While reading a preprint [Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process](https://arxiv.org/abs/1903.04522) I came across the following SDE in Section 3:
$$d X\_t=d B\_t+\nabla \log P\_{1-t} f\left(X\_t\right) d t$$
where $B\_t$ is the Brownian motion and $P\_t$ denotes the heat semigroup.


Given a probability measure $\mu(dx)=f(x)\gamma(dx)$ where $\gamma$ is the gaussian density, one can consider $X\_t$ to be the process that is close to a Brownian motion while having the law $\mu$ at time $t=1$.


I would like to understand why the following change of measure formula holds for every test function $u$,


$$\mathbb{E}\left[u\left(X\_1\right) \mid \mathcal{F}\_t\right]=\frac{P\_{1-t}(u f)\left(X\_t\right)}{P\_{1-t} f\left(X\_t\right)}$$


where $\mathcal{F}\_t$ denotes the natural filtration associated with the process $X\_t$. I looked into Girsanov's theorem in this regard but I don't see any exponential martingales in the above expression, so I am not sure what is going on here.



Change of measure formula for the Föllmer process
One can show it using the Feynman-Kac formula. In particular, from Feynman-Kac, we know that $h(x, t) := \mathbb{E}[u(X\_1) \mid \mathcal{F}\_t ]$ solves the following PDE:


$ \frac{\partial}{\partial t} h(x, t) + \nabla \log P\_{1-t} f(X\_t)^T \nabla\_x h(x, t) + \frac{1}{2} \Delta\_x h(x, t) = 0.$


with the terminal condition $h(x, 1) = u(x)$.


Now, note that your proposed function for $h(x, t)$, namely $\frac{P\_{1-t} (uf)}{P\_{1-t}(f) }$ satisfies the terminal condition, so you just need to plug your proposed function to the above equation to show that it satisfies the PDE above.


