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Local fields, Gaussian measures, and Brownian motions. (English) Zbl 0989.60039
Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 28, 11-50 (2001).
Let \(K\) be a local field with discrete valuation \(v\) of its ring \(A\), \(|\cdots |\) is the absolute value of \(K\) derived using \(v\). Let \(E\) be a vector space over \(K\), and we intoduce its norm function \(\|\cdot \|\): (1) \(\|x \|= 0\) if and only if \(x = 0 ,x \in E\), (2) for \(x \in E\) and \(\lambda \in K\), \(\|\lambda x \|\) = \(|\lambda|\) \(\|x\|\), (3) for \( x, y \in E\), \(\|x + y \|\) \(\leq\) \(\max(\|x\|, \|y\|)\) (ultrametric property). A complete normed \(K\)-vector space \(E\) is a (\(K\)-)Banach space. For example, let \(D\) be the unit ball of \(K\) with center \(0\), and \(C(D^{d},K^{n})\) be the function space of \(K^{n}\)-valued continuous functions defined over \(D^{d}\). To consider \(K\)-Gaussian random variables \(X\) and to define \(n\)-parameter, \(d\)-dimensional \(K\)-Brownian process, we need to define the “orthonormal basis” of the \(K\)-vector space \(E\).
Definition: A system of basis of \(E\), \(\{ e_{j} : j \in J \}\), is an orthonormal basis of \(E\) if and only if for any finite \(\{ e_{k} :k =1,2, \ldots,n \}\) and \(\{ a_{k} \in K : k = 1,2, \ldots ,n \}\), \(\|a_{k} e_{k} \|= \bigvee_{1}^{n} \{ |a_{i}|\|e_{i} \|\}\) and for each \(j \in J\), \(\|e_{j} \|= 1\).
Definition of \(K\)-Gaussian random variable: Let \(E\) be a Banach space over \(K\), and \(X\) be an \(E\)-valued random variable. \(X\) is a \(K\)-Gaussian random variable if and only if for any orthonormal \( a_{j} = (a_{j,1}, a_{j,2}) \in K^{2}\), \(j = 1,2\), \(\text{Law}( Y) =\text{Law}( a_{1} \cdot Y, a_{2} \cdot Y)\), where \( Y = (X_{1}, X_{2})^{t} \), and \(X_{j}\) \((j = 1,2)\) are independent copies of \(X\).
Section 4 describes characterizations of \(K\)-Gauusian random measures. Now, let us define \(n\)-parameter, \(d\)-dimensional \(K\)-Brownian process: An \(n\)-parameter, \(d\)-dimensional \(K\)-Brownian process \(B(D^{n}, K^{d})\) is a \(C(D^{n}, K^{d})\)-valued Gaussian random variable such that the closure of its law in \(C(D^{n},\) \(K^{d})\) is compact \(( f \in C(D^{n}, K^{d}) : \|f(t) \|\leq 1,\) and \(\|f(t)-f(s) \|/ \|t -s \|\leq q^{-1}),\) where \(q\) is the order of residual field. Section 5 descibes a construction of \(B(D^{n}, K^{d}),\) and Sections 6 and 7 describe its analytic properties.
For the entire collection see [Zbl 0970.00015].

60G15 Gaussian processes
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60J45 Probabilistic potential theory
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