A general version of the fundamental theorem of asset pricing.

Diffusions on fractals

The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations

/pdf/harmonic-calculus-on-p-c-f-self-similar-sets-12leoy3dvx.pdf

Harmonic calculus on p.c.f. self-similar sets

A complete description of the eigenvalues of the Laplacian on the finite Sierpinski gasket is presented. We then demonstrate highly oscillatory behaviours of the distribution function of the eigenvalues, the integrated density of states (for the infinite gasket) and the spectrum of the Laplacian on the infinite gasket. The method has two ingredients: the decimation method in calculating eigenvalues due to Rammal and Toulouse and a simple description of the Dirichlet form associated with the Laplacian.

On a spectral analysis for the Sierpinski gasket

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

/pdf/brownian-motion-and-harmonic-analysis-on-sierpinski-carpets-10w4834oq3.pdf

Brownian Motion and Harmonic Analysis on Sierpinski Carpets

The author studies Dirichlet forms on fractals. He constructs some local Dirichlet forms on abstract fractal sets by using products of random matrices. Also, he studies the martingale dimension of the associated diffusion processes and its self-similarity.

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=25&iss=4&rank=7

Dirichlet forms on fractals and products of random matrices

We study Dirichlet forms associated with random walks on fractal-like finite grahs. We consider related Poincare constants and resistance, and study their asymptotic behaviour. We construct a Markov semi-group on fractals as a subsequence of random walks, and study its properties. Finally we construct self-similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries.

Dirichlet forms on fractals: Poincaré constant and resistance

Analysis on Wiener spaces. I. Nonlinear maps

We study a statistical mechanics of self-avoiding paths on the pre-Sierpinski gasket. We first show the existence of the thermodynamic limit of the (appropriately scaled) free energy. Then we show that there are two domains in the weight parameters (i.e. two phases) between which the scaling differs; i.e. there is a certain kind of phase transition in our model, and we find the critical exponents of the free energy at the phase transition point. We also show the convergence of the distribution of the scaled length of the paths at thermodynamic limit.

Self-avoiding paths on the pre-Sierpinski gasket

The authors rigorously prove that the exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket is

$$\log 2/\log \left( {\frac{{7 - \sqrt 5 }}{2}} \right) = 0.79862 > 0.5 > \log 2/\log 5.$$

Shigeo Kusuoka

Papers

Dirichlet forms on fractals and products of random matrices

Dirichlet forms on fractals: Poincaré constant and resistance

Analysis on Wiener spaces. I. Nonlinear maps

Self-avoiding paths on the pre-Sierpinski gasket

The exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket