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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].

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