William Krebs - Academia.edu (original) (raw)
Uploads
Papers by William Krebs
Stochastic Processes and their Applications, 1991
In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine co... more In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine contractions in Iw'. We show that the Vicsek snowflake is a nested fractal in the sense of Lindstrom (1990). We define random walks on the Vicsek snowflake and explicitly find an invariant probability for random walk. From this invariant probability, we construct a Brownian motion on the Vicsek snowflake. We show that this Brownian motion is the unique diffusion limit under weak convergence of resealed random walks with any probability parameter. We show that Brownian motion on the Vicsek snowflake has a scaling property reminiscent of Brownian motion in &I'. Using a coupling argument, we show that our Brownian motion has transition densities with respect to Hausdorff measure on the snowflake.
Proceedings of the American Mathematical Society, 1993
We calculate a bound on hitting times for Brownian motion defined on any nested fractal. We apply... more We calculate a bound on hitting times for Brownian motion defined on any nested fractal. We apply this bound to show that any such process is point recurrent. We then show that any diffusion on a nested fractal must have a transition density with respect to Hausdorff measure on the underlying fractal. We also prove that any Brownian motion on a nested fractal has a jointly continuous local time with a simple modulus of space-time continuity.
Probability Theory and Related Fields, 1995
We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit t... more We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.
Stochastic Processes and their Applications, 1991
In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine co... more In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine contractions in Iw'. We show that the Vicsek snowflake is a nested fractal in the sense of Lindstrom (1990). We define random walks on the Vicsek snowflake and explicitly find an invariant probability for random walk. From this invariant probability, we construct a Brownian motion on the Vicsek snowflake. We show that this Brownian motion is the unique diffusion limit under weak convergence of resealed random walks with any probability parameter. We show that Brownian motion on the Vicsek snowflake has a scaling property reminiscent of Brownian motion in &I'. Using a coupling argument, we show that our Brownian motion has transition densities with respect to Hausdorff measure on the snowflake.
Proceedings of the American Mathematical Society, 1993
We calculate a bound on hitting times for Brownian motion defined on any nested fractal. We apply... more We calculate a bound on hitting times for Brownian motion defined on any nested fractal. We apply this bound to show that any such process is point recurrent. We then show that any diffusion on a nested fractal must have a transition density with respect to Hausdorff measure on the underlying fractal. We also prove that any Brownian motion on a nested fractal has a jointly continuous local time with a simple modulus of space-time continuity.
Probability Theory and Related Fields, 1995
We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit t... more We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.