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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