arXiv subjects

Michael Hinz

Publications and source records attributed to Michael Hinz.

44 records · Page 3Linked to original sources

arXiv2014

Sup-norm-closable bilinear forms and Lagrangians

We consider symmetric non-negative definite bilinear forms on algebras of bounded real valued functions and investigate closability with respect to the supremum norm. In particular, any Dirichlet form gives rise to a sup-norm closable bilinear form. Under mild conditions a sup-norm closable bilinear form admits finitely additive energy measures. If, in addition, there exists a (countably additive) energy dominant measure, then a sup-norm closable bilinear form can be turned into a Dirichlet form admitting a carré du champ. Moreover, we can always transfer the bilinear form to an isometrically isomorphic algebra of bounded functions on the Gelfand spectrum, where these measures exist. Our results complement a former closability study of Mokobodzki for the locally compact and separable case.

math.FA↗

arXiv2013

Elementary pathwise methods for nonlinear parabolic and transport type SPDE with fractal noise

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.

math.PR↗

arXiv2013

Metrics and spectral triples for Dirichlet and resistance forms

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet form, we consider Dirac operators within the framework of differential 1-forms proposed by Cipriani and Sauvageot, and comment on its spectral properties. If the Dirichlet form admits a carré operator and the generator has discrete spectrum, then we can construct a related spectral triple, and in the compact and strongly local case the associated Connes distance coincides with the intrinsic metric. We finally give a description of the intrinsic metric in terms of vector fields.

math.OA↗

arXiv2013

Measures and Dirichlet forms under the Gelfand transform

Using the standard tools of Daniell-Stone integrals, Stone-Čech compactification and Gelfand transform, we discuss how any 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.

math.FA↗

arXiv2013

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.

math.FA↗

arXiv2012

Energy measure closability for Dirichlet forms

We consider symmetric Dirichlet forms on locally compact and non-locally compact spaces and provide an elementary proof for their closability with respect to energy dominant measures. We also discuss how to use known potential theoretic results to furnish an alternative proof of this theorem.

math.FA↗

arXiv2012

Vector analysis on fractals and applications

The paper surveys some recent results concerning vector analysis on fractals. We start with a local regular Dirichlet form and use the framework of 1-forms and derivations introduced by Cipriani and Sauvageot to set up some elements of a related vector analysis in weak and non-local formulation. This allows to study various scalar and vector valued linear and non-linear partial differential equations on fractals that had not been accessible before. Subsequently a stronger (localized, pointwise or fiberwise) version of this vector analysis can be developed, which is related to previous work of Kusuoka, Kigami, Eberle, Strichartz, Hino, Ionescu, Rogers, Röckner, and the authors.

math.AP↗

arXiv2012

Dirac and magnetic Schrödinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schrödinger 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 Röckner, 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.

math-ph↗