I am reading a paper on the generalization of Caffarelli's contraction theorem. I came across following statements that is stated as obvious but I am not able to prove it.
Let $V:\mathbb{R}^n\to \mathbb{R}$ be a convex function and let $f=e^{-V}$ be a probability density with respect to the standard gaussian measure $d\gamma$. Define $$f_t(x):=P_tf(x)=\mathbb{E}f(e^{-t}x+\sqrt{1-e^{-2t}}Z)$$ where $Z$ is a standard normal random variable and let $V_t:=-\log(f_t)$. Consider the flow defined by $$\frac{dS_t(x)}{dt}=\nabla V_t(S_t(x)).$$
The author remarks "assuming that such a flow exists, it is not hard to see that $(S_t)\sharp fd\gamma =f_td\gamma$" where $\gamma$ is standard gaussian measure.
I tried sing brute-force, that is, take a $C^2$ function $g$ and compute $\mathbb{E}(g(S_t(x)))$ (where the expectation is taken with respect to the measure $fd\gamma$). But this does not give me anything useful. I wanted to show $$\mathbb{E}(g(S_t(x)))=\int g(y)f_t(y)d\gamma(y),$$ but my approach does not give me anything close to this! Any help would be appreciated! I would prefer a direct proof that does not involve too many properties of Semigroups and so on (if that's possible).
Here's a formal argument that relies on two facts:
(a). The semigroup $P_t$ you have defined is the Ornstein--Uhlenbeck semigroup with generator $L$ given by $$ Lf = \Delta f - x\cdot\nabla f \, . $$ This is just an explicit computation and involves nothing deep about semigroups.
(b). Given this information, both $(S_t)_\# (f \mathrm{d}\gamma)$ and $f_t \mathrm{d}\gamma$ are weak (i.e. distributional) solutions of the following continuity equation $$ \partial_t \rho_t = -\nabla \cdot(\rho_t \nabla V_t )\, , $$ with initial datum $\rho_0 = f \mathrm{d}\gamma$. Again this is an explicit computation. Indeed, for $(S_t)_{\#}(f \mathrm{d}\gamma)$, we have for any $\varphi \in C_c^\infty(\mathbb{R}^n)$ \begin{align} \int_0^T\int_{\mathbb{R}^n}\nabla \varphi(x) \cdot \nabla V_t(x) (S_t)_{\#}(f \mathrm{d}\gamma)(x) \mathrm{d}t =& \int_0^T \int_{\mathbb{R}^n}\nabla \varphi(S_t (x)) \cdot \nabla V_t(S_t(x)) f(x) \mathrm{d}\gamma(x) \mathrm{d}t \\ =& \int_0^T \frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^n} \varphi(S_t (x)) f(x) \mathrm{d}\gamma(x)\mathrm{d}t \\ =& \int_{\mathbb{R}^n} \varphi(S_T (x)) f(x) \mathrm{d}\gamma(x)- \int_{\mathbb{R}^n} \varphi(x) f(x) \mathrm{d}\gamma(x) \, . \end{align} A similar calculation can be carried out for the other measure where we need to use the fact that $$ \partial_t f_t =L f_t \, . $$ Solutions to the continuity equation should be unique under rather mild conditions on $V$; I am not sure what the minimal conditions are, I would guess globally Lipschitz is sufficient but there are countless papers which study uniqueness for such equations under much more restrictive assumptions on $V$.