Polynomials of the occupation field and related random fields

To every symmetric Markov process there correspond two random fields over the state space: a Gaussian (“free”) field φ_x and the occupation field T_x which describes the amount of time spent by a particle at each state. For the Brownian motion in d ? 2 dimensions both fields are generalized. Using a relation between T_x and the field ξ_x = :φ_x²:/2 established in a previous publication, polynomials of the fields T and ξ are investigated. In particular, polynomials of T characterize self-intersections of the process.