Krzysztof Burdzy
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Krzysztof Burdzy has a total of 49 co-authors. Here are the top co-authors by number of shared publications:
- Zhen-Qing Chen
Department of Mathematics, University of Washington, Seattle, Washington 98195
9 shared publications - Richard F. Bass
University of Connecticut
8 shared publications - Martin T. Barlow
University of British Columbia
3 shared publications - Rene L. Schilling
Fakultät Mathematik, Institut für Mathematische Stochastik, TU Dresden, Dresden, Germany
3 shared publications - Robert Hołyst
Institute of Physical Chemistry Polish Academy of Sciences Kasprzaka 44/52 01-224 Warszawa Poland
3 shared publications - Tadeusz Kulczycki
Polish Academy of Sciences
3 shared publications
Latest Publications
- Open Access
Gravitation versus Brownian motion
We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.
Open AccessGravitation versus Brownian motion
We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.
- 11 July 2016
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 31 (1) , 466-493
Random Reflections in a High-Dimensional Tube
- 11 July 2016
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 31 (1) , 466-493
Random Reflections in a High-Dimensional Tube
- Open Access
A random flight process associated to a Lorentz gas with variable density in a gravitational field
We investigate the random flight process that arises as the Boltzmann-Grad limit of a random scatterer Lorentz gas with variable scatterer density in a gravitational field. For power function densities we show how the parameters of the model determine recurrence or transience of the vertical component of the trajectory. Finally, our methods show that, with appropriate scaling of space, time and the density of obstacles, the trajectory of the particle converges to a diffusion with explicitly given parameters.
Open AccessA random flight process associated to a Lorentz gas with variable density in a gravitational field
We investigate the random flight process that arises as the Boltzmann-Grad limit of a random scatterer Lorentz gas with variable scatterer density in a gravitational field. For power function densities we show how the parameters of the model determine recurrence or transience of the vertical component of the trajectory. Finally, our methods show that, with appropriate scaling of space, time and the density of obstacles, the trajectory of the particle converges to a diffusion with explicitly given parameters.
- Open Access
Can one make a laser out of cardboard?
We consider two dimensional and three dimensional semi-infinite tubes made of "Lambertian" material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three dimensional tube will see "infinitely bright" spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away.
Open AccessCan one make a laser out of cardboard?
We consider two dimensional and three dimensional semi-infinite tubes made of "Lambertian" material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three dimensional tube will see "infinitely bright" spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away.
- Open Access
Obliquely reflected Brownian motion in non-smooth planar domains
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.
Open AccessObliquely reflected Brownian motion in non-smooth planar domains
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.
- Open Access
The distribution of the spine of a Fleming-Viot type process
We show uniqueness of the spine of a Fleming-Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.
Open AccessThe distribution of the spine of a Fleming-Viot type process
We show uniqueness of the spine of a Fleming-Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.
- 12 February 2015
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 164 (3-4) , 741-770
Comparison of quenched and annealed invariance principles for random conductance model
- 12 February 2015
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 164 (3-4) , 741-770
Comparison of quenched and annealed invariance principles for random conductance model
- 1 January 2015
- Published by Springer Nature
- p. 189-220
Reflections on Reflections
- Open AccessMeteor process on ${\mathbb Z}^d$The meteor process is a model for mass redistribution on a graph. The case of finite graphs was analyzed in \cite{BBPS}. This paper is devoted to the meteor process on ${\mathbb Z}^d$. The process is constructed and a stationary distribution is found. Convergence to this stationary distribution is proved for a large family of initial distributions. The first two moments of the mass distribution at a vertex are computed for the stationary distribution. For the one-dimensional lattice ${\mathbb Z}$, the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity. An alternative representation of the process on ${\mathbb Z}$ as a collection of non-crossing paths is presented. The distributions of a "tracer particle" in this system of non-crossing paths are shown to be tight as time goes to infinity.Open AccessMeteor process on ${\mathbb Z}^d$The meteor process is a model for mass redistribution on a graph. The case of finite graphs was analyzed in \cite{BBPS}. This paper is devoted to the meteor process on ${\mathbb Z}^d$. The process is constructed and a stationary distribution is found. Convergence to this stationary distribution is proved for a large family of initial distributions. The first two moments of the mass distribution at a vertex are computed for the stationary distribution. For the one-dimensional lattice ${\mathbb Z}$, the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity. An alternative representation of the process on ${\mathbb Z}$ as a collection of non-crossing paths is presented. The distributions of a "tracer particle" in this system of non-crossing paths are shown to be tight as time goes to infinity.
- Open Access
Stirring two grains of sand
Consider two unit balls in a $d$-dimensional flat torus with edge length $r$, for $d\geq 2$. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect---they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when $r\to \infty$, assuming that we use a proper clock and proper scaling. The diffusion coefficient of the limit process depends on the dimension. The positions of the balls are asymptotically independent also in the following sense. The rescaled stationary distributions of the centers of the balls converge to the product of the stationary (hence uniform) distributions for each ball separately, as $r\to\infty$.Open AccessStirring two grains of sand
Consider two unit balls in a $d$-dimensional flat torus with edge length $r$, for $d\geq 2$. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect---they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when $r\to \infty$, assuming that we use a proper clock and proper scaling. The diffusion coefficient of the limit process depends on the dimension. The positions of the balls are asymptotically independent also in the following sense. The rescaled stationary distributions of the centers of the balls converge to the product of the stationary (hence uniform) distributions for each ball separately, as $r\to\infty$.