J. Fractal Geom., vol. 3 (2016), 245-263 2016 [Refereed]

We confirm, in a more general framework, a part of the conjecture posed by R.
Bell, C.-W. Ho, and R. S. Strichartz [Energy measures of harmonic functions on
the Sierpiński gasket, Indiana Univ. Math. J. 63 (2014), 831--868] on the
distribution o...

Festschrift Masatoshi Fukushima: In Honor of Masatoshi Fukushima's Sanju 219-236 Jan 2015 [Refereed]

We introduce the concept of functions of locally bounded variation on
abstract Wiener spaces and study their properties. Some nontrivial examples and
applications to stochastic analysis are also discussed.

Given strong local Dirichlet forms and -valued functions on a
metrizable space, we introduce the concepts of geodesic distance and intrinsic
distance on the basis of these objects. They are defined in a geometric and an
analytic way,...

We introduce Riemannian-like structures associated with strong local
Dirichlet forms on general state spaces. Such structures justify the principle
that the pointwise index of the Dirichlet form represents the effective
dimension of the virtual ta...

Probab. Theory Related Fields 156 (2013), 739-793 2013 [Refereed]

We study upper estimates of the martingale dimension of diffusion
processes associated with strong local Dirichlet forms. By applying a general
strategy to self-similar Dirichlet forms on self-similar fractals, we prove
that for natu...

We consider the -Sobolev space on subsets in an
abstract Wiener space, which is regarded as a canonical Dirichlet space on .
We prove that has smooth cylindrical functions as a dense subset
if is -con...

We prove the neo-classical inequality with the optimal constant, which was
conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For
the proof, we introduce the fractional order Taylor's series with residual
terms. Their applica...

Journal of Functional Analysis, vol. 258 (2010), 1656-1681. 2010 [Refereed]

In Euclidean space, the integration by parts formula for a set of finite
perimeter is expressed by the integration with respect to a type of surface
measure. According to geometric measure theory, this surface measure is
realized by the one-codime...

Proceedings of the London Mathematical Society, vol. 100 (2010), 269-302. 2010 [Refereed]

We introduce the concept of index for regular Dirichlet forms by means of
energy measures, and discuss its properties. In particular, it is proved that
the index of strong local regular Dirichlet forms is identical with the
martingale dimension of...

We prove that the martingale dimensions for canonical diffusion processes on
a class of self-similar sets including nested fractals are always one. This
provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res.
Inst. Math. Sci. 25...

Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 111-128, RIMS Kokyuroku Bessatsu, B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008 2008 [Refereed]

Consider a set of continuous maps from the interval to a domain in
. Although the topological boundary of this set in the path
space is not smooth in general, by using the theory of functions of bounded
variation (BV functio...

J. Func. Anal., 238 (2006), 578--611 2006 [Refereed]

We consider a trace theorem for self-similar Dirichlet forms on self-similar
sets to self-similar subsets. In particular, we characterize the trace of the
domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets
to their bou...

Probabilistic approach to geometry. Papers from the 1st Conference of the Seasonal Institute of the Mathematical Society of Japan held at Kyoto University, Kyoto, July 28-August 8, 2008

The first Seasonal Institute of the Mathematical Society of Japan (MSJ-SI)—“Probabilistic Approach to Geometry”— was held at Kyoto University, Japan, on July 28–August 8, 2008. The conference aimed to make interactions between geometry and probabi...