian motion. The components of twodimensional reflected Brownian motion in D converge to onedimensional Brownian motion and iterated Brownian motion, ... 
Brownian motion reflected on Brownian motion
Burdzy, Krzysztof; Nualart, David (SpringerVerlag GmbH, 200204)We study Brownian motion reflected on an "independent" Brownian path. We prove results on the joint distribution of both processes and the support of the parabolic measure in the spacetime domain bounded by a Brownian ... 
Censored stable processes
Burdzy, Krzysztof; Bogdan, Krzysztof; Chen, ZhenQing (SpringerVerlag GmbH, 200309)We present several constructions of a "censored stable process" in an open set D [is an element of the subset] R [to the power of] n, i.e., a symmetric stable process which is not allowed to jump outside D. We address ... 
Coalescence of skew Brownian motions
Burdzy, Krzysztof; Barlow, Martin T.; Kaspi, Haya; Mandelbaum, Avi (SpringerVerlag, 2001)The purpose of this short note is to prove almost sure coalescence of two skew Brownian motions starting from different initial points, assuming that they are driven by the same Brownian motion. The result is very simple ... 
Coalescence of synchronous couplings
Burdzy, Krzysztof; Chen, ZhenQing (SpringerVerlag GmbH, 200208)We consider a pair of reflected Brownian motions in a Lipschitz planar domain starting from different points but driven by the same Brownian motion. First we construct such a pair of processes in a certain weak sense, since ... 
Comparison of potential theoretic properties of rough domains
Burdzy, Krzysztof; Chen, ZhenQing (2005)We discuss the relationships between the notion of intrinsic ultracontractivity, parabolic Harnack principle, compactness of the 1resolvent of the Neumann Laplacian, and nontrap property for Euclidean domains with finite ... 
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. 
Configurational transition in a FlemingViottype model and probabilistic interpretation of Laplacian eigenfunctions
Burdzy, Krzysztof; Holyst, Robert; Ingerman, David; March, Peter (Institute of Physics, 19960607)We analyze and simulate a twodimensional Brownian multitype particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in the twodimensional box, ... 
A counterexample to the "hot spots" conjecture
Burdzy, Krzysztof; Werner, Wendelin (Princeton University and Institute for Advanced Study, 199901)We construct a counterexample to the "hot spots" conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions ... 
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 ... 
Curvature of the convex hull of planar Brownian motion near its minimum point
Burdzy, Krzysztof; San Martin, Jaime (NorthHolland (Elsevier), 198910)Let f be a (random) realvalued function whose graph represents the boundary of the convex hull of planar Brownian motion run until time 1 near its lowest point in a coordinate system so that f is nonnegative and f(0) = ... 
Cut points on Brownian paths
Burdzy, Krzysztof (Institute of Mathematical Statistics, 198907)Let X be a standard twodimensional Brownian motion. There exists a.s. t [is an element of the set] (0; 1) such that X([0; t))[intersected with] X((t; 1]) = [empty set]. It follows that X([0; 1]) is not homeomorphic to ... 
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 ... 
Diffusion on curved, periodic surfaces
Burdzy, Krzysztof; Holyst, Robert; Plewezynski, D.; Aksimentiev, A. (American Physical Society, 199907)We present a simulation algorithm for a diffusion on a curved surface given by the equation [omega](r)50. The algorithm is tested against analytical results known for diffusion on a cylinder and a sphere, and applied to ...