Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2011 (vol 64 pp. 805)
Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 |
2. CJM 2005 (vol 57 pp. 204)
On the Duality between Coalescing Brownian Motions A duality formula is found for coalescing Brownian motions on the
real line. It is shown that the joint distribution of a coalescing
Brownian motion can be determined by another coalescing Brownian
motion running backward. This duality is used to study a
measure-valued process arising as the high density limit of the
empirical measures of coalescing Brownian motions.
Keywords:coalescing Brownian motions, duality, martingale problem,, measure-valued processes Categories:60J65, 60G57 |
3. CJM 2003 (vol 55 pp. 292)
Infinitely Divisible Laws Associated with Hyperbolic Functions The infinitely divisible distributions on $\mathbb{R}^+$ of random
variables $C_t$, $S_t$ and $T_t$ with Laplace transforms
$$
\left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left(
\frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and}
\quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t
$$
respectively are characterized for various $t>0$ in a number of
different ways: by simple relations between their moments and
cumulants, by corresponding relations between the distributions and
their L\'evy measures, by recursions for their Mellin transforms, and
by differential equations satisfied by their Laplace transforms. Some
of these results are interpreted probabilistically via known
appearances of these distributions for $t=1$ or $2$ in the description
of the laws of various functionals of Brownian motion and Bessel
processes, such as the heights and lengths of excursions of a
one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$
are also known to appear in the Mellin representations of two
important functions in analytic number theory, the Riemann zeta
function and the Dirichlet $L$-function associated with the quadratic
character modulo~4. Related families of infinitely divisible laws,
including the gamma, logistic and generalized hyperbolic secant
distributions, are derived from $S_t$ and $C_t$ by operations such as
Brownian subordination, exponential tilting, and weak limits, and
characterized in various ways.
Keywords:Riemann zeta function, Mellin transform, characterization of distributions, Brownian motion, Bessel process, LÃ©vy process, gamma process, Meixner process Categories:11M06, 60J65, 60E07 |
4. CJM 1999 (vol 51 pp. 673)
Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.
Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions Categories:60J60, 60B05, 60J35 |