Expand all Collapse all | Results 26 - 41 of 41 |
26. CJM 2004 (vol 56 pp. 431)
Group Actions and Singular Martingales II, The Recognition Problem We continue our investigation in [RST] of a martingale formed by picking a
measurable set $A$ in a compact group $G$, taking random rotates of $A$, and
considering measures of the resulting intersections, suitably normalized. Here
we concentrate on the inverse problem of recognizing $A$ from a small amount of
data from this martingale. This leads to problems in harmonic analysis on $G$,
including an analysis of integrals of products of Gegenbauer polynomials.
Categories:43A77, 60B15, 60G42, 42C10 |
27. CJM 2004 (vol 56 pp. 77)
High-Dimensional Graphical Networks of Self-Avoiding Walks We use the lace expansion to analyse networks of mutually-avoiding
self-avoiding walks, having the topology of a graph. The networks are
defined in terms of spread-out self-avoiding walks that are permitted
to take large steps. We study the asymptotic behaviour of networks in
the limit of widely separated network branch points, and prove
Gaussian behaviour for sufficiently spread-out networks on
$\mathbb{Z}^d$ in dimensions $d>4$.
Categories:82B41, 60K35 |
28. CJM 2004 (vol 56 pp. 209)
A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations |
A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations In \cite{P69}, Walter Philipp wrote that ``\dots the law of the
iterated logarithm holds for any process for which the Borel-Cantelli
Lemma, the central limit theorem with a reasonably good remainder and
a certain maximal inequality are valid.'' Many authors \cite{DW80},
\cite{I68}, \cite{N91}, \cite{OY71}, \cite{Y79} have followed this
plan in proving the law of the iterated logarithm for sequences (or
fields) of dependent random variables.
We carry on this tradition by proving the law of the iterated
logarithm for a random field whose correlations satisfy an exponential
decay condition like the one obtained by Spohn \cite{Sp86} for
certain Gibbs measures. These do not fall into the $\phi$-mixing or
strong mixing cases established in the literature, but are needed for
our investigations \cite{SS01} into diffusions on configuration
space.
The proofs are all obtained by patching together standard results from
\cite{OY71}, \cite{Y79} while keeping a careful eye on the
correlations.
Keywords:law of the iterated logarithm Categories:60F99, 60G60 |
29. 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 |
30. CJM 2003 (vol 55 pp. 3)
An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a
multiple tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time,
with independent mutation events at the sites, and crossover events
between them. It admits a closed solution of the corresponding
differential equation on the basis of the original state space, and
also closed expressions for the linkage disequilibria, derived by means
of M\"obius inversion. As an extra benefit, the approach can be extended
to a model with selection of additive type across sites. We also derive
a necessary and sufficient criterion for the mean fitness to be a Lyapunov
function and determine the asymptotic behaviour of the solutions.
Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |
31. CJM 2002 (vol 54 pp. 533)
Approximations fortes pour des processus bivariÃ©s Nous \'etablissons un r\'esultat d'approximation forte pour des
processus bivari\'es ayant une partie gaus\-sien\-ne et une partie
empirique. Ce r\'esultat apporte un nouveau point de vue sur deux
th\'eor\`emes hongrois bidimensionnels \'etablis pr\'ec\'edemment,
concernant l'approximation par un processus de Kiefer d'un
processus empirique uniforme unidimensionnel et l'approximation par
un pont brownien bidimensionnel d'un processus empirique uniforme
bidimensionnel. Nous les enrichissons un peu et montrons que sous leur
nouvelle forme ils ne sont que deux \'enonc\'es d'un m\^eme r\'esultat.
We establish a strong approximation result for bivariate processes
containing a Gaussian part and an empirical part. This result leads
to a new point of view on two Hungarian bidimensional theorems
previously established, about the approximation of an unidimensional
uniform empirical process by a Kiefer process and the approximation of
a bidimensional uniform empirical process by a bidimensional Brownian
bridge. We enrich them slightly and we prove that, under their new
fashion, they are but two statements of the same result.
Categories:60F17, 60G15, 62G30 |
32. CJM 2001 (vol 53 pp. 382)
Building a Stationary Stochastic Process From a Finite-Dimensional Marginal If $\mathfrak{A}$ is a finite alphabet, $\sU \subset\mathbb{Z}^D$, and
$\mu_\sU$ is a probability measure on $\mathfrak{A}^\sU$ that ``looks like''
the marginal projection of a stationary stochastic process on
$\mathfrak{A}^{\mathbb{Z}^D}$, then can we ``extend''
$\mu_\sU$ to such a process? Under what conditions can we make this
extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying
classical work on this problem when $D=1$, we provide some sufficient
conditions and some necessary conditions for $\mu_\sU$ to be extendible
for $D>1$, and show that, in general, the problem is not formally decidable.
Categories:37A50, 60G10, 37B10 |
33. 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 |
34. 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 |
35. CJM 2000 (vol 52 pp. 92)
A Stochastic Calculus Approach for the Brownian Snake We study the ``Brownian snake'' introduced by Le Gall, and also
studied by Dynkin, Kuznetsov, Watanabe. We prove that It\^o's
formula holds for a wide class of functionals. As a consequence,
we give a new proof of the connections between the Brownian snake
and super-Brownian motion. We also give a new definition of the
Brownian snake as the solution of a well-posed martingale problem.
Finally, we construct a modified Brownian snake whose lifetime is
driven by a path-dependent stochastic equation. This process gives
a representation of some super-processes.
Categories:60J25, 60G44, 60J80, 60J60 |
36. 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 |
37. CJM 1999 (vol 51 pp. 372)
Uniqueness for a Competing Species Model We show that a martingale problem associated with a competing
species model has a unique solution. The proof of uniqueness of the
solution for the martingale problem is based on duality
technique. It requires the construction of dual probability
measures.
Keywords:stochastic partial differential equation, Martingale problem, duality Categories:60H15, 35R60 |
38. CJM 1998 (vol 50 pp. 1163)
Gradient estimates for harmonic Functions on manifolds with Lipschitz metrics We introduce a distributional Ricci curvature on complete smooth
manifolds with Lipschitz continuous metrics. Under an assumption
on the volume growth of geodesics balls, we obtain a gradient
estimate for weakly harmonic functions if the distributional Ricci
curvature is bounded below.
Categories:60D58, 28D05 |
39. CJM 1998 (vol 50 pp. 487)
On the Liouville property for divergence form operators In this paper we construct a bounded strictly positive
function $\sigma$ such that the Liouville property fails for the
divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in
addition $\Delta \sigma/\sigma$ is bounded, this example also gives a
negative answer to a problem of Berestycki, Caffarelli and Nirenberg
concerning linear Schr\"odinger operators.
Categories:31C05, 60H10, 35J10 |
40. CJM 1997 (vol 49 pp. 24)
Spatial branching processes and subordination We present a subordination theory for spatial branching processes. This
theory is developed in three different settings, first for branching Markov
processes, then for superprocesses and finally for the path-valued process
called the {\it Brownian snake}. As a common feature of these three
situations, subordination can be used to generate new branching
mechanisms. As an application, we investigate the compact support
property for superprocesses with a general branching mechanism.
Categories:60J80, 60J25, 60J27, 60J55, 60G57 |
41. CJM 1997 (vol 49 pp. 3)
Sweeping out properties of operator sequences Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach
Spaces of real valued functions on a complete non-atomic
probability space. Let $(T_1,\ldots,T_{K})$ be
$L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a
function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if
$\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where
$Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned
by $(T_1f,\ldots,T_kf)$. Call a function $h$ a
$(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$,
$\|h\|_1<\varepsilon$, and if
$\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of
measure greater than $1-\varepsilon$. The following is the main
technical result, which is obtained by elementary estimates. There
is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a
$\delta$-spanning function, and if the joint distribution
of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge
M)\vee(-M)\bigr)/M$
is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$.
Furthermore, if $T_k$s are the averages of operators induced by
the iterates of a measure preserving ergodic transformation, then a
similar result is true without requiring that the joint distribution
is normal. This gives the following theorem on a sequence $(T_i)$ of
these averages. Assume that for each $K\geq1$ there is a subsequence
$(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning
function $f_K$ for this subsequence. Then for each $\varepsilon>0$
there is a function $h$,
$0\leq h\leq1$,
$\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e..
Another application of the main result gives a refinement of a part
of Bourgain's ``Entropy Theorem'', resulting in a
different, self contained proof of that theorem.
Keywords:Strong and $\delta$-sweeping out, Gaussian distributions, Bourgain's entropy theorem. Categories:28D99, 60F99 |