Follmer’s drift, Ito’s lemma, and the log-Sobolev inequality

~ James

1. 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 \rightarrow \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

\displaystyle W_t = B_t + \int_0^t v_s\,ds\,, \ \ \ \ \ (1)

where {\{v_s\}} is a progressively measurable drift and such that {W_1} has law {f\,d\gamma_n} .

Theorem 1 (Föllmer) It holds that

\displaystyle D(f d\gamma_n \,\|\, d\gamma_n) = \min D(W_{[0,1]} \,\|\, B_{[0,1]}) = \min \frac12 \int_0^1 \mathop{\mathbb E}\,\|v_t\|^2\,dt\,,

where the minima are over all processes of the form (1).

Proof: In the preceding post (Lemma 2), we have already seen that for any drift of the form (1), it holds that

\displaystyle D(f d\gamma_n \,\|\,d\gamma_n) \leq \frac12 \int_0^1 \mathop{\mathbb E}\,\|v_t\|^2\,dt = D(W_{[0,1]} \,\|\, B_{[0,1]})\,,

thus we need only exhibit a drift {\{v_t\}} achieving equality.

We define

\displaystyle 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

\displaystyle P_t f(x) = \mathop{\mathbb E}[f(x + B_t)]\,.

As we saw in the previous post (Lemma 2), the chain rule yields

\displaystyle D(W_{[0,1]} \,\|\, B_{[0,1]}) = \frac12 \int_0^1 \mathop{\mathbb E}\,\|v_t\|^2\,dt\,. \ \ \ \ \ (2)

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}} .

Remark 1 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.

Lemma 2 Let {\mu_t} denote the law of {\{W_s : s \in [0,t]\}} . If we define

\displaystyle 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

\displaystyle 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]\}} .

Proof: 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} :

\displaystyle 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\}} :

\displaystyle 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\}} :

\displaystyle \begin{array}{lll} \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{array}

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

\displaystyle \begin{array}{lll} \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{array}

In the last line, we used the fact that {p} is the infinitesimal transition kernel for Brownian motion. \Box

Now we will show that

\displaystyle P_{1-t} f(W_t) = \exp\left(\frac12 \int_0^t \|v_s\|^2\,ds + \int_0^t \langle v_s, dB_s\rangle\right) = \frac{1}{M_t}\,. \ \ \ \ \ (3)

From Lemma 2, it will follow that {W_t} has the law {(P_{1-t} f)\cdot d\nu_t} where {d\nu_t} is the law of {B_t} . In particular, {W_1} has the law {f\,d\nu_1 = f\,d\gamma_n} which was our first goal.

Given our preceding less formal arguments, let us use a proper stochastic calculus argument to establish (3). To do that we need a way to calculate

\displaystyle d \log P_{1-t} f(W_t) \quad \textrm{``}= \log P_{1-t-dt} f(W_{t+dt}) - \log P_{1-t} f(W_t)\textrm{''} \ \ \ \ \ (4)

Notice that this involves both time and space derivatives.

Itô’s lemma. Suppose we have a continuously differentiable function {F : \mathbb R \times [0,1] \rightarrow \mathbb R} that we write as {F(x,t)} where {x} is a space variable and {t} is a time variable. We can expand {d F} via its Taylor series:

\displaystyle d F = \partial_t F \,dt + \partial_x F\,dx + \frac12 \partial_x^2 F\,dx^2 + \frac12 \partial_x \partial_t F\,dx\,dt + \cdots\,.

Normally we could eliminate the terms {dx^2, dx\, dt} , etc. since they are lower order as {dx,dt \rightarrow 0} . But recall that for Brownian motion we have the heuristic {\mathop{\mathbb E}[dB_t^2]=dt} . Thus we cannot eliminate the second-order space derivative if we plan to plug in {x=B_t} (or {x=W_t} , a process driven by Brownian motion). Itô’s lemma says that this consideration alone gives us the correct result:

\displaystyle d F(W_t,t) = \partial_t F(W_t,t)\,dt + \partial_x F(W_t,t)\,dW_t + \frac12 \partial_x^2 F(W_t,t)\,dt\,.

This generalizes in a straightforward way to the higher dimensional setting {F : \mathbb R^n \times [0,1] \rightarrow \mathbb R} .

With Itô’s lemma in hand, let us continue to calculate the derivative

\displaystyle \begin{array}{lll} d P_{1-t} f(W_t) &= - \Delta P_{1-t} f(W_t)\,dt + \langle \nabla P_{1-t} f(W_t), dW_t\rangle + \Delta P_{1-t} f(W_t) \,dt \\ &= \langle \nabla P_{1-t} f(W_t), dW_t\rangle \\ &= P_{1-t} f(W_t) \,\langle v_t, dW_t\rangle\,. \end{array}

For the time derivative (the first term), we have employed the heat equation

\displaystyle \partial_t P_{1-t} f = - \Delta P_{1-t} f\,,

where {\Delta = \frac12 \sum_{i=1}^n \partial_{x_i}^2} is the Laplacian on {\mathbb R^n} .

Note that the heat equation was already contained in our “infinitesimal density” {p} in the proof of Lemma 2, or in the representation {P_t = e^{t \Delta}} , and Itô’s lemma was also contained in our heuristic that {dB_t} has covariance {dt \cdot I} .

Using Itô’s formula again yields

\displaystyle d \log P_{1-t} f(W_t) = \langle v_t, dW_t\rangle - \frac12 \|v_t\|^2\,dt = \frac12 \|v_t\|^2\,dt + \langle v_t,dB_t\rangle\,.

giving our desired conclusion (3).

Our final task is to establish optimality: {D\left(W_{[0,1]} \,\|\, B_{[0,1]}\right) = D(W_1\,\|\,B_1)} . We apply the formula (3):

\displaystyle D(W_1\,\|\,B_1) = \mathop{\mathbb E}[\log f(W_1)] = \mathop{\mathbb E}\left[\frac12 \int_0^1 \|v_t\|^2\,dt\right],

where we used {\mathop{\mathbb E}[\langle v_t,dB_t\rangle]=0} . Combined with (2), this completes the proof of the theorem. \Box

2. The Gaussian log-Sobolev inequality

Consider again a measurable {f : \mathbb R^n \rightarrow \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

\displaystyle \mathrm{Ent}_{\gamma_n}(f) \leq \frac12 \int \frac{\|\nabla f\|^2}{f}\,d\gamma_n\,. \ \ \ \ \ (5)

First, we discuss the correct way to interpret this. Define the Ornstein-Uhlenbeck semi-group {\{U_t\}} by its action

\displaystyle U_t f(x) = \mathop{\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

\displaystyle U_t f \rightarrow \int f d\gamma_n \quad \textrm{ as } t \to \infty

or equivalently

\displaystyle \mathrm{Ent}_{\gamma_n}(U_t f) \rightarrow 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

\displaystyle I(f) = \int \frac{\|\nabla f\|^2}{f} \,d\gamma_n\,.

One can check that

\displaystyle \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:

\displaystyle - \frac{d}{dt} \mathrm{Ent}_{\gamma_n}(U_t f)\Big|_{t=0} \geq 2 \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.

Martingale property of the optimal drift. Now for the proof of (5). Let {dW_t = dB_t + v_t\,dt} be the entropy-optimal process with {W_1 \sim f \,d\gamma_n} . We need one more fact about {\{v_t\}} : The optimal drift is a martingale, i.e. {\mathop{\mathbb E}[v_t \mid v_s] = v_s} for {s < t} .

Let’s give two arguments to support this.

Argument one: Brownian bridges. First, note that by the chain rule for relative entropy, we have:

\displaystyle D(W_{[0,1]} \,\|\, B_{[0,1]}) = D(W_1 \,\|\, B_1) + \int D(W_{[0,1]} \,\|\, B_{[0,1]} \mid W_1=B_1=x) f(x) d\gamma_n(x)\,.

But from optimality, we know that the latter expectation is zero. Therefore {f \,d\gamma_n} -almost surely, we have

\displaystyle D(W_{[0,1]} \,\| B_{[0,1]} \mid W_1=B_1=x) = 0\,.

This implies that if we condition on the endpoint {x} , then {W_{[0,1]}} is a Brownian bridge (i.e., a Brownian motion conditioned to start at {0} and end at {x} ).

This implies that {\mathop{\mathbb E}[v_t \mid v_s, W_1=x] = v_s} , as one can check that a Brownian bridge {\{\hat B_t\}} with endpoint {x} is described by the drift process {d\hat B_t = dB_t + \frac{x-\hat B_t}{1-t}\,dt} , and

\displaystyle \mathop{\mathbb E}\left[\frac{x-\hat B_t}{1-t} \,\Big|\, B_{[0,s]}\right] = \frac{x-\hat B_s}{1-s}\,.

That seemed complicated. There is a simpler way to see this: Given {\hat B_s} and any bridge {\gamma} from {\hat B_s} to {x} , every “permutation” of the infinitesimal steps in {\gamma} has the same law (by commutativity, they all land at {x} ). Thus the marginal law of {dB_t + v_t\,dt} at every point {t \geq s} should be the same. In particular,

\displaystyle \mathop{\mathbb E}[v_t\,dt \mid v_s] = \mathop{\mathbb E}[dB_t + v_t\,dt \mid v_s] = \mathop{\mathbb E}[dB_s + v_s \,ds \mid v_s] = v_s\,ds\,.

Argument two: Change of measure. There is a more succinct (though perhaps more opaque) way to see that {\{v_t\}} is a martingale. Note that the process {\nabla P_{1-t} f(B_t) = P_{1-t} \nabla f(B_t)} is a Doob martingale. But we have {v_t = \frac{\nabla P_{1-t} f(W_t)}{P_{1-t} f(W_t)}} and we also know that {\frac{1}{P_{1-t} f(W_t)} = \frac{1}{M_t}} is precisely the change of measure that makes {\{W_t\}} into Brownian motion.

Proof of the log-Sobolev inequality. In any case, now we are ready for the proof of (5). It also comes straight from Lehec’s paper. Since {\{v_t\}} is a martingale, we have {\mathop{\mathbb E}\,\|v_t\|^2 \leq \mathop{\mathbb E}\,\|v_1\|^2} . So by Theorem 1:

\displaystyle \mathrm{Ent}_{\gamma_n}(f) = \frac12 \int_0^1 \mathop{\mathbb E}\,\|v_t\|^2\,dt \leq \frac12 \mathop{\mathbb E}\,\|v_1\|^2 = \frac12 \mathop{\mathbb E}\, \frac{\|\nabla f(W_1)\|^2}{f(W_1)^2} = \frac12 \mathop{\mathbb E}\, \frac{\|\nabla f(B_1)\|^2}{f(B_1)}\,.

The latter quantity is \frac12 I(f) . In the last equality, we used the fact that {\frac{1}{f(W_1)}} is precisely the change of measure that turns {\{W_t\}} into Brownian motion.

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