Journal of Mathematics of Kyoto University
- J. Math. Kyoto Univ.
- Volume 45, Number 2 (2005), 307-327.
Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces
We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion processes on a locally compact separable metric measure space $(M,\rho , \mu )$. These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.
J. Math. Kyoto Univ., Volume 45, Number 2 (2005), 307-327.
First available in Project Euclid: 14 August 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 28A80: Fractals [See also 37Fxx] 31C25: Dirichlet spaces 49Q20: Variational problems in a geometric measure-theoretic setting 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Kumagai, Takashi; Sturm, Karl-Theodor. Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces. J. Math. Kyoto Univ. 45 (2005), no. 2, 307--327. doi:10.1215/kjm/1250281992. https://projecteuclid.org/euclid.kjm/1250281992