## Lecture 7. Integration by parts formula and log-Sobolev inequality

Let $\mathbb M$ be a smooth, connected manifold with dimension $n+m$. We assume that $\mathbb M$ is equipped with a Riemannian foliation $\mathcal{F}$ with bundle like metric $g$ and totally geodesic $m$-dimensional leaves.

We will assume that $\mathfrak{Ric}_{\mathcal{H}}$ is bounded from below and that $-\mathbf{J}^2$ and $\delta_\mathcal{H} T$ are bounded from above.
In that case, for every $\varepsilon >0$.
$\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge - \left( K+\frac{\kappa}{ \varepsilon} \right) \| \alpha \|^2_\varepsilon$
As before, we denote by $(X_t)_{t \ge 0}$ the horizontal Brownian motion. The stochastic parallel transport for the connection $\nabla$ along the paths of $(X_t)_{t \ge 0}$ will be denoted by $\zeta_{0,t}$. Since the connection $\nabla$ is horizontal, the map $\zeta_{0,t}: T_{X_0} \mathbb M \to T_{X_t} \mathbb M$ is an isometry that preserves the horizontal bundle, that is, if $u \in \mathcal{H}_{X_0}$, then $\zeta_{0,t} u \in \mathcal{H}_{X_t}$. We see then that the anti-development of $(X_t)_{t \ge 0}$,

$B_t=\int_0^t \zeta_{0,s}^{-1} \circ dX_s,$
is a Brownian motion in the horizontal space $\mathcal{H}_{X_0}$. The following integration by parts formula will play an important role in the sequel.

Lemma:  Let $x \in \mathbb M$. For any $C^1$ adapted process $\gamma:\mathbb{R}_{\ge 0} \to \mathcal{H}_{x}$ such that

$\mathbb{E}_x\left(\int_0^{T} \| \gamma'(s) \|_\mathcal{H}^2 ds\right)<+\infty$

and any $f \in C_0^\infty(\mathbb M)$,

$\mathbb{E}_x \left( f(X_T) \int_0^T \langle \gamma'(s),dB_s\rangle_{\mathcal{H}} \right)=\mathbb{E}_x \left(\left\langle \tau^\varepsilon_T df (X_T) ,\int_0^T (\tau^{\varepsilon,*}_s)^{-1} \zeta_{0,s} \gamma'(s) ds \right\rangle \right).$
Proof:
We consider the martingale process

$N_s=\tau_s^\varepsilon (dP_{T-s} f) (X_s).$
We have then for $f \in C_0^\infty(\mathbb M)$,

$\mathbb{E}_x \left( f(X_t) \int_0^t \langle \gamma'(s),dB_s\rangle_{\mathcal{H}} \right)$

$=\mathbb{E}_x \left( f(X_t) \int_0^t \langle \zeta_{0,s} \gamma'(s),\zeta_{0,s}dB_s\rangle_{\mathcal{H}} \right)$
$=\mathbb{E}_x \left( ( f(X_t) -\mathbb{E}_x \left( f(X_t)\right)) \int_0^t \langle \zeta_{0,s} \gamma'(s),\zeta_{0,s}dB_s\rangle_{\mathcal{H}} \right)$
$=\mathbb{E}_x \left(\int_0^t \langle dP_{t-s}f (X_s), \zeta_{0,s} dB_s \rangle \int_0^t \langle \zeta_{0,s} \gamma'(s),\zeta_{0,s}dB_s\rangle_{\mathcal{H}} \right)$
$=\mathbb{E}_x \left(\int_0^t \langle dP_{t-s}f (X_s), \zeta_{0,s} \gamma'(s) \rangle ds \right)$
$=\mathbb{E}_x \left(\int_0^t \langle \tau_s^\varepsilon dP_{t-s}f (X_s), (\tau^{\varepsilon,*}_s)^{-1} \zeta_{0,s} \gamma'(s) \rangle ds \right)$
$=\mathbb{E}_x \left(\int_0^t \langle N_s , (\tau^{\varepsilon,*}_s)^{-1} \zeta_{0,s} \gamma'(s) \rangle ds \right)$
$=\mathbb{E}_x \left(\left\langle N_t ,\int_0^t (\tau^{\varepsilon,*}_s)^{-1} \zeta_{0,s} \gamma'(s) ds \right\rangle \right),$

where we integrated by parts in the last equality. $\square$

As an immediate consequence of the integration by parts formula, we obtain the following Clark-Ocone type representation.

Proposition: Let $X_0=x \in \mathbb M$. For every $\in C_0^\infty(\mathbb M)$, and every $t \ge 0$,

$f(X_t)=P_tf(x) +\int_0^t \left\langle \mathbb{E}_x \left( (\tau^\varepsilon_s)^{-1} \tau^\varepsilon_t df(X_t) \mid \mathcal{F}_s \right), \zeta_{0,s} dB_s \right \rangle ,$
where $(\mathcal{F}_t)_{t \ge 0}$ is the natural filtration of $(B_t)_{t \ge 0}$.

Proof:
Let $t \ge 0$. From Ito’s integral representation theorem, we can write

$f(X_t)=P_tf(x) +\int_0^t \left\langle a_s, dB_s \right \rangle_{\mathcal{H}} ,$
for some adapted and square integrable $(a_s)_{0 \le s \le t}$. Using the integration by parts formula, we obtain therefore,

$\mathbb{E}_x \left( \int_0^t \langle \gamma'(s),a_s\rangle_{\mathcal{H}} ds \right)=\mathbb{E}_x \left(\left\langle \tau^\varepsilon_t df (X_t) ,\int_0^t (\tau^{\varepsilon,*}_s)^{-1} \zeta_{0,s} \gamma'(s) ds \right\rangle \right).$
Since $\gamma'$ is arbitrary, we obtain that

$a_s= \mathbb{E}_x \left( \zeta^{-1}_{0,s} (\tau^\varepsilon_s)^{-1} \tau^\varepsilon_t df(X_t) \mid \mathcal{F}_s \right).$

$\square$

We deduce first the following Poincare inequality for the heat kernel measure.

Proposition: For every $f \in C_0^\infty(\mathbb M)$, $t \ge 0$, $x \in \mathbb M$, $\varepsilon >0$,

$P_t(f^2)(x) -(P_t f)^2(x) \le \frac{ e^{\left(K+\frac{\kappa}{\varepsilon} \right)t} -1}{K+\frac{\kappa}{\varepsilon}} \left[ P_t (\| \nabla_\mathcal{H} f \|^2)(x) + \varepsilon P_t (\| \nabla_\mathcal{V} f \|^2)(x) \right]$
Proof: From the previous proposition we have
$\mathbb{E}_x\left((f(X_t)-P_tf(x))^2 \right) \le \int_0^t e^{\left( K+\frac{\kappa}{ \varepsilon} \right)(t-s)} ds P_t (\| df \|_{\varepsilon}^2)(x).$

$\square$

We also get the log-Sobolev inequality for the heat kernel measure.

Proposition: For every $f \in C_0^\infty(\mathbb M)$, $t \ge 0$, $x \in \mathbb M$, $\varepsilon >0$,

$P_t(f^2\ln f^2 )(x) -P_t (f^2)(x)\ln P_t (f^2)(x) \le 2 \frac{ e^{\left(K+\frac{\kappa}{\varepsilon} \right)t} -1}{K+\frac{\kappa}{\varepsilon}} \left[ P_t (\| \nabla_\mathcal{H} f \|^2)(x) + \varepsilon P_t (\| \nabla_\mathcal{V} f \|^2)(x) \right]$

Proof: The method for proving the log-Sobolev inequality from the representation theorem is due to Capitaine-Hsu-Ledoux and the argument is easy to reproduce in our setting. Denote $G=f(X_t)^2$ and consider the martingale $N_s= \mathbb{E}( G | \mathcal{F}_s)$. Applying now Ito’s formula to $N_s \ln N_s$ and taking expectation yields

$\mathbb{E}_x( N_t \ln N_t)-\mathbb{E}_x( N_0 \ln N_0)=\frac{1}{2} \mathbb{E}_x\left( \int_0^t \frac{d[N]_s}{N_s} \right),$
where $[N]$ is the quadratic variation of $N$. From the Clark-Ocone representation theorem applied with $f^2$, we have

$dN_s=2 \left\langle \mathbb{E} \left( f(X_t)(\tau^\varepsilon_s)^{-1} \tau^\varepsilon_t df(X_t) \mid \mathcal{F}_s \right), \zeta_{0,s} dB_s \right \rangle_{\mathcal{H}}.$
Thus we have from Cauchy-Schwarz inequality

$\mathbb{E}_x( N_t \ln N_t)-\mathbb{E}_x( N_0 \ln N_0) \le 2 \mathbb{E}_x\left( \int_0^t \frac{\|\mathbb{E} \left( f(X_t)(\tau^\varepsilon_s)^{-1} \tau^\varepsilon_t df(X_t) \mid \mathcal{F}_s \right)\|_{\varepsilon}^2}{N_s} ds \right)$
$\le 2 \int_0^t e^{\left( K+\frac{\kappa}{ \varepsilon} \right)(t-s)} dsP_t (\| df \|_{\varepsilon}^2)(x).$

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