We know how to find a path/function that extremises the value of a functional using the Euler-Lagrange equation. (Functional here means something that takes a function and spits out a number.) I'd like to ask a slightly different question - can we find the average value of a functional, taken over all possible paths (maybe with some weightage attached to each path)?
As a somewhat concrete example, suppose our functional returns the path length of a path, and the domain under consideration is the set of all monotonically increasing, convex, rectifiable paths from $[0,1]$ to $[0,1]$. Let's also impose that the paths must pass through $(0,0)$ and $(1,1)$. (I would suppose that the value of the path length functional on this domain must be somewhere between $\sqrt{2}$ and $2$.)
Now, I want to find the 'average value' for this functional, taken over all feasible paths. In some sense, if I choose such a path randomly, how long would I expect to walk?
For some optional added complexity, let's say I'm more likely to choose a path that involves less 'turning', i.e. paths with more Dirichlet energy are chosen with less probability. How would this average calculation then be approached?
Is this even a well-posed question, in the first place?
I imagine that for a simple enough domain, one could parametrize the set of paths of interest using a small set of real parameters, and then solve the problem using regular calculus. But I wonder how this problem could be approached using the machinery of the calculus of functionals, if such is needed.