I might not have stated the question as clearly as I could have in the title.
Basically my thought process is the following; If I have a brownian process that I'm going to let run from, say, $0$ to $1$ minute, then this brownian process will trace out some non-differentiable continuous path in $\mathbb{R}^2$. If I let the process run again, then I will get a different path. This suggests to me that there is a random variable on the set of paths a brownian process can trace out.
My question is; is the set of all these paths a well defined probability space, and if so, do we know what that distribution looks like (does it have a mean? or variance? etc)? Or, does this set of paths exhibit certain pathologies that prevent it from being a probability space?
Related (perhaps easier): if we just restrict our attention to the set of values that a brownian process hits (so, ignoring the actual paths), do we have a way of answering something like "What is the probability that this brownian process hits a value of $1000$ in the first $30$ seconds?", or similar questions?
Thanks in advance.