Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there is an obvious way to identify this with $\mathbb{Z}_2^\infty$, and this is in fact a nice compact abelian group.

So, the functional of a simple random walk which, say, gives the hitting time of some $k$, or the indicator function of the event that we hit $n$ before $-n$, can be seen as functions from $\mathbb{Z}_2^\infty$ into $\mathbb{R}$. Thus, these functions have Fourier transforms.

Unless I am completely mistaken, the Fourier transform of these functions should look like (taking our omegas to be -1 or 1, for convenience) $$f(\omega_1,\omega_2,\ldots) = \sum_{S\subset\mathbb{N},\ |S|<\infty} \hat{f}(S)\prod_{i\in S}\omega_i$$ with $\hat{f}(S)$ real-valued Fourier coefficients, analogously to the Fourier analysis of functions of finitely many $\omega$.

Has this been studied before? Do the Fourier coefficients of these functions encode anything interesting about the simple random walk?

I tried to analyse the second example a bit, trying to compute the level one Fourier coefficients (i.e. the ones corresponding to singleton subsets of $\mathbb{N}$). Letting $\tau_{x,y}$ be the first time the random walk hits $y$ after it has hit $x$, and let $\tau_x$ be the hitting time of $x$ from $0$, and further letting $\tau^i_x$ be the first time after time $i$ at which the random walk hits $x$, I get the following formula: $$\hat{f}(\{i\}) = \mathbb{P}\left(\omega_i = 1, i \leq \tau_n, \tau^i_{-(n-1)} < \tau_{n+1}-i\ \middle|\ \tau_n < \tau_{-n}\right)$$ which I don't really immediately see how to compute.