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Search: MSC category 60J60 ( Diffusion processes [See also 58J65] )

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1. CJM 2010 (vol 63 pp. 104)

Feng, Shui; Schmuland, Byron; Vaillancourt, Jean; Zhou, Xiaowen
Reversibility of Interacting Fleming-Viot Processes with Mutation, Selection, and Recombination
Reversibility of the Fleming--Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming--Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

Categories:60J60, 60J70

2. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

3. CJM 2006 (vol 58 pp. 691)

Bendikov, A.; Saloff-Coste, L.
Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups
On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if $(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that $\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega$ can be represented by a continuous function in $\Omega$. Examples are discussed.

Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60

4. CJM 2000 (vol 52 pp. 412)

Varopoulos, N. Th.
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

5. CJM 2000 (vol 52 pp. 92)

Dhersin, Jean-Stéphane; Serlet, Laurent
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

6. CJM 1999 (vol 51 pp. 673)

Barlow, Martin T.; Bass, Richard F.
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

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