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

Dohmann JMN. Diffusions on path spaces over the real line with singular interaction via Dirichlet forms. Bielefeld (Germany): Bielefeld University; 2007.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)). In 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. The 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. With 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. Finally we show that E is a quasi-regular Dirichlet form and show that the process associated to E is a conservative diffusion