- 96-152 Hattori, T., Watanabe, H.
- Anisotropic random walks and the asymptotically one-dimensional
diffusions on the abc-gasket
Apr 25, 96
(auto. generated ps),
of related papers
Abstract. A new class of fractals, abc-gaskets, is defined and
asymptotically one-dimensional diffusion processes are studied on them.
The class contains the Sierpinski gasket as well as infinitely many fractals
which lack non-degenerate fixed points of renormalization maps
(hence are not in the class of nested fractals).
The lack of non-degenerate fixed points implies that
the ``standard'' diffusions are degenerate on such fractals and
the standard construction of (non-degenerate) diffusion processes fails.
The asymptotically one-dimensional diffusion, in contrast to
the standard diffusion, is constructed on any abc-gasket
by means of an unstable degenerate fixed point.
To this end, the generating functions for numbers of steps
of anisotropic random works on the \ags\ are analyzed,
according to the line of authors' previous studies,
and relevant scaling factors are calculated explicitly.
In addition, a general
strategy of handling random walk sequences with more than one parameters
for the construction of asymptotically one-dimensional diffusion is proposed.