In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the infinite-dimensional configuration space of the polymer chain e.g. in A Wiener integral model for stiff polymer chains and The Statistical Mechanical Theory of Stiff Chains. In both the linked papers, the integral is said to be the Wiener integral. If they are using the term the way I understand it, the Wiener integral is the integral with respect to the Wiener measure. I have recently gone through the proof of the existence and uniqueness of the Wiener measure using Kolmogorov's extension theorem, and am familiar with Lebesgue integration.
However, I am finding it hard to understand how the Wiener measure is appropriate for this particular application in polymer physics, because the functions used to represent particular spatial configurations of the polymer chains are space curves that are at least once differentiable, and in many cases the calculation of the bending energy of the polymer requires the use of the second derivative of the space curve. If I have understood the Wiener measure correctly, it results in sample functions being almost surely nowhere differentiable.
How does one rigorously justify the use of the Wiener measure in the calculation of these partition functions (and more generally when integrating over a space of differentiable functions)? I'd appreciate resources that lay this out systematically and rigorously, because I've spent quite a bit of time looking and haven't found any. There is a lot of literature available on path integrals used in quantum mechanics (and many papers in statistical mechanics now use the terminology of path integrals rather than Wiener integrals), but I do not need to worry about imaginary exponents here.
