Penalizations of Two-Dimensional Simple Random Walk by Direction-Dependent Excursion Counts and Edge-Crossing Counts

Source abstract

We study penalizations of the two-dimensional simple random walk by functions of direction-dependent excursion counts and edge-crossing counts. The former record completed excursions with prescribed pairs of departure and return directions at the origin, whereas the latter record crossings of a prescribed set of unoriented edges incident to the origin. We determine the distributions of their almost surely finite terminal values and derive explicit path decomposition formulas involving the random walk conditioned to avoid the origin. Moreover, both families of penalized measures can be represented by weighting and normalizing the same universal $σ$-finite measure associated with local-time penalization at the origin. Thus, direction-dependent excursion counts, edge-crossing counts, and local time at the origin give rise to penalizations belonging to a common universality class.