Masatoshi Fukushima

On a spectral analysis for the Sierpinski gasket

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.

Regular representations of Dirichlet spaces

We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space.

Capacity and quantum mechanical tunneling

We connect the notion of capacity of sets in the theory of symmetric Markov process and Dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics. In particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero. We also discuss several examples in ℝd.

Traces of symmetric Markov processes and their characterizations

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling–Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the Levy system) for the time-changed process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process.

Reflecting Diffusions on Lipschitz Domains with Cusps -Analytic Construction and Skorohod Representation-

The reflecting Brownian motion on a bounded Lipschitz domain D ⊂ R d was constructed by Bass and Hsu [3] as a conservative, symmetric (with respect to the Lebesgue measure) and strong Feller diffusion process (X t ,P x ) on the closure \( \overline D \) of D whose transition semigroup on L 2 (D) is associated with the Sobolev space H 1 (D) with inner product

Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions

\varepsilon \left( {u,v} \right) = \frac{1}{2}\int_D {\nabla u \cdot \nabla vdx,} \quad \quad u,v \in {H^1}\left( D \right)

An invariance result for capacities on Wiener space

(1.1) .

On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals

Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k de- fined on E×E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associ- ated regular lower bounded semi-Dirichlet formon L 2 (E;m) pro- ducing a Hunt process X 0 on E whose jump behaviours are governed by k. For an arbitrary open subset DE, we also construct a Hunt process X D,0 on D in an analogous manner. When D is relatively compact, we show that X D,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X 0 is the stable-like process that will be also identified with the solution of a martingale prob- lem up to an �-polar set of starting points. Approachability to the boundary @D in finite time of its censored process X D,0 on a bounded open subset D will be examined in terms of the polarity of @D for the symmetric stable processes with indices that bound the variable exponent �(x).