35 research outputs found
Continuous-time quantum walk on integer lattices and homogeneous trees
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and
infinite homogeneous trees. By using the generating function method, we compute
the limit of the average probability distribution for the general isotropic
walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous
trees. In addition, we compute the asymptotic approximation for the probability
of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page
Billiards in a general domain with random reflections
We study stochastic billiards on general tables: a particle moves according
to its constant velocity inside some domain until it hits the boundary and bounces randomly inside according to some
reflection law. We assume that the boundary of the domain is locally Lipschitz
and almost everywhere continuously differentiable. The angle of the outgoing
velocity with the inner normal vector has a specified, absolutely continuous
density. We construct the discrete time and the continuous time processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains. Then, we
prove exponential ergodicity of these two Markov processes, we study their
invariant distribution and their normal (Gaussian) fluctuations. Of particular
interest is the case of the cosine reflection law: the stationary distributions
for the two processes are uniform in this case, the discrete time chain is
reversible though the continuous time process is quasi-reversible. Also in this
case, we give a natural construction of a chord "picked at random" in
, and we study the angle of intersection of the process with a
-dimensional manifold contained in .Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics
and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains
Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces
A branching process of particles moving at finite velocity over the geodesic
lines of the hyperbolic space (Poincar\'e half-plane and Poincar\'e disk) is
examined. Each particle can split into two particles only once at Poisson paced
times and deviates orthogonally when splitted. At time , after
Poisson events, there are particles moving along different geodesic
lines. We are able to obtain the exact expression of the mean hyperbolic
distance of the center of mass of the cloud of particles. We derive such mean
hyperbolic distance from two different and independent ways and we study the
behavior of the relevant expression as increases and for different values
of the parameters (hyperbolic velocity of motion) and (rate of
reproduction). The mean hyperbolic distance of each moving particle is also
examined and a useful representation, as the distance of a randomly stopped
particle moving over the main geodesic line, is presented
Some recent developments in quantization of fractal measures
We give an overview on the quantization problem for fractal measures,
including some related results and methods which have been developed in the
last decades. Based on the work of Graf and Luschgy, we propose a three-step
procedure to estimate the quantization errors. We survey some recent progress,
which makes use of this procedure, including the quantization for self-affine
measures, Markov-type measures on graph-directed fractals, and product measures
on multiscale Moran sets. Several open problems are mentioned.Comment: 13 page
Chernoff's distribution and parabolic partial differential equations
We give an alternative route to the derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift, using the Feynman-Kac formula with stopping times. The derivation also uses an interesting relations between integrals of special functions, in particular involving integrals with respect to functions which can be called ``incomplete Scorer functions". The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solution must therefore have the stated relation