Sample Path Properties of Gaussian Stochastic Processes Indexed by a Local Field

Local fields, Gaussian measures, and Brownian motions

“…Proof. Given Theorem 6.5, the result follows immediately from a local field version of Frostman's theorem connecting Riesz-type capacities and Hausdorff dimension (see Theorem 2.3 of [Evans, 1988b] for more details.) ⊓ ⊔…”

“…When h(r) = r α for α > 0, we will write r α − m for h − m. The Hausdorff dimension of a set A is given by inf{α : Proof. Given Theorem 6.5, the result follows immediately from a local field version of Frostman's theorem connecting Riesz-type capacities and Hausdorff dimension (see Theorem 2.3 of [Evans, 1988b] for more details.) ⊓ ⊔…”

“…The functions (x, y) → u(|x − y|) appear in the local field Riesz potential theory of [Taibleson, 1975] (see, also, [Evans, 1988b[Evans, , 1992). They will turn out to play the role of the object variously referred to in Markov process theory as the 0-potential kernel density, the 0-resolvent density, or the Green's function.…”

“…In contrast, there has not been a similarly extensive study of probability on local field objects. Most of the small body of work that we are aware of may be found in Karwowski, 1991, 1994], [Brillinger, 1991], [Evans, 1988a[Evans, , 1988b[Evans, , 1989a[Evans, , 1989b[Evans, , 1991[Evans, , 1993, [Guimier, 1989], [Madrecki, 1983[Madrecki, , 1985[Madrecki, , 1990[Madrecki, , 1991, and [Missarov, 1989[Missarov, , 1991. We should remark, however, that if one ignores the algebraic structure of local fields and thinks of them merely as ultrametric spaces or sets with a tree-like structure, then this work can be seen as part of the large and growing literature on probability in such a setting.…”

The maximum distribution of a Gaussian stochastic process indexed by a local field

“…Steven N. Evans [2] results in Evans (1986Evans ( ,1988) was a fairly crude upper bound on a probability that X will exceed a high level (see Corollary (9-2) of Evans (1988) which is restated as Lemma (2-7) below). In the present paper we show that it is sometimes possible to say considerably more about the tail of the maximum of X.…”