An entropy optimal drift

## Construction of Föllmer's drift In a previous post, we saw how an entropy-optimal drift process could be used to prove the Brascamp-Lieb inequalities. Our main tool was a result of Föllmer that we now recall and justify. Afterward, we will use it to prove the Gaussian log-Sobolev inequality. Consider $f : \mathbb R^n \to \mathbb R_+$ with $\int f \,d\gamma_n = 1$, where $\gamma_n$ is the standard Gaussian measure on $\mathbb R^n$. Let $\\{B_t\\}$ denote an $n$-dimensional Brownian motion with $B_0=0$. We consider all processes of the form \begin{equation}\label{eq:drift} W_t = B_t + \int_0^t v_s\,ds\,, \end{equation} where $\\{v_s\\}$ is a progressively measurable drift and such that $W_1$ has law $f\,d\gamma_n$.
It holds that \[ D(f d\gamma_n \,\|\, d\gamma_n) = \min D(W_{[0,1]} \,\|\, B_{[0,1]}) = \min \frac12 \int_0^1 \mathbb{E}\,\|v_t\|^2\,dt\,, \] where the minima are over all processes of the form \eqref{eq:drift}.
In the preceding post (Lemma 2), we have already seen that for any drift of the form \eqref{eq:drift}, it holds that \[ D(f d\gamma_n \,\|\,d\gamma_n) \leq \frac12 \int_0^1 \mathbb{E}\,\|v_t\|^2\,dt \leq D(W_{[0,1]} \,\|\, B_{[0,1]})\,, \] thus we need only exhibit a drift $\\{v_t\\}$ achieving equality.

We define \[ v_t = \nabla \log P_{1-t} f(W_t) = \frac{\nabla P_{1-t} f(W_t)}{P_{1-t} f(W_t)}\,, \] where $\\{P_t\\}$ is the Brownian semigroup defined by \[ P_t f(x) = \mathbb{E}[f(x + B_t)]\,. \]

Note that $v_t$ is almost surely constant conditioned on the past, hence the chain rule yields \begin{equation}\label{eq:chain} D(W_{[0,1]} \,\|\, B_{[0,1]}) = \frac12 \int_0^1 \mathbb{E}\,\|v_t\|^2\,dt\,. \end{equation} (See line (7) of Lemma 2 in the previous post. Note that $h(v_t)=0$ since $v_t$ is deterministic given the past.) We are left to show that $W_1$ has law $f \,d\gamma_n$ and $D(W_{[0,1]} \,\|\, B_{[0,1]}) = D(f d\gamma_n \,\|\,d\gamma_n)$.

We will prove the first fact using Girsanov's theorem to argue about the change of measure between $\{W_t\}$ and $\{B_t\}$. As in the previous post, we will argue somewhat informally using the heuristic that the law of $dB_t$ is a Gaussian random variable in $\mathbb R^n$ with covariance $dt \cdot I$. Itô's formula states that this heuristic is justified (see our use of the formula below).

The following lemma says that, given any sample path $\{W_s : s \in [0,t]\}$ of our process up to time $s$, the probability that Brownian motion (without drift) would have "done the same thing is $\frac{1}{M_t}$.
I chose to present various steps in the next proof at varying levels of formality. The arguments have the same structure as corresponding formal proofs, but I thought (perhaps naïvely) that this would be instructive.
Let $\mu_t$ denote the law of $\\{W_s : s \in [0,t]\\}$. If we define \[ M_t = \exp\left(-\int_0^t \langle v_s,dB_s\rangle - \frac12 \int_0^t \|v_s\|^2\,ds\right)\,, \] then under the measure $\nu_t$ given by \[ d\nu_t = M_t \,d\mu_t\,, \] the process $\\{W_s : s \in [0,t]\\}$ has the same law as $\\{B_s : s \in [0,t]\\}$.
We argue by analogy with the discrete proof. First, let us define the infinitesimal ``transition kernel'' of Brownian motion using our heuristic that $dB_t$ has covariance $dt \cdot I$: \[ p(x,y) = \frac{e^{-\|x-y\|^2/2dt}}{(2\pi dt)^{n/2}}\,. \]

We can also compute the (time-inhomogeneous) transition kernel $q_t$ of $\\{W_t\\}$: \[ q_t(x,y) = \frac{e^{-\|v_t dt + x - y\|^2/2dt}}{(2\pi dt)^{n/2}} = p(x,y) e^{-\frac12 \|v_t\|^2 dt} e^{-\langle v_t, x-y\rangle}\,. \] Here we are using that $dW_t = dB_t + v_t\,dt$ and $v_t$ is deterministic conditioned on the past, thus the law of $dW_t$ is a normal with mean $v_t\,dt$ and covariance $dt \cdot I$.

To avoid confusion of derivatives, let's use $\alpha_t$ for the density of $\mu_t$ and $\beta_t$ for the density of Brownian motion (recall that these are densities on paths). Now let us relate the density $\alpha_{t+dt}$ to the density $\alpha_{t}$. We use here the notations $\\{\hat W_t, \hat v_t, \hat B_t\\}$ to denote a (non-random) sample path of $\\{W_t\\}$: \begin{align*} \alpha_{t+dt}(\hat W_{[0,t+dt]}) &= \alpha_t(\hat W_{[0,t]}) q_t(\hat W_t, \hat W_{t+dt}) \\ &= \alpha_t(\hat W_{[0,t]}) p(\hat W_t, \hat W_{t+dt}) e^{-\frac12 \|\hat v_t\|^2\,dt-\langle \hat v_t,\hat W_t-\hat W_{t+dt}\rangle} \\ &= \alpha_t(\hat W_{[0,t]}) p(\hat W_t, \hat W_{t+dt}) e^{-\frac12 \|\hat v_t\|^2\,dt+\langle \hat v_t,d \hat W_t\rangle} \\ &= \alpha_t(\hat W_{[0,t]}) p(\hat W_t, \hat W_{t+dt}) e^{\frac12 \|\hat v_t\|^2\,dt+\langle \hat v_t, d \hat B_t\rangle}\,, \end{align*} where the last line uses $d\hat W_t = d\hat B_t + \hat v_t\,dt$.

Now by ``heuristic'' induction, we can assume $\alpha_t(\hat W_{[0,t]})=\frac{1}{M_t} \beta_t(\hat W_{[0,t]})$, yielding \begin{align*} \alpha_{t+dt}(\hat W_{[0,t+dt]}) &= \frac{1}{M_t} \beta_t(\hat W_{[0,t]}) p(\hat W_t, \hat W_{t+dt}) e^{\frac12 \|\hat v_t\|^2\,dt+\langle \hat v_t, d \hat B_t\rangle} \\ &= \frac{1}{M_{t+dt}} \beta_t(\hat W_{[0,t]}) p(\hat W_t, \hat W_{t+dt}) \\ &= \frac{1}{M_{t+dt}} \beta_{t+dt}(\hat W_{[0,t+dt]})\,. \end{align*} In the last line, we used the fact that $p$ is the infinitesimal transition kernel for Brownian motion.
## The Gaussian log-Sobolev inequality Consider again a measurable $f : \mathbb R^n \to \mathbb R_+$ with $\int f\,d\gamma_n=1$. Let us define $\mathrm{Ent}_{\gamma_n}(f) = D(f\,d\gamma_n \,\\|\,d\gamma_n)$. Then the classical log-Sobolev inequality in Gaussian space asserts that \begin{equation}\label{eq:logsob} \mathrm{Ent}_{\gamma_n}(f) \leq \frac12 \int \frac{\|\nabla f\|^2}{f}\,d\gamma_n\\,. \end{equation} First, we discuss the correct way to interpret this. Define the Ornstein-Uhlenbeck semi-group $\\{U_t\\}$ by its action \\[ U_t f(x) = \mathbb{E}[f(e^{-t} x + \sqrt{1-e^{-2t}} B_1)]\\,. \\] This is the natural stationary diffusion process on Gaussian space. For every measurable $f$, we have \\[ U_t f \to \int f d\gamma_n \quad \textrm{ as $t \to \infty$}\,, \\] or equivalently \\[ \mathrm{Ent}_{\gamma_n}(U_t f) \to 0 \quad \textrm{ as $t \to \infty$}\,. \\]

The log-Sobolev inequality yields quantitative convergence in the relative entropy distance as follows: Define the Fisher information \[ I(f) = \int \frac{\|\nabla f\|^2}{f} \,d\gamma_n\,. \]

One can check that $$ \frac{d}{dt} \mathrm{Ent}_{\gamma_n} (U_t f)\Big|_{t=0} = - I(f)\,, $$ thus the Fisher information describes the instantaneous decay of the relative entropy of $f$ under diffusion.

So we can rewrite the log-Sobolev inequality as: \[ - \frac{d}{dt} \mathrm{Ent}_{\gamma_n}(U_t f)\Big|_{t=0} \geq \mathrm{Ent}_{\gamma_n}(f)\,. \] This expresses the intuitive fact that when the relative entropy is large, its rate of decay toward equilibrium is faster.