## Curve interactions in R^2: An analytical and stochastical approach

- In the last few years a lot of work has been done in the investigation of Brownian motion with point interaction(s) in one and higher dimensions. Roughly speaking a Brownian motion with point interaction is nothing else than a Brownian motion whose generator is disturbed by a measure supported in just one point. The purpose of the present work is the introducing of curve interactions of the two dimensional Brownian motion for a closed curve \(\mathcal{C}\). We will understand a curve interaction as a self-adjoint extension of the restriction of the Laplacian to the set of infinitely often continuously differentiable functions with compact support in \(\mathbb{R}^{2}\) which are constantly 0 at the closed curve. We will give a full description of all these self-adjoint extensions. In the second chapter we will prove a generalization of Tanaka's formula to \(\mathbb{R}^{2}\). We define \(g\) to be a so-called harmonic single layer with continuous layer function \(\eta\) in \(\mathbb{R}^{2}\). For such a function \(g\) we prove \begin{align} g\left(B_{t}\right)=g\left(B_{0}\right)+\int\limits_{0}^{t}{\nabla g\left(B_{s}\right)\mathrm{d}B_{s}}+\int\limits_{0}^{t}\eta\left(B_{s}\right)\mathrm{d}L\left(s,\mathcal{C}\right) \end{align} where \(B_{t}\) is just the usual Brownian motion in \(\mathbb{R}^{2}\) and \(L\left(t,\mathcal{C}\right)\) is the connected unique local time process of \(B_{t}\) on the closed curve \(\mathcal{C}\). We will use the generalized Tanaka formula in the following chapter to construct classes of processes related to curve interactions. In a first step we get the generalization of point interactions in a second step we get processes which behaves like a Brownian motion in the complement of \(\mathcal{C}\) and has an additional movement along the curve in the time- scale of \(L\left(t,\mathcal{C}\right)\). Such processes do not exist in the one point case since there we cannot move when the Brownian motion is in the point. By establishing an approximation of a curve interaction by operators of the form Laplacian \(+V_{n}\) with "nice" potentials \(V_{n}\) we are able to deduce the existence of superprocesses related to curve interactions. The last step is to give an approximation of these superprocesses by a sytem of branching particles. This approximation gives a better understanding of the related mass creation.

Author: | Benedikt Heinrich |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-36467 |

Advisor: | Heinrich von Weizsäcker |

Document Type: | Doctoral Thesis |

Language of publication: | English |

Publication Date: | 2013/11/11 |

Year of Publication: | 2013 |

Publishing Institute: | Technische Universität Kaiserslautern |

Granting Institute: | Technische Universität Kaiserslautern |

Acceptance Date of the Thesis: | 2013/10/31 |

Date of the Publication (Server): | 2013/11/13 |

Number of page: | IV, 97 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 510 Mathematik |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vom 10.09.2012 |