What is a generalized stochastic process? I've found two different definitions. Are they equivalent? Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$.

What is a generalized stochastic process? I've found two different definitions in some textbooks:
  
  
*
  
*A stochastic process $(\xi_\varphi)_{\varphi\in\mathcal D}$. But where does $\xi$ take its values? In $\mathbb R$?
  
*A $\mathcal D'$-valued random variable
  

Are these definitions somehow equivalent? Moreover, which topology on $\mathcal D$ do they assume? Without specifying this topology, we can't even talk about $\mathcal D'$.
 A: For fixed compact $K \subset \def\R{\mathbf R}\R^d$, we topologize $C^\infty(K)$ by the semi-norms 
$$ \|u\|_{\alpha,K} := \sup_{x \in K} \def\abs#1{\left|#1\right|}\def\norm#1{\left\|#1\right\|}\abs{D^\alpha u(x)}, \qquad \alpha \in \def\N{\mathbf N}\N^d $$
The usual topology on $\def\D{\mathcal D}\D := C^\infty_c(\R^d)$ is the direct limit topology of the $C^\infty(K)$, that is the largest topology that makes all inclusion maps $\D \to C^\infty(K)$, continuous. 
Given a $\D'$-valued random variable $X \colon \Omega \to \D'$, for each $\phi \in D$, we have a $\def\C{\mathbf C}\C$-valued variable $\xi_\phi := \def\<#1>{\left<#1\right>}\<\phi, X(-)> \colon \Omega \to \C$. But note, that the $\xi$'s which arrise like that have some special properties, both algebraic, e. g. $$ \xi_\phi + \xi_\psi = \xi_{\phi + \psi} $$ and analytic, e. g. 
$$ \phi_n \to \phi \implies \xi_{\phi_n} \to \xi_\phi $$
$(\xi_\phi)$ without these properties cannot be respresented as an $\D'$-valued random variable. 
