To pose my question, I first have to describe the construction which I use (due to Polletta): Assume that $\Omega = \Pi_{t \in [0, \infty)} \dot{\mathbb{R}}$ with topology of uniform convergence and sigma algebra $\Sigma$ generated by this topology. The way one can build a Wiener measure defined on the subset of continuous functions on this space is as follows: Let $\Omega'= \Pi_{t \in \mathbb{Q}^+} \dot{\mathbb{R}}$ and its sigma algebra $\Sigma'$ the sigma algebra generated by simple cylinders of the form $C'(t_1,...,t_n,a_1,b_1,...,a_n,b_n)=\{\omega(t_i) \in [a_i,b_i)\}$. We can define the Wiener measure on simple cylinders as $\mu'(C'(T,A,B))= \prod_{i=1}^n \int_{a_i}^{b_i} g(x_i,t_{i+1}-t_i)dx_i$ with g being the Gaussian distribution. This can be extended to $\Sigma'$. Defining Holder continuous functions on $\Omega'$ (denoted as $\Omega'_c$) one shows that this set has measure 1. Then define the embedding $E:\Omega' \rightarrow \Omega$ by: if continuous take it to is extension, if not take it $0$. Show that $E$ is measurable with respect to $\Sigma'$, $\Sigma$. This allows you to define a probability measure on $\Omega$ by $\mu(A)=\mu'(E^{-1}(A))$.

My first question is if I wanted to write down the measure for a cylinder set say $C(t,U) \in \Omega$ how would I do that? When I take $t \in \mathbb{Q}$, $E^{-1}(C(t,U)) \neq C'(t,U)$ since the inverse image does not contain the discontinuous functions, however discontinuous functions have measure $0$ so one has that the measure of this set should be $\mu(C'(t,U))$ which we know what it is. But if we take $t$ irrational, I would expect $\mu(E^{-1}(C(t,U)))$ still to be equal to $\int_{U}g(x,t)dx$. For this I tried writing $E^{-1}(C(t,U))$ as a limit of decreasing sets but could not. It is equal to $\{ \omega' \in \Omega'_c | \lim_{t_i \rightarrow t} \omega'(t_i) \in U\}$. Even when you attempt to write this as an intersection of sets, I could not find a way in which they would be decreasing and lead to some sort of limit. Could it could be written as some sort of "limit" $C(t_i,U)$ where $t_i \rightarrow t$ where $t_i$ are in $\mathbb{Q}$

My second question is how would you compute the expectation of a random variable that is not constant on the cylinders? Those which are constant is easy to do by writing $\Omega$ as union of cylinders and then using the measure define on each cylinder (like $X(\omega)=\omega^2(t)$)). I tried thinking about $X(\omega)=\int_{0}^t\omega^2(s)ds$ but could not even see if it was measurable or not.



1 Answer 1


Okay, I think I got the answer. For $t$ irrational, let $t_n$ be an increasing sequence of rationals converging to $t$ and define the sets $C'_n = C'(t_n,U)$. If $\gamma \in C'(t,U)$ (defined using limits in $\Omega'$) then one can s,how that $\gamma \in \text{liminf}(C'_n)$ (assuming $U$ is open if $\gamma(t) \in U$ then for all $t_n$ large enough $\gamma(t_n) \in U$). So $C'(t,U) \subset \text{liminf}(C'_n)$. On the other hand of $\gamma \in \text{limsup}(C'_n)$ then, $\gamma(t_n) \in U$ for infinitely many $t_n$. Taking the limit $\gamma(t) \in U$ so $\text{limsup}(C'_n) \subset C'(t,U)$. This proves set theoretic limit of $C'_n$ is $C(t,U)$ hence $p_{\Omega}(C'(t,U))=\lim_{n \rightarrow \infty} C'(t_n,U) = \int_{U} g(x,t) dx$.

For the second one assume $X(\gamma)=\int_0^t (\gamma(s))^2ds$. Let $X_n(\gamma)=\frac{1}{n}\sum_{i=1}^n \gamma(\frac{ti}{n})$. Then $X_n(\gamma)=f_n \circ M_{\frac{1}{n},...,t}$ where $M_{t}$ is the process map of the Wiener process and $M_T=M_{t_1,...,t_n}=(M_{t_1},...,M_{t_n})$ and $f(x_1,...,x_n)=\frac{1}{n}\sum_{i=1}^n x^2_i$. So $$ \mathbb{E}(X_n)=\int_{\Omega}f_n \circ M_T dp_{\Omega}= \int_{\mathbb{R}^n}(\frac{1}{n}\sum_{i=1}^n x^2_i) g(\frac{1}{n},x_1)g(\frac{1}{n},x_2-x_1).... $$ One can do the calculations and show that this is actually equal to $$ \mathbb{E}(X_n)=\frac{1}{n}\sum_{i=1}^n \mathbb{E}(M^2_{\frac{ti}{n}}) $$ which converges to the integral $$ \int_{0}^t \mathbb{E}(M^2_s)ds=\int_0^t s = \frac{t^2}{2}. $$ I guess this could be derived using additivity of $E$ as well but $X$ was just a particular choice I wanted to see how to calculate expectations of more complex random variables and now I see that it is not really an issue whether if it is constant on cylinders or not.


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