Dirichlet forms for singular one-dimensional operators and on graphs

Abstract

We treat the time evolution of states on a finite directed graph, with singular diffusion on the edges of the graph and glueing conditions at the vertices. The operator driving the evolution is obtained by the method of quadratic forms on a suitable Hilbert space. Using the Beurling-Deny criteria we describe glueing conditions leading to positive and to submarkovian semigroups, respectively. IntroductionThe intentions of this paper are twofold. The first aim is to present a treatment of one-dimensional "singular" diffusion in the framework of Dirichlet forms. The second is to present suitable boundary or glueing conditions on graphs (quantum graphs) leading to positive or submarkovian C 0 -semigroups.Concerning the first topic we assume that µ is a finite Borel measure on a bounded interval [a, b]. We assume that particles move in [a, b] according to "Brownian motion" but are only allowed to be located in the support of µ, and in fact are slowed down or accelerated by the "speed measure" µ. (Incidentally, the support of µ is allowed to have gaps, what sometimes is referred to by "gap diffusion".) More generally, instead of starting with Brownian motion, one also can include a drift in the diffusion. This leads to including a scale function. The treatment of the corresponding process has a longer history (cf. [1,3-5,12-14,19,20,22]), but there appears to be no treatment of the arising evolution in the context of Dirichlet forms.Concerning the second topic, we assume that finitely many intervals, with diffusion as described above, are arranged in a graph, and we treat the question how boundary conditions (glueing conditions) at the vertices can be posed in a way to describe diffusing particles. These topics have also been treated in a recent paper by Kostrykin et al. [16]. Since we pose the glueing conditions in a different form (following [17]) it does not seem evident to us to establish the connection between the conditions given in [16] and our conditions.There are many motivations from applications for this work. Since we were not primarily motivated by specific applications we refer to [17,18,23] for some of these Mathematics Subject Classification (2000): 47D06, 60J60, 47E05, 35Q99, 05C99

An inverse spectral problem for a star graph of Krein strings

“…Consequently, we will introduce the associated self-adjoint linear relation in the Sobolev space H 1 0 (G) as well as those relations associated with the individual edges of the graph. This is somewhat in contrast to most of the existing literature (for example, see [13], [25] for intervals and [5], [26] for graphs) which usually considers the spectral problem in a weighted L 2 (G; ω) Hilbert space. Consequently, after deriving few necessary conditions for our spectral data, we will introduce the coupling matrices in Section 3.…”

Note on basic features of large time behaviour of heat kernels

“…Then Q is a Dirichlet form and −L generates a positivity preserving semigroup (see, e.g., [17,31]; we refer to [22,32] for more general boundary conditions). Proof.…”

Generalized Eigenfunctions and Spectral Theory for Strongly Local Dirichlet Forms

“…Hence, there exists a unique associated self-adjoint operator which we denote by H P . It can be explicitly characterized by (H K f ) e : = −f ′′ e for all e ∈ E. It is possible to define quantum graphs with more general generalised boundary conditions at the vertices but not all reasonable choices will lead to Dirichlet forms; in [40] a characterization of those boundary conditions for which the form is a Dirichlet form is given. However the setup is somewhat different from ours.…”