Michael Hinz

Dirac and magnetic Schrödinger operators on fractals

Abstract In this paper we define (local) Dirac operators and magnetic Schrodinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Rockner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.

Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals

We consider finite energy and

Measures and Dirichlet Forms Under the Gelfand Transform

L^2

Magnetic Energies and Feynman–Kac–Itô Formulas for Symmetric Markov Processes

differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion harmonicity for finite energy 1-forms. As as corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classical \v{C}ech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence-free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions.

Gradient-type noises I–partial and hybrid integrals

Using the standard tools of Daniell–Stone integrals, Stone–Čech compactification, and Gelfand transform, we show explicitly that any closed Dirichlet form defined on a measurable space can be transformed into a regular Dirichlet form on a locally compact space. This implies existence, on the Stone–Čech compactification, of the associated Hunt process. As an application, we show that for any separable resistance form in the sense of Kigami there exists an associated Markov process. Bibliography: 29 titles.

Finite Energy Coordinates and Vector Analysis on Fractals

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman–Kac–Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.

Closability, Regularity, and Approximation by Graphs for Separable Bilinear Forms

The final objective of our study is to propose a finite-dimensional approach to systems of parabolic partial differential equations perturbed by low-order noises of Brownian or fractional Brownian type. The present article is the preparatory first part, where we introduce partial pathwise integrals over D and (a, b) × D, where D is a smooth bounded domain in ℝ n . Corresponding stochastic versions appear as limit cases.

Elementary Pathwise Methods for Nonlinear Parabolic and Transport Type Stochastic Partial Differential Equations with Fractal Noise

We consider local finite energy coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.

Fractal snowflake domain diffusion with boundary and interior drifts

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact, separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e., energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e., strong resolvent convergence.

From Non-symmetric Particle Systems to Non-linear PDEs on Fractals

We survey some of our recent results on existence, uniqueness, and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.