Functional derivative under a path integral sign

Question

Is it possible to have the analogous version of the Leibniz integral rule for the path integral? Something like

$$\frac{\delta}{\delta\chi}\int_{\phi|_{x=x'}= \chi} [\mathcal{D}\phi]e^{-S[\phi]} \sim e^{-S[\chi]}~? $$

Answer 1

For what it's worth, to gain some intuition, consider the following crude discretization of OP's path integral: $$\frac{\partial}{\partial\chi^k}\underbrace{\left[\prod_{i\in I} \int d\phi^{i} e^{-{\cal L}(\phi^{i})} \right]}_{\rm bulk}\underbrace{\left[\prod_{j\in J}e^{-{\cal L}(\chi^j)}\right]}_{\rm boundary}~=~- \frac{\partial {\cal L}(\chi^k)}{\partial\chi^k}\left[\prod_{i\in I} \int d\phi^{i} e^{-{\cal L}(\phi^{i})} \right]\left[\prod_{j\in J}e^{-{\cal L}(\chi^j)}\right].$$ This should give an indication of what to expect.