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Cut points on Brownian paths
(Institute of Mathematical Statistics, 1989-07)
Let X be a standard two-dimensional 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 the Sierpinski carpet a.s.
Non-polar points for reflected Brownian motion
(Elsevier, 1993)
Our main results are (i) a new construction of reflected Brownian motion X in a half-plane with non-smooth angle of oblique reflection and
(ii) a theorem on existence of some "exceptional" points on the paths of the standard two-dimensional Brownian motion. The link between these two seemingly disparate results will be formed ...
Sets avoided by Brownian motion
(Institute of Mathematical Statistics, 1998-04)
A fixed two-dimensional projection of a three-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the two-dimensional projections with probability one? Equivalently: three-dimensional Brownian motion hits any infinite cylinder with probability one; does it hit all cylinders? ...
On minimal parabolic functions and time-homogenous parabolic h-transforms
(American Mathematical Society, 1999-03-29)
Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin
semi-infinite tubes D [is an element of the subset of real numbers to the power of] d of variable width and minimal harmonic functions h corresponding to the boundary point of D "at ...
Non-intersection exponents for Brownian paths. Part II: Estimations and applications to a random fractal.
(Institute of Mathematical Statistics, 1990-07)
Let X and Y be independent two-dimensional Brownian motions, X(0) = (0; 0); Y(0) = ([epsilon]; 0), and let p([epsilon]) = P(X[0; 1] [intersected with] Y [0; 1] = [empty set], q([epsilon]) = {Y [0; 1] does not contain a closed loop around 0}. Asymptotic estimates (when [epsilon] --> 0) of p([epsilon]); q([epsilon]),
and some ...
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 ...
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.
A Fleming-Viat particle representation of Dirichlet Laplacian
(Springer-Verlag GmbH, 2000-11)
We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D ...
Brownian motion reflected on Brownian motion
(Springer-Verlag GmbH, 2002-04)
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 space-time domain bounded by a Brownian path. We show that there exist
two different natural local times for a Brownian path reflected on a Brownian path.
On non-increase of Brownian motion
(Institute of Mathematical Statistics, 1990-07)
A new proof of the non-increase of Brownian paths is given.