As far as I have been able to see, the Feynman path integral can be given a precise mathematical meaning via analytic continuation of Wiener Gaussian integrals. First you define the Wiener measure on cylindrical sets of paths and extend it to all paths via the Caratheodory extension theorem. Then M. Kac realized that the kernel of the heat equation could be written as this path integral. The book I am reading (Mathematical Feynman path integrals and their applications - Sonia Mazzuchi) then states:
"The starting point of this approach is the transformation of variable formula for the Wiener integral with covariance $\sigma$ \begin{equation} \int f(\omega) dW_{\sigma}(\omega) = \int f(\sqrt{\sigma}\omega)dW(\omega) \end{equation} As Cameron proved, the left hand side is not defined when $\sigma$ is complex. The leading idea of analytic continuation approach is to give meaning to the right hand side in the case where $\sigma =i $"
The book later states that under suitable conditions the analytically continued expression can be shown, in a completely rigorous manner, to be a solution of Schrödinger's equations.
I do not pretend to know all the measure theory involved in the construction of the Wiener measure (in fact this book is not even about this approach), nor all the analytical requirements necessary to make the whole thing work. But by reading other things, including the answers here (Path integral vs. measure on infinite dimensional space), I have been lead to believe that this whole thing fails in the case of fields in $3+1$. And that a rigorous construction of the path integral for fields has not yet been worked out.
I would like to know why. Does the measure theory involved fail? Is the problem in the analytic continuation? Does it fail solely for gauge theories? Either an expository answer or a list of references regarding the question would be very appreciated.
