Expand all Collapse all | Results 1 - 5 of 5 |
1. CJM 2010 (vol 63 pp. 153)
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the on-diagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.
Categories:35K05, 28A80, 35B40, 60J65 |
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 2000 (vol 52 pp. 961)
Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations |
Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations Explicit evaluations of the symmetric Euler integral $\int_0^1
u^{\alpha} (1-u)^{\alpha} f(u) \,du$ are obtained for some particular
functions $f$. These evaluations are related to duplication formulae
for Appell's hypergeometric function $F_1$ which give reductions of
$F_1 (\alpha, \beta, \beta, 2 \alpha, y, z)$ in terms of more
elementary functions for arbitrary $\beta$ with $z = y/(y-1)$ and for
$\beta = \alpha + \half$ with arbitrary $y$, $z$. These duplication
formulae generalize the evaluations of some symmetric Euler integrals
implied by the following result: if a standard Brownian bridge is
sampled at time $0$, time $1$, and at $n$ independent random times
with uniform distribution on $[0,1]$, then the broken line
approximation to the bridge obtained from these $n+2$ values has a
total variation whose mean square is $n(n+1)/(2n+1)$.
Keywords:Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functions Categories:33C65, 60J65 |
5. CJM 2000 (vol 52 pp. 412)
Geometric and Potential Theoretic Results on Lie Groups The main new results in this paper are contained in the geometric
Theorems 1 and~2 of Section~0.1 below and they are related to
previous results of M.~Gromov and of myself (\cf\
\cite{1},~\cite{2}). These results are used to prove some general
potential theoretic estimates on Lie groups (\cf\ Section~0.3) that
are related to my previous work in the area (\cf\
\cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos
(\cf\ \cite{5},~\cite{21}).
Categories:22E30, 43A80, 60J60, 60J65 |