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## 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 urn:nbn:de:hbz:386-kluedo-36467 Heinrich von Weizsäcker Doctoral Thesis English 2013/11/11 2013 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2013/10/31 2013/11/13 IV, 97 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vom 10.09.2012