Brownian motion on the Sierpinski gasket and related stochastic differential equations
Preprint
Abstract
In this paper, we study the Cauchy problem of a non-linear parabolic PDE subject to the Dirichlet boundary condition on the Sierpinski gasket by means of backward stochastic differential equations. We prove the existence and uniqueness of solutions of two classes of backward stochastic differential equations driven by the Brownian motion on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. A representation formula for weak solutions of quasi-linear parabolic PDEs is also established.Keywords
All Related Versions
This publication has 14 references indexed in Scilit:
- Gradients on FractalsJournal of Functional Analysis, 2000
- The theory of generalized Dirichlet forms and its applications in analysis and stochasticsMemoirs of the American Mathematical Society, 1999
- Distributions of Localized Eigenvalues of Laplacians on Post Critically Finite Self-Similar SetsJournal of Functional Analysis, 1998
- Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractalsCommunications in Mathematical Physics, 1993
- On a spectral analysis for the Sierpinski gasketPotential Analysis, 1992
- Adapted solution of a backward stochastic differential equationSystems & Control Letters, 1990
- Diffusion processes with non-smooth diffusion coefficients and their density functionsProceedings of the Royal Society of Edinburgh: Section A Mathematics, 1990
- A harmonic calculus on the Sierpinski spacesJapan Journal of Applied Mathematics, 1989
- Brownian motion on the Sierpinski gasketProbability Theory and Related Fields, 1988
- Random walks on fractal structures and percolation clustersJournal de Physique Lettres, 1983