Local field brownian motion

Abstract

A local field is any locally compact, non-discrete field other than the field of real numbers or the field of complex numbers. There is a natural notion of Gaussian measures on a local field vector space. We construct and study a specific local field Gaussian stochastic process taking values in a finite dimensional local field vector space and indexed by another finite dimensional local field vector space. This process has a structure that strongly reflects the algebraic and geometric structure of the underlying index space and, as such, plays the same role in the local field setting that standard Brownian motion and the related multiparameter processes such as Levy's multiparameter Brownian motion play in a Euclidean context. We investigate the theory of ‘additive functionals’ and the related potential theory for this process and show that it strongly resembles the Euclidean prototype. As a particular consequence of this investigation, we find that a local time process exists when the process hits points. We give two intrinsic constructions of the local time at a given level. These constructions are analogous to the dilation construction of Kingman and the Hausdorff measure construction of Taylor and Wendel in the Euclidean case. Finally, the local time is shown to be continuous as a measure valued stochastic process indexed by the level at which it is evaluated.