A trace theorem for Dirichlet forms on fractals
نویسندگان
چکیده
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 boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields, which behave as the appropriate fractal diffusion within each fractal component of the field. MSC: 46E35; 28A80; 31C25; 60J60
منابع مشابه
Riesz Potentials and Liouville Operators on Fractals
An analogue to the theory of Riesz potentials and Liouville operators in R for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As ...
متن کاملSelf-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کاملConstruction of diffusion processes on fractals, d-sets, and general metric measure spaces
We give a sufficient condition to construct non-trivial μ-symmetric diffusion processes on a locally compact metric measure space (M,ρ, μ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of Γ-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that ...
متن کاملEnergy measures and indices of Dirichlet forms, with applications to derivatives on some fractals
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 the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order...
متن کاملDerivations, Dirichlet Forms and Spectral Analysis
We study derivations and Fredholm modules on metric spaces with a Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree. This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals.
متن کامل