Diffusions on path spaces over the real line with singular interaction via Dirichlet forms

The thesis deals with closability and quasiregularity of classical Dirichlet forms on the space L^2(C(R,R),mu), i.e. E(u,v):=1/2 int _H d mu, where u,v are in F C_b^infty (C(R,R)).\nIn the investigated case mu is a Gibbs measure on C(R,R) defined by the specification pi_r^H(xi,f):=1/Z pi_r(xi,e^(-H_r) f), where pi_r(xi) is the image measure of m_r under the shift by the path which is equal to xi outside the interval [-r,r] and inside it is the affine linear function g with g(r)=xi(r) and g(-r)=xi(-r). The measure m_r describes a Brownian bridge on the interval [-r,r] and has its support on the functions which are zero outside of this interval.\nThe main results are the proof that the specification from above is a specification, then that the measure mu is k-quasiinvariant for all k in C_0^1(R,R), i.e. the measure mu shifted by s*k is absolutely continuous w.r.t mu for all s in R, with the Radon-Nikodym density a_sk.\nWith these densities we get a criterion for closability: If for an ONB of H such that mu is k_n-quasi-invariant for all n in N, the densities a_sk fulfill the Hamza-condition, then the form E from above is closable.\nFinally we show that E is a quasi-regular Dirichlet form and show that the process associated to E is a conservative diffusion.