## 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

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

## Lecture 6. Transverse Weitzenbock formula and heat equation on one-forms

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 define the canonical variation of $g$ as the one-parameter family of Riemannian metrics:

$g_{\varepsilon}=g_\mathcal{H} \oplus \frac{1}{\varepsilon }g_{\mathcal{V}}, \quad \varepsilon >0.$
We now introduce some tensors and definitions that will play an important role in the sequel.

For $Z \in \Gamma^\infty(T\mathbb M)$, there is a unique skew-symmetric endomorphism $J_Z:\mathcal{H}_x \to \mathcal{H}_x$ such that for all horizontal vector fields $X$ and $Y$,

$g_\mathcal{H} (J_Z (X),Y)= g_\mathcal{V} (Z,T(X,Y)).$
where $T$ is the torsion tensor of $\nabla$. We then extend $J_{Z}$ to be 0 on $\mathcal{V}_x$. If $Z_1,\cdots,Z_m$ is a local vertical frame, the operator $\sum_{l=1}^m J_{Z_l}J_{Z_l}$ does not depend on the choice of the frame and shall concisely be denoted by $\mathbf{J}^2$. For instance, if $\mathbb M$ is a K-contact manifold equipped with the Reeb foliation, then $\mathbf{J}$ is an almost complex structure, $\mathbf{J}^2=-\mathbf{Id}_{\mathcal{H}}$.
The horizontal divergence of the torsion $T$ is the $(1,1)$ tensor which is defined in a local horizontal frame $X_1,\cdots,X_n$ by

$\delta_\mathcal{H} T (X)=- \sum_{j=1}^n(\nabla_{X_j} T) (X_j,X), \quad X \in \Gamma^\infty(\mathbb M).$
The $g$-adjoint of $\delta_\mathcal{H}T$ will be denoted $\delta_\mathcal{H} T^*$.

In the sequel, we shall need to perform computations on one-forms. For that purpose we introduce some definitions and notations on the cotangent bundle.

We say that a one-form to be horizontal (resp. vertical) if it vanishes on the vertical bundle $\mathcal{V}$ (resp. on the horizontal bundle $\mathcal{H}$). We thus have a splitting of the cotangent space

$T^*_x \mathbb M= \mathcal{H}^*(x) \oplus \mathcal{V}^*(x)$

The metric $g_\varepsilon$ induces then a metric on the cotangent bundle which we still denote $g_\varepsilon$. By using similar notations and conventions as before we have for every $\eta$ in $T^*_x \mathbb M$,

$\| \eta \|^2_{\varepsilon} =\| \eta \|_\mathcal{H}^2+\varepsilon \| \eta \|_\mathcal{V}^2.$

By using the duality given by the metric $g$, $(1,1)$ tensors can also be seen as linear maps on the cotangent bundle $T^* \mathbb M$. More precisely, if $A$ is a $(1,1)$ tensor, we will still denote by $A$ the fiberwise linear map on the cotangent bundle which is defined as the $g$-adjoint of the dual map of $A$. The same convention will be made for any $(r,s)$ tensor.

We define then the horizontal Ricci curvature $\mathfrak{Ric}_{\mathcal{H}}$ as the fiberwise symmetric linear map on one-forms such that for every smooth functions $f,g$,

$\langle \mathfrak{Ric}_{\mathcal{H}} (df), dg \rangle=\mathbf{Ricci} (\nabla_\mathcal{H} f ,\nabla_\mathcal{H} g),$
where $\mathbf{Ricci}$ is the Ricci curvature of the connection $\nabla$.
If $V$ is a horizontal vector field and $\varepsilon >0$, we consider the fiberwise linear map from the space of one-forms into itself which is given for $\eta \in \Gamma^\infty(T^* \mathbb M)$ and $Y \in \Gamma^\infty(T \mathbb M)$ by

$\mathfrak{T}^\varepsilon_V \eta (Y) = \begin{cases} \frac{1}{\varepsilon} \eta (J_Y V), \quad Y \in \Gamma^\infty(\mathcal{V}) \\ -\eta (T(V,Y)), Y \in \Gamma^\infty(\mathcal{H}) \end{cases}$
We observe that $\mathfrak{T}^\varepsilon_V$ is skew-symmetric for the metric $g_\varepsilon$ so that $\nabla -\mathfrak{T}^\varepsilon$ is a $g_\varepsilon$-metric connection.

If $\eta$ is a one-form, we define the horizontal gradient of $\eta$ in a local frame as the $(0,2)$ tensor

$\nabla_\mathcal{H} \eta =\sum_{i=1}^n \nabla_{X_i} \eta \otimes \theta_i.$

Similarly, we will use the notation

$\mathfrak{T}^\varepsilon_\mathcal{H} \eta =\sum_{i=1}^n \mathfrak{T}^\varepsilon_{X_i} \eta \otimes \theta_i.$

Finally, we will still denote by $\Delta_\mathcal{H}$ the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus

$\Delta_{\mathcal{H}}=-\nabla_{\mathcal{H}}^* \nabla_{\mathcal{H}}=\sum_{i=1}^n \nabla_{X_i}\nabla_{X_i} -\nabla_{\nabla_{X_i} X_i}.$
For $\varepsilon >0$, we consider the following operator which is defined on one-forms by

$\square_\varepsilon=-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)-\frac{1}{ \varepsilon}\mathbf{J}^2+\frac{1}{\varepsilon} \delta_\mathcal{H} T- \mathfrak{Ric}_{\mathcal{H}},$
where the adjoint is understood with respect to the metric $g_{\varepsilon}$. It is easily seen that, in a local horizontal frame,
$-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) =\sum_{i=1}^n (\nabla_{X_i} -\mathfrak{T}^\varepsilon_{X_i})^2 - ( \nabla_{\nabla_{X_i} X_i}- \mathfrak{T}^\varepsilon_{\nabla_{X_i} X_i}),$

We can also consider the operator which is defined on one-forms by

$\square_\infty:=\sum_{i=1}^n (\nabla_{X_i} -\mathfrak{T}^\infty_{X_i})^2 - ( \nabla_{\nabla_{X_i} X_i}- \mathfrak{T}^\infty_{\nabla_{X_i} X_i}) - \mathfrak{Ric}_{\mathcal{H}}$

It is clear that for every smooth one-form $\alpha$ on $\mathbb M$ and every $x \in \mathbb M$ the following holds

$\lim_{\varepsilon \to \infty} \square_\varepsilon \alpha (x)=\square_\infty \alpha (x).$

The following theorem that was proved in this paper is the main result of the lecture:

Theorem: Let $0 < \varepsilon \le +\infty$. For every $f \in C^\infty(\mathbb M)$, we have
$d \Delta_{\mathcal{H}} f=\square_\varepsilon df.$

Proof:
We only sketch the proof and refer to the original paper for the details. If $Z_1,\cdots,Z_m$ is a local vertical frame of the leaves, we denote

$\mathfrak J(\eta)=-\sum_{l=1}^mJ_{Z_l}(\iota_{Z_l}d\eta_{\mathcal V}),$
where $\eta_{\mathcal V}$ is the the projection of $\eta$ onto the vertical cotangent bundle. It does not depend on the choice of the frame and therefore defines a globally defined tensor.
Also, let us consider the map $\mathcal{T} \colon \Gamma^\infty(\wedge^2 T^*\mathbb M)\to \Gamma^\infty( T^*\mathbb M)$ which is given in a local coframe $\theta_i \in \Gamma^\infty(\mathcal{H}^*)$, $\nu_k \in \Gamma^\infty(\mathcal{V}^*)$

$\mathcal{T}(\theta_i\wedge\theta_j)=-\gamma_{ij}^l \nu_l,\quad \mathcal{T}(\theta_i\wedge\nu_k)=\mathcal{T}(\nu_k\wedge\nu_l)=0.$
A direct computation shows then that

$-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) =$
$\Delta_{\mathcal H} +2\mathfrak J-\frac{2}{\varepsilon}\mathcal{T}\circ d+\delta_\mathcal{H} T^*-\frac{1}{\varepsilon}\delta_\mathcal{H} T+\frac{1}{\varepsilon}\mathbf{J}^2.$
Thus, we just need to prove that if $\square_\infty$ is the operator defined on one-forms by

$\square_\infty=\Delta_{\mathcal H}+2\mathfrak J-\mathfrak{Ric}_{\mathcal{H}}+\delta_\mathcal{H} T^* ,$
then for any $f\in C^\infty(\mathbb M)$,

$d\Delta_{\mathcal H} f=\square_\infty df.$
A computation in local frame shows that

$d\Delta_{\mathcal H} f- d\Delta_{\mathcal H} f = 2\mathfrak J(df) -\mathfrak{Ric}_{\mathcal{H}}(df) +\delta_\mathcal{H} T^* (df),$
which completes the proof $\square$

We also have the following Bochner’s type identity.

Theorem: For any $\eta \in \Gamma^\infty(T^* \mathbb M)$,
$\frac{1}{2} \Delta_\mathcal{H} \| \eta \|_{\varepsilon}^2 -\langle \square_\varepsilon \eta , \eta \rangle_{\varepsilon}= \| \nabla_{\mathcal{H}} \eta -\mathfrak{T}^\varepsilon_{\mathcal{H}} \eta \|_{\varepsilon}^2 + \left\langle \mathfrak{Ric}_{\mathcal{H}} (\eta), \eta \right\rangle_\mathcal{H} -\left \langle \delta_\mathcal{H} T (\eta) , \eta \right\rangle_\mathcal{V} +\frac{1}{\varepsilon} \langle \mathbf{J}^2 (\eta) , \eta \rangle_\mathcal{H}.$

We now turn to probabilistic applications.

We denote by $(X_t)_{t\geq 0}$ the horizontal Brownian motion on $\mathbb M$. The lifetime of the process is denoted by $\mathbf{e}$. We assume that the metric $g$ is complete and $\mathcal{H}$ is bracket generating. As a consequence, one can define $(P_t)_{t \ge 0}$ the heat semigroup associated to $(X_t)_{t\geq 0}$ as being the semigroup generated by the self-adjoint extension of $\frac{1}{2} \Delta_\mathcal{H}$.

We define a process $\tau^\varepsilon_t:T_{X_t}^*\mathbb{M}\rightarrow T^*_{X_0}\mathbb{M}$ by the formula
$\tau^{\varepsilon}_t=\mathcal{M}_{t}^{\varepsilon}\Theta_{t}^{\varepsilon}, \quad t < \mathbf{e}$
where the process $\Theta_t^{\varepsilon}: T_{X_t}^{*}\mathbb{M}\rightarrow T_{X_0}^{*}\mathbb{M}$ is the stochastic parallel transport with respect to the connection $\nabla -\mathfrak{T}^\varepsilon$ along the paths of $(X_t)_{t\geq 0}$. The multiplicative functional $(\mathcal{M}_t^{\varepsilon})_{t\geq 0}$ is defined as the solution of the following ordinary differential equation

$\frac{d\mathcal{M}_t^{\varepsilon}}{dt}=-\frac{1}{2}\mathcal{M}_t^{\varepsilon}\Theta_t^{\varepsilon}\left(\frac{1}{\varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+\mathfrak{Ric}_{\mathcal{H}} \right)(\Theta_t^{\varepsilon})^{-1}, ~~\mathcal{M}_0^{\varepsilon}=\mathbf{Id}.$

Observe that the process $\tau^\varepsilon_t:T_{X_t}^*\mathbb{M}\rightarrow T^*_{X_0}\mathbb{M}$ is a solution of the following covariant Stratonovitch stochastic differential equation:

$d[\tau^\varepsilon_t \alpha(X_t)]=\tau^\varepsilon_t\left( \nabla_{\circ dX_t}-\mathfrak{T}_{\circ dX_t}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right)dt\right) \alpha(X_t),~~\tau_0=\mathbf{Id},$
where $\alpha$ is any smooth one-form.

From Gronwall’s lemma and the fact that $\Theta_t^{\varepsilon}$ is an isometry, we easily deduce that

Lemma: Let $\varepsilon >0$. Assume that there exists a constant $C_\varepsilon \ge 0$ such that for every $\alpha \in \Gamma^\infty (T^*\mathbb{M})$,

$\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon$
Then, there exists a constant $\tilde{C}_\varepsilon \ge 0$, such that for every $t \ge 0$,

$\| \tau^\varepsilon_t \alpha(X_t) \|_\varepsilon \le e^{ \tilde{C}_\varepsilon t }\| \alpha(X_t) \|_\varepsilon$

For $x \in \mathbb M$, as usual we will denote

$\mathbb{P}_x=\mathbb{P} \left( \cdot \mid X_0=x \right).$

Theorem: Assume that there exists a constant $C_\varepsilon \ge 0$ such that for every $\alpha \in \Gamma^\infty (T^*\mathbb{M})$,

$\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon$
Let $\eta$ be a one-form on $\mathbb{M}$ which is smooth and compactly supported. The unique solution in $L^2$ of the Cauchy problem:

$\begin{cases} \phi (0,x)=\eta (x) \\ \frac{\partial \phi}{\partial t} =\frac{1}{2} \square_\varepsilon \phi \end{cases}$
is given by

$\phi(t,x)=\mathbb{E}_x \left( \tau^\varepsilon_t \eta (X_t) 1_{t <\mathbf{e}}\right) .$

Proof:

This is Feynman-Kac formula.
Sketch of the proof.

It is proved in this course, that the operator

$-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)$

is essentially self-adjoint on the space of smooth and compactly supported one-forms. Thus, from the assumption, $\frac{1}{2} \square_\varepsilon$ is the generator of a bounded semigroup $Q_t^\varepsilon$ in $L^2$ that uniquely solves the above Cauchy problem.

From the Bochner’s identity, one has

$\frac{1}{2} \Delta_\mathcal{H} \| \eta \|_{\varepsilon}^2 -\langle \square_\varepsilon \eta , \eta \rangle_{\varepsilon} \ge -C_\varepsilon \| \eta \|^2_\varepsilon$.

From Shigekawa (L^p contraction for vector valued semigroups), this implies the a priori pointwise bound

$\| Q_t^\varepsilon \eta \|^2_\varepsilon \le e^{2C_\varepsilon t} P_t (\| \eta\|^2_\varepsilon)$.

We now claim that the process

$N_s=\tau^\varepsilon_s(Q_{T-s}^\varepsilon \eta) (X_s)1_{T <\mathbf{e}}, \quad 0 \le s \le T$,

is a local martingale. Indeed, from Ito’s formula and the definition of $\tau^\varepsilon$, we have

$dN_s=\tau^\varepsilon_s \left( \nabla_{\circ dX_s}-\mathfrak{T}_{\circ dX_s}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+\mathfrak{Ric}_{\mathcal{H}}\right)ds\right) (Q_{T-s}^\varepsilon \eta) (X_s)$
$+\tau^\varepsilon_s \frac{d}{ds} (Q_{T-s}^\varepsilon \eta) (X_s) ds.$
We now  conclude from the fact that the bounded variation part of

$\tau^\varepsilon_s \left( \nabla_{\circ dX_s}-\mathfrak{T}_{\circ dX_s}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right)ds\right)(Q_{T-s}^\varepsilon \eta) (X_s)$

is given by $\frac{1}{2} \tau^\varepsilon_s \square_\varepsilon (Q_{T-s}^\varepsilon \eta) (X_s) (X_s) ds$.

Form the previous estimates, we conclude that $N$ is a martingale $\square$

Corollary:  Let $\varepsilon >0$. Assume that there exists a constant $C_\varepsilon \ge 0$ such that for every $\alpha \in \Gamma^\infty (T^*\mathbb{M})$,

$\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon$

Then, for $f \in C^\infty_0(\mathbb M)$, and $t \ge 0$

$dP_tf (x)=\mathbb{E}_x ( \tau^\varepsilon_t df (X_t) 1_{t <\mathbf{e}})$
As a consequence, $\mathbb{P}_x(\mathbf{e}=+\infty)=1$.

Proof: Let $\phi(t,x)=dP_t f(x)$.
We have

$\frac{\partial \phi}{\partial t} =d\Delta_{\mathcal{H}} P_t f= \frac{1}{2} \square_\varepsilon \phi.$
Thus, from the previous theorem

$dP_tf (x)=\mathbb{E}_x ( \tau^\varepsilon_t df (X_t) 1_{t <\mathbf{e}})$
This representation implies the bound

$\| dP_t f (x) \|_\varepsilon \le e^{\frac{C_\varepsilon}{2} t} (P_t \| df \|_\varepsilon) (x).$

It is well-known that this type of gradient bound implies the stochastic completeness of $P_t$. More precisely, we can adapt an argument of Bakry. Let $f,g \in C^\infty_0(\mathbb M)$, we have

$\int_{\mathbb M} (P_t f -f) g d\mu = \int_0^t \int_{\mathbb M}\left( \frac{\partial}{\partial s} P_s f \right) g d\mu ds$
$=\frac{1}{2} \int_0^t \int_{\mathbb M}\left(\Delta_{\mathcal{H}} P_s f \right) g d\mu ds$
$- \int_0^t \int_{\mathbb M} \langle \nabla P_s f , \nabla g\rangle_\mathcal{H} d\mu ds.$

By means of Cauchy-Schwarz inequality we
find
$\left| \int_{\mathbb M} (P_t f -f) g d\mu \right| \le 2 \left(\int_0^t e^{\frac{C_\varepsilon}{2} s} ds\right) \| df \|_{\varepsilon,\infty} \int_{\mathbb M}\| \nabla_\mathcal{H} g \|^{\frac{1}{2}}d\mu.$

We now apply the previous inequality with $f = h_n$, where $h_n$ is an increasing sequence in $C_0^\infty(\mathbb M)$, $0 \le h_n \le 1$, such that $h_n\nearrow 1$ on $\mathbb{M}$, and $||\Gamma (h_n)||_{\infty} \to 0$, as $n\to \infty$.

By monotone convergence theorem we have $P_t h_k(x)\nearrow P_t 1(x)$ for every $x\in \mathbb M$. We conclude that the
left-hand side converges to $\int_{\mathbb M} (P_t 1 -1) g d\mu$. Since the right-hand side converges to zero, we reach the conclusion

$\int_{\mathbb M} (P_t 1 -1) g d\mu=0,\ \ \ g\in C^\infty_0(\mathbb M).$
Since it is true for every $g\in C^\infty_0(\mathbb M)$, it follows that $P_t 1 =1$.

$\square$

## Lecture 5. Riemannian foliations and horizontal Brownian motion

In many interesting cases, we do not actually have a globally defined Riemannian sumersion but a Riemannian foliation.

Definition:  Let $\mathbb M$ be a smooth and connected $n+m$ dimensional manifold. A $m$-dimensional foliation $\mathcal{F}$ on $\mathbb M$ is defined by a maximal collection of pairs $\{ (U_\alpha, \pi_\alpha), \alpha \in I \}$ of open subsets $U_\alpha$ of $\mathbb M$ and submersions $\pi_\alpha: U_\alpha \to U_\alpha^0$ onto open subsets of $\mathbb{R}^n$ satisfying:

•  $\cup_{\alpha \in I} U_\alpha =\mathbb M$;
•  If $U_\alpha \cap U_\beta \neq \emptyset$, there exists a local diffeomorphism $\Psi_{\alpha \beta}$ of $\mathbb{R}^n$ such that $\pi_\alpha=\Psi_{\alpha \beta} \pi_\beta$ on $U_\alpha \cap U_\beta$.

The maps $\pi_\alpha$ are called disintegrating maps of $\mathcal{F}$. The connected components of the sets $\pi_\alpha^{-1}(c)$, $c \in \mathbb{R}^n$, are called the plaques of the foliation. A foliation arises from an integrable sub-bundle of $T\mathbb M$, to be denoted by $\mathcal{V}$ and referred to as the vertical distribution. These are the vectors tangent to the leaves, the maximal integral sub-manifolds of $\mathcal{V}$.

Foliations have been extensively studied and numerous books are devoted to them. We refer in particular to the book by Tondeur.

In the sequel, we shall only be interested in Riemannian foliations with bundle like metric.

Definition: Let $\mathbb M$ be a smooth and connected $n+m$ dimensional Riemannian manifold. A $m$-dimensional foliation $\mathcal{F}$ on $\mathbb M$ is said to be Riemannian with a bundle like metric if the disintegrating maps $\pi_\alpha$ are Riemannian submersions onto $U_\alpha^0$ with its given Riemannian structure. If moreover the leaves are totally geodesic sub-manifolds of $\mathbb M$, then we say that the Riemannian foliation is totally geodesic with a bundle like metric.

Observe that if we have a Riemannian submersion $\pi : (\mathbb M,g) \to (\mathbb{B},j)$, then $\mathbb M$ is equipped with a Riemannian foliation with bundle like metric whose leaves are the fibers of the submersion. Of course, there are many Riemannian foliations with bundle like metric that do not come from a Riemannian submersion.

Example: (Contact manifolds) Let $(\mathbb M,\theta)$ be a $2n+1$-dimensional smooth contact manifold. On $\mathbb M$ there is a unique smooth vector field $T$, the so-called Reeb vector field, that satisfies

$\theta(T)=1,\quad \mathcal{L}_T(\theta)=0,$
where $\mathcal{L}_T$ denotes the Lie derivative with respect to $T$. On $\mathbb M$ there is a foliation, the Reeb foliation, whose leaves are the orbits of the vector field $T$. As it is well-known, it is always possible to find a Riemannian metric $g$ and a $(1,1)$-tensor field $J$ on $\mathbb M$ so that for every vector fields $X, Y$

$g(X,T)=\theta(X),\quad J^2(X)=-X+\theta (X) T, \quad g(X,JY)=(d\theta)(X,Y).$
The triple $(\mathbb M, \theta,g)$ is called a contact Riemannian manifold. We see then that the Reeb foliation is totally geodesic with bundle like metric if and only if the Reeb vector field $T$ is a Killing field, that is,

$\mathcal{L}_T g=0.$
In that case $(\mathbb M, \theta,g)$ is called a K-contact Riemannian manifold.

Since Riemannian foliations with a bundle like metric can locally be desribed by a Riemannian submersion, we can define a horizontal Laplacian $\Delta_{\mathcal H}$. This allows to define horizontal Brownian motions.
Let $\mathbb M$ be a smooth and connected manifold with dimension $n+m$. In the sequel, we assume that $\mathbb M$ is equipped with a Riemannian foliation $\mathcal{F}$ with bundle-like metric $g$ and totally geodesic $m$-dimensional leaves.
The sub-bundle $\mathcal{V}$ formed by vectors tangent to the leaves is referred to as the set of  vertical directions. The sub-bundle $\mathcal{H}$ which is normal to $\mathcal{V}$ is referred to as the set of horizontal directions. The metric $g$ can be split as

$g=g_\mathcal{H} \oplus g_{\mathcal{V}}.$
On the Riemannian manifold $(\mathbb M,g)$ there is the Levi-Civita connection that we denote by $D$, but this connection is not adapted to the study of follations because the horizontal and the vertical bundle may not be parallel. More adapted to the geometry of the foliation is the Bott’s connection that we now define. It is an easy exercise to check that, since the foliation is totally geodesic, there exists a unique affine connection $\nabla$ such that:

• $\nabla$ is metric, that is, $\nabla g =0$;
• For $X,Y \in \Gamma^\infty(\mathcal H)$, $\nabla_X Y \in \Gamma^\infty(\mathcal H)$;
• For $U,V \in \Gamma^\infty(\mathcal V)$, $\nabla_U V \in \Gamma^\infty(\mathcal V)$
•  For $X,Y \in \Gamma^\infty(\mathcal H)$, $T(X,Y) \in \Gamma^\infty(\mathcal V)$ and for $U,V \in \Gamma^\infty(\mathcal V)$, $T(U,V) \in \Gamma^\infty(\mathcal H)$, where $T$ denotes the torsion tensor of $\nabla$
• For $X \in \Gamma^\infty(\mathcal H), U \in \Gamma^\infty(\mathcal V)$, $T(X,U)=0$.

In terms of the Levi-Civita connection, the Bott connection writes

$\nabla_X Y = \begin{cases} ( D_X Y)_{\mathcal{H}} , \quad X,Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{H}}, \quad X \in \Gamma^\infty(\mathcal{V}), Y \in \Gamma^\infty(\mathcal{H}) \\ [X,Y]_{\mathcal{V}}, \quad X \in \Gamma^\infty(\mathcal{H}), Y \in \Gamma^\infty(\mathcal{V}) \\ ( D_X Y)_{\mathcal{V}}, \quad X,Y \in \Gamma^\infty(\mathcal{V}) \end{cases}$
Observe that for horizontal vector fields $X,Y$ the torsion $T(X,Y)$ is given by

$T(X,Y)=-[X,Y]_\mathcal{V}.$
Also observe that for $X,Y \in \Gamma^\infty(\mathcal{V})$ we actually have $( D_X Y)_{\mathcal{V}}= D_X Y$ because the leaves are assumed to be totally geodesic.

For local computations, it is convenient to work in normal frames.

Lemma: [B., Kim, Wang 2016]  Let $x \in \mathbb M$. Around $x$, there exist a local orthonormal horizontal frame $\{X_1,\cdots,X_n \}$ and a local orthonormal vertical frame $\{Z_1,\cdots,Z_m \}$ such that the following structure relations hold

$[X_i,X_j]=\sum_{k=1}^n \omega_{ij}^k X_k +\sum_{k=1}^m \gamma_{ij}^k Z_k$
$[X_i,Z_k]=\sum_{j=1}^m \beta_{ik}^j Z_j,$
where $\omega_{ij}^k, \gamma_{ij}^k, \beta_{ik}^j$ are smooth functions such that:

$\beta_{ik}^j=- \beta_{ij}^k.$
Moreover, at $x$, we have
$\omega_{ij}^k=0, \beta_{ij}^k=0.$

For later use, we record the fact that in this frame the Christofell symbols of the Bott connection are given by
$\begin{cases} \nabla_{X_i} X_j =\frac{1}{2} \sum_{k=1}^n \left( \omega_{ij}^k +\omega_{ki}^j+\omega_{kj}^i\right)X_k \\ \nabla_{Z_j} X_i =0 \\ \nabla_{X_i} Z_j=\sum_{k=1}^m \beta_{ij}^{k} Z_k \end{cases}$

Also observe that, since the foliation is totally geodesic, the horizontal Laplacian is locally given by

$\Delta_\mathcal{H}=\sum_{i=1}^n X_i^2 -\nabla_{X_i} X_i$

A horizontal orthonormal map at $x \in \mathbb M$ is an isometry $u: \mathbb{R}^n \to \mathcal{H}_x$. The horizontal orthonormal map bundle will be denoted by $\mathcal{O}_\mathcal{H} (\mathbb M)$.
The Bott connection allows to lift vector fields on $\mathbb M$ into vector fields on $\mathcal{O}_\mathcal{H} (\mathbb M)$. Let $e_1,\cdots,e_n$ be the canonical basis of $\mathbb{R}^n$. We denote by $A_i$ the vector field on $\mathcal{O}_\mathcal{H} (\mathbb M)$ such that $A_i (x,u)$ is the lift of $u(e_i) \in \mathcal{H}_x$.

We can locally write the vector fields $A_i$‘s in terms of the normal frames constructed in the previous subsection. We consider $x \in \mathbb M$ and a normal horizontal orthonormal frame $X_1,\cdots,X_n$ around $x$ as in the previous section.

If $u: \mathbb{R}^n \to \mathcal{H}_y$ is an isometry, we can find an orthogonal matrix $e_i^j$ such that $u(e_i)=\sum_{j=1}^n e_i^j X_j$. It is then easy to prove that

$A_i=\sum_{j=1}^n e_i^j \bar{X}_j-\sum_{j,k,l,m=1}^n e_i^j e_m^l \Gamma_{jl}^k \frac{\partial}{\partial e_m^k}$
where the $\Gamma_{jl}^k$‘s are the Christoffels symbols of the Bott connection and $\bar{X}_j$ is the vector field on $\mathcal{O}_\mathcal{H} (\mathbb M)$ defined by

$\bar{X}_j f (y,u)=\lim_{t \to 0} \frac{ f( e^{tX_j}(y), u)-f(y,u)}{t}$
In particular, at the center $x$ of the frame we have,

$A_i=\sum_{j=1}^n e_i^j \bar{X}_j.$

The main result is the following.

Proposition:  Let $\pi: \mathcal{O}_\mathcal{H} (\mathbb M) \to \mathbb M$ be the bundle projection map. For a smooth $f:\mathbb M \to \mathbb{R}$,
$\left( \sum_{i=1}^n A_i^2 \right)(f\circ \pi)=\Delta_\mathcal{H} f \circ \pi.$

Proof: It is enough to prove this identity at the center of the frame $x$. Using the fact that $\Gamma_{jl}^k(x)=0$, we see that, at $x$,

$\sum_{i=1}^n A_i^2 =\sum_{j=1}^n \bar{X}_j^2.$
The conclusion follows then easily $\square$

As a corollary, we obtain:

Corollary: Let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion and let $(U_t)_{t \ge 0}$ be a solution of the stochastic differential equation

$dU_t =\sum_{i=1}^n A_i (U_t) \circ dB^i_t, \quad t < \mathbf{e}$
then $X_t=\pi(U_t)$ is a horizontal Brownian motion on $\mathbb M$, that is a diffusion process with generator $\frac{1}{2}\Delta_\mathcal{H}$.

## Lecture 4. Horizontal Brownian motions on bundles and Hopf fibrations

Let us now turn to some examples of some horizontal Brownian motions associated with submersions.

We come back first to an example studied earlier that encompasses the Heisenberg group. Let
$\alpha=\sum_{i=1}^n \alpha^i(x)dx_i$
be a smooth one-form on $\mathbb{R}^n$ and let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion. The process

$X_t=\left(B_t, \int_{B[0,t]} \alpha \right)$
is a diffusion process in $\mathbb{R}^n \times \mathbb{R}$ with generator

$L=\frac{1}{2} \sum_{i=1} X_i^2,$
where
$X_i=\frac{\partial}{\partial x_i} +\alpha^i(x) \frac{\partial}{\partial z} .$
We can then interpret $(X_t)_{t \ge 0}$ as a horizontal Brownian motion. Indeed, consider the Riemannian metric $g$ on $\mathbb{R}^n \times \mathbb{R}$ that makes $X_1,\cdots, X_n , \frac{\partial}{\partial z}$ orthonormal. The map

$\pi : (\mathbb{R}^n \times \mathbb{R}, g) \to (\mathbb{R}^n, \mathbf{eucl.})$
such that $\pi (x,z)=x$ is then a Riemannian submersion and $\sum_{i=1} X_i^2$ is the horizontal Laplacian of this submersion. Therefore $(X_t)_{t \ge 0}$ is a horizontal Brownian motion for this submersion.

A second class of examples that naturally arise in stochastic calculus are horizontal Brownian motions on vector bundles. We present here the case of the tangent bundle, but the construction may be extended to any vector bundle. Let $(\mathbb{M},g)$ be a smooth and connected Riemannian manifold with dimension $n$. Let $(B_t)_{t \ge 0}$ be a Brownian motion on $\mathbb M$ started at $x$. Let $\tau_{0,t} : T_x \mathbb M \to T_{B_t} \mathbb{M}$ be the stochastic parallel transport along the paths of $(B_t)_{t \ge 0}$. Let now $v \in T_x \mathbb M$ and consider the tangent bundle $T\mathbb M$ valued process:

$X_t=\left( B_t , \tau_{0,t} v \right).$
The process $(X_t)_{t \ge 0}$ can be interpreted as a horizontal Brownian for some Riemannian submersion. The submersion is simply the bundle projection map $\pi : T\mathbb{M} \to \mathbb{M}$. One then needs to construct a Riemannian metric on $T\mathbb{M}$ that makes $\pi$ a Riemannian submersion. Call a $C^1$ curve $\gamma(t)=(x(t),v(t))$ to be horizontal if $v(t)$ is parallelly transported along $x$. This uniquely determines the rank $n$ horizontal bundle in $TT\mathbb{M}$. Now, if $X$ is a vector field on $\mathbb{M}$, define its horizontal lift $X^h$ as the unique horizontal vector field on $TT\mathbb{M}$ that projects onto $X$. Define its vertical lift as the unique vertical vector field $X^v$ on $TT\mathbb{M}$ such that for every smooth $f:\mathbb{M} \to \mathbb{R}$ one has

$X^v (f^*)=Xf,$
where $f^*(x,v)=df_x (v)$. The Sasaki metric $g_S$ on $T\mathbb{M}$ is then the unique metric such that if $X_1,\cdots, X_n$ is a local orthonormal frame on $\mathbb{M}$, then $X^h_1,\cdots, X^h_n,X_1^v,\cdots,X_n^v$ is a local orthonormal frame on $T\mathbb{M}$. It is then easy to check that $\pi$ is then a Riemannian submersion with totally geodesic fibers and that $(X_t)_{t \ge 0}$ is a horizontal Brownian motion for this submersion.

A similar construction works in the orthonormal frame bundle. Let $(\mathbb{M},g)$ be a smooth and connected Riemannian manifold with dimension $n$ and let $(U_t)_{t \ge 0}$ be the horizontal Brownian motion on the orthonormal frame bundle $O(\mathbb{M})$, that is $(U_t)_{t \ge 0}$ solves the stochastic differential equation

$dU_t=\sum_{i=1}^n H_i(U_t) \circ dB^i_t,$
where $H_1,\cdots,H_n$ are the fundamental horizontal vector fields. Then, similarly as before, one can easily interpret $(U_t)_{t \ge 0}$ as the horizontal Brownian motion of a Riemannian submersion.

The most general construction on bundles is the following. Let $\mathbb{M}$ be a principal bundle over $\mathbb B$ with fiber $\mathbf F$ and structure group $\mathbb G$. Then, given a Riemannian metric $j$ on $\mathbb B$, a $\mathbb G$-invariant metric $k$ on $\mathbf F$ and a $\mathbb G$ connection form $\omega$, there exists a unique Riemannian metric $g$ on $\mathbb M$ such that the bundle projection map $\pi: \mathbb M \to \mathbb B$ is a Riemannian submersion with totally geodesic fibers isometric to $(\mathbf{F},k)$ and such that the horizontal distribution of $\omega$ is the orthogonal complement of the vertical distribution.

We finish the lecture with two canonical examples of horizontal Brownian motions which are related to the Hopf fibrations.

The complex projective space $\mathbb{CP}^n$ can be defined as the set of complex lines in $\mathbb{C}^{n+1}$. To parametrize points in $\mathbb{CP}^n$, it is convenient to use the local inhomogeneous coordinates given by $w_j=z_j/z_{n+1}$, $1 \le j \le n$, $z \in \mathbb{C}^{n+1}$, $z_{n+1}\neq 0$. In these coordinates, the Riemannian structure of $\mathbb{CP}^n$ is easily worked out from the standard Riemannian structure of the Euclidean sphere. Indeed, if we consider the unit sphere

$\mathbb S^{2n+1}=\lbrace z=(z_1,\cdots,z_{n+1})\in \mathbb{C}^{n+1}, \| z \| =1\rbrace,$
then, at each point, the differential of the map $\mathbb S^{2n+1} -\{z_{n+1}=0 \} \to \mathbb{CP}^n$, $(z_1,\cdots,z_{n+1}) \to (z_1/z_{n+1},\cdots,z_n/z_{n+1})$ is an isometry between the orthogonal space of its kernel and the corresponding tangent space to $\mathbb{CP}^n$. This map actually is the local description of a globally defined Riemannian submersion $\mathbb S^{2n+1} \to \mathbb{CP}^n$, that can be constructed as follows. There is an isometric group action of $\mathbb{S}^1=\mathbf{U}(1)$ on $\mathbb S^{2n+1}$ which is defined by

$e^{i\theta}\cdot(z_1,\cdots, z_n) = (e^{i\theta} z_1,\cdots, e^{i\theta} z_n).$

The quotient space $\mathbb S^{2n+1} / \mathbf{U}(1)$ can be identified with $\mathbb{CP}^n$ and the projection map $\pi : \mathbb S^{2n+1} \to \mathbb{CP}^n$ is a Riemannian submersion with totally geodesic fibers isometric to $\mathbf{U}(1)$. The fibration

$\mathbf{U}(1) \to \mathbb S^{2n+1} \to \mathbb{CP}^n$
is called the Hopf fibration.

The submersion $\pi$ allows to construct the Brownian motion on $\mathbb{CP}^n$ from the Brownian motion on $\mathbb S^{2n+1}$. Indeed, let $(z(t))_{t \ge 0}$ be a Brownian motion on $\mathbb S^{2n+1}$ started at the north pole. Since $\mathbb{P}( \exists t \ge 0, z_{n+1}(t)=0 )=0$, one can use the local description of the submersion $\pi$ to infer that

$w(t)= \left( \frac{z_1(t)}{z_{n+1}(t)} , \cdots, \frac{z_n(t)}{z_{n+1}(t)}\right), \quad t \ge 0,$
is a Brownian motion on $\mathbb{CP}^n$.

Consider now the one-form $\alpha$ on $\mathbb{CP}^n$ which is the pushforward by $\pi$ of the standard contact form of $\mathbb S^{2n+1}$. In local inhomogeneous coordinates, we have

$\alpha=\frac{i}{2(1+|w|^2)}\sum_{j=1}^n(w_jd\overline{w_j}-\overline{w_j}dw_j)$
where $|w|^2=\sum_{j=1}^n |w_j|^2$. It is easy to compute that

$d\alpha=\frac{i}{(1+|w|^2)^2}\left((1+|w|^2)\sum_{j=1}^ndw_j\wedge d\overline{w}_j-\sum_{j,k=1}^n\overline{w}_jw_k dw_j\wedge d\overline{w}_k \right).$
Thus $d\alpha$ is almost everywhere the Kahler form that induces the standard Fubini-Study metric on $\mathbb{CP}^n$. The following definition is therefore natural:

Definition: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CP}^n$ started at 0. The generalized stochastic area process of $(w(t))_{t \ge 0}$ is defined by
$\theta(t)=\int_{w[0,t]} \alpha=\frac{i}{2}\sum_{j=1}^n \int_0^t \frac{w_j(s) d\overline{w_j}(s)-\overline{w_j}(s) dw_j(s)}{1+|w(s)|^2},$
where the above stochastic integrals are understood in the Stratonovitch, or equivalently in the Ito sense.

We have then the following representation for the horizontal Brownian motion of the submersion $\pi$.

Theorem: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CP}^n$ started at 0 and $(\theta(t))_{t\ge 0}$ be its stochastic area process. The $\mathbb{S}^{2n+1}$-valued diffusion process
$X_t=\frac{e^{-i\theta(t)} }{\sqrt{1+|w(t)|^2}} \left( w(t),1\right), \quad t \ge 0$
is a horizontal Brownian motion for the submersion $\pi$.

A similar construction works on the complex hyperbolic space. As a set, the complex hyperbolic space $\mathbb{CH}^n$ can be defined as the open unit ball in $\mathbb{C}^n$. Its Riemannian structure can be constructed as follows. Let

$\mathbb H^{2n+1}=\{ z \in \mathbb{C}^{n+1}, | z_1|^2+\cdots+|z_n|^2 -|z_{n+1}|^2=-1 \}$
be the $2n+1$ dimensional anti-de Sitter space. We endow $\mathbb H^{2n+1}$ with its standard Lorentz metric with signature $(2n,1)$. The Riemannian structure on $\mathbb{CH}^n$ is then such that the map
$\begin{array}{llll} \pi :& \mathbb H^{2n+1} & \to & \mathbb{CH}^n \\ & (z_1,\cdots,z_{n+1}) & \to & \left( \frac{z_1}{z_{n+1}}, \cdots, \frac{z_n}{z_{n+1}}\right) \end{array}$
is an indefinite Riemannian submersion whose one-dimensional fibers are definite negative. This submersion is associated with a fibration. Indeed, the group $\mathbf{U}(1)$ acts isometrically on $\mathbb H^{2n+1}$, and the quotient space of $\mathbb H^{2n+1}$ by this action is isometric to $\mathbb{CH}^n$. The fibration

$\mathbf{U}(1)\to\mathbb H^{2n+1}\to\mathbb{CH}^n$
is called the anti-de Sitter fibration.

To parametrize $\mathbb{CH}^n$, we will use the global inhomogeneous coordinates given by $w_j=z_j/z_{n+1}$ where $(z_1,\dots, z_n)\in M$ with $M=\{z\in \mathbb{C}^{n,1}, \sum_{k=1}^n|z_{k}|^2-|z_{n+1}|^2<0 \}$. Let $\alpha$ be the one-form on $\mathbb{CH}^n$ which is the push-forward by the submersion $\pi$ of the standard contact form on $\mathbb H^{2n+1}$. In inhomogeneous coordinates, we compute

$\alpha=\frac{i}{2(1-|w|^2)}\sum_{j=1}^n(w_jd\overline{w_j}-\overline{w_j}dw_j),$
where $|w|^2=\sum_{j=1}^n|w_j|^2<1$. A simple computation yields

$d\alpha=\frac{i}{(1-|w|^2)^2}\left((1-|w|^2)\sum_{j=1}^ndw_j\wedge d\overline{w}_j-\sum_{j,k=1}^n\overline{w}_jw_k dw_j\wedge d\overline{w}_k \right).$
Thus $d\alpha$ is exactly the Kahler form which induces the standard Bergman metric on $\mathbb{CH}^n$. We can then naturally define the stochastic area process on $\mathbb{CH}^n$ as follows:
Definition:  Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CH}^n$ started at 0. The generalized stochastic area process of $(w(t))_{t \ge 0}$ is defined by

$\theta(t)=\int_{w[0,t]} \alpha=\frac{i}{2}\sum_{j=1}^n \int_0^t \frac{w_j(s) d\overline{w_j}(s)-\overline{w_j}(s) dw_j(s)}{1-|w(s)|^2},$
where the above stochastic integrals are understood in the Stratonovitch sense or equivalently Ito sense.

As in in the Heisenberg group case or the Hopf fibration case, the stochastic area process is intimately related to the horizontal Brownian motion on the total space of the fibration.

Theorem: Let $(w(t))_{t \ge 0}$ be a Brownian motion on $\mathbb{CH}^n$ started at 0 and $(\theta(t))_{t\ge 0}$ be its stochastic area process. The $\mathbb H^{2n+1}$-valued diffusion process

$Y_t=\frac{e^{i\theta_t} }{\sqrt{1-|w(t)|^2}} \left( w(t),1\right), \quad t \ge 0$
is the horizontal lift at $(0,1)$ of $(w(t))_{t \ge 0}$ by the submersion $\pi$.

## Lecture 3. Horizontal Brownian motions and submersions

From now on, we will assume knowledge of some basic Riemannian geometry.

We start by reminding the definition of Brownian motions on Riemannian manifolds. Let $(\mathbb{M},g)$ be a smooth and connected Riemannian manifold. In a local orthonormal frame $X_1,\cdots,X_n$, one can compute the length of the gradient of a smooth function $f$:

$\| \nabla f \|^2=\sum_{i=1}^n (X_i f)^2.$
Let us denote by $\mu$ the Riemannian volume measure. We can then consider the pre-Dirichlet form

$\mathcal{E}(f,g)=\int \langle \nabla f , \nabla g \rangle d\mu, \quad f,g \in C_0^\infty(\mathbb{M})$
There exists a unique second order operator $\Delta$ such that for every $f,g \in C_0^\infty(\mathbb{M})$,

$\mathcal{E}(f,g)=-\int f \Delta g d\mu =-\int g \Delta f d\mu.$
The operator $\Delta$ is called the Laplace-Beltrami operator. Locally, we have

$\Delta=\sum_{i=1}^n X_i^2 -D_{X_i} X_i,$
where $D$ denotes the Levi-Civita connection on $\mathbb{M}$.

Definition:  A Brownian motion $(X_t)_{ t \ge 0}$ on $\mathbb{M}$ is a diffusion process with generator $\frac{1}{2} \Delta$, that is for every $f \in C^\infty(\mathbb{M})$,

$f(X_t)-\frac{1}{2} \int_0^t \Delta f(X_s) ds, \quad 0 \le t < \mathbf{e}$
is a local martingale, where $\mathbf{e}$ is the lifetime of $(X_t)_{ t \ge 0}$ on $\mathbb{M}$.

One can construct Brownian motions by using the theory of Dirichlet forms by using the minimal closed extension of $\mathcal{E}$. If the metric $g$ is complete (which we always assume for Riemannian metrics in this course), then one can prove that $\Delta$ is essentially self-adjoint on $C_0^\infty(\mathbb{M})$ and there is a unique closed extension of $\mathcal{E}$. Note that even in the complete case $(X_t)_{t \ge 0}$ may have a finite lifetime. One can equivalently define Brownian motion by solving a stochastic differential equation in the frame bundle.

In a local orthonormal frame $X_1,\cdots,X_n$, we have

$dX_t=\sum_{i=1}^n X_i(X_t) \circ dB^i_t -\frac{1}{2} \sum_{i=1}^nD_{X_i} X_i (X_t) dt,$
where $B^1,...,B^n$ is a Brownian motion in $\mathbb{R}^n$. For further details on Riemannian Brownian motions, we refer to Elton’s book.

We now turn to the notion of horizontal Brownian motion. For this, we need to distinguish a particular set of directions within the tangent spaces. This can be done by using the notion of submersion.

Let $(\mathbb{M} , g)$ and $(\mathbb{B},j)$ be smooth and connected complete Riemannian manifolds.
Definition:  A smooth surjective map $\pi: (\mathbb M , g)\to (\mathbb B,j)$ is called a Riemannian submersion if its derivative maps $T_x\pi : T_x \mathbb M \to T_{\pi(x)} \mathbb B$ are orthogonal projections, i.e. for every $x \in \mathbb M$, the map $T_{x} \pi (T_{x} \pi)^*: T_{p(x)} \mathbb B \to T_{p(x)} \mathbb B$ is the identity.

Example: (Warped products) Let $(\mathbb M_1 , g_1)$ and $(\mathbb M_2,g_2)$ be Riemannian manifolds and $f$ be a smooth and positive function on $\mathbb M_1$. Then the first projection $(\mathbb M_1 \times \mathbb M_2,g_1 \oplus f g_2) \to (\mathbb M_1, g_1)$ is a Riemannian submersion.

If $\pi$ is a Riemannian submersion and $b \in \mathbb B$, the set $\pi^{-1}(\{ b \})$ is called a fiber.

For $x \in \mathbb M$, $\mathcal{V}_x =\mathbf{Ker} (T_x\pi)$ is called the vertical space at $x$. The orthogonal complement of $\mathcal{H}_x$ shall be denoted $\mathcal{H}_x$ and will be referred to as the horizontal space at $x$. We have an orthogonal decomposition

$T_x \mathbb M=\mathcal{H}_x \oplus \mathcal{V}_x$
and a corresponding splitting of the metric

$g=g_{\mathcal{H}} \oplus g_{\mathcal{V}}.$
The vertical distribution $\mathcal V$ is of course integrable since it is the tangent distribution to the fibers, but the horizontal distribution is in general not integrable. Actually, in almost all the situations we will consider, the horizontal distribution is everywhere bracket-generating in the sense that for every $x \in \mathbb M$, $\mathbf{Lie} (\mathcal{H}) (x)=T_x \mathbb M$.

If $f \in C^\infty(\mathbb M)$ we define its vertical gradient $\nabla_{\mathcal{V}}$ as the projection of its gradient onto the vertical distribution and its horizontal gradient $\nabla_{\mathcal{H}}$ as the projection of the gradient onto the horizontal distribution. We define then the vertical Laplacian $\Delta_{\mathcal{V}}$ as (minus) the generator of the pre-Dirichlet form

$\mathcal{E}_{\mathcal{V}}(f,g)=\int_{\mathbb M} \langle \nabla_{\mathcal{V}} f , \nabla_{\mathcal{V}} g \rangle d\mu, \quad f,g \in C_0^\infty(\mathbb{M}),$
where $\mu$ is the Riemannian volume measure on $\mathbb M$. Similarly, we define the horizontal Laplacian $\Delta_{\mathcal{H}}$ as (minus) the generator of the pre-Dirichlet form

$\mathcal{E}_{\mathcal{H}}(f,g)=-\int_{\mathbb M} \langle \nabla_{\mathcal{H}} f , \nabla_{\mathcal{H}} g \rangle d\mu \quad f,g \in C_0^\infty(\mathbb{M}).$

Definition: A horizontal Brownian motion $(X_t)_{ t \ge 0}$ on $\mathbb{M}$ is a diffusion process with generator $\frac{1}{2} \Delta_{\mathcal{H}}$, that is for every $f \in C^\infty(\mathbb{M})$,

$f(X_t)-\frac{1}{2} \int_0^t \Delta_{\mathcal{H}} f(X_s) ds, \quad 0 \le t < \mathbf{e}$
is a local martingale, where $\mathbf{e}$ is the lifetime of $(X_t)_{ t \ge 0}$ on $\mathbb{M}$.

If $X_1,\cdots,X_n$ is a local orthonormal frame of basic vector fields and $Z_1,\cdots,Z_m$ a local orthonormal frame of the vertical distribution, then we have

$\Delta_{\mathcal{H}}=-\sum_{i=1}^n X_i^* X_i$
and

$\Delta_{\mathcal{V}}=-\sum_{i=1}^m Z_i^* Z_i,$
where the adjoints are (formally) understood in $L^2(\mu)$. Classically, we have

$X_i^*=-X_i+\sum_{k=1}^n \langle D_{X_k} X_k, X_i\rangle +\sum_{k=1}^m \langle D_{Z_k} Z_k, X_i\rangle,$
where $D$ is the Levi-Civita connection. As a consequence, we obtain

$\Delta_{\mathcal{H}}=\sum_{i=1}^n X_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}} -\sum_{i=1}^m (D_{Z_i}Z_i)_{\mathcal{H}},$
where $(\cdot)_{\mathcal{H}}$ denotes the horizontal part of the vector. In a similar way we obviously have

$\Delta_{\mathcal{V}}=\sum_{i=1}^m Z_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{V}} -\sum_{i=1}^m (D_{Z_i}Z_i)_{\mathcal{V}}.$

We note that from Hormander’s theorem, the operator $\Delta_{\mathcal{H}}$ is locally subelliptic if the horizontal distribution is bracket generating. Of course, the vertical Laplacian is never subelliptic because the vertical distribution is always integrable. We have the following theorem.
Proposition: Assume that the horizontal distribution $\mathcal{H}$ is everywhere bracket-generating. The horizontal Laplacian $\Delta_{\mathcal{H}}$ is essentially self-adjoint on the space $C_0^\infty(\mathbb M)$.
In the sequel, we will often assume that the horizontal distribution $\mathcal{H}$ is everywhere bracket-generating.

As in the Riemannian case, one can construct horizontal Brownian motions by globally solving a stochastic differential equation on a frame bundle. The construction will be shown later. However, in many instances, one can construct the horizontal Brownian motion on $\mathbb{M}$ from the Brownian motion on $\mathbb{B}$. It uses the notion of horizontal lift.

A vector field $X \in \Gamma^\infty(T\mathbb M)$ is said to be projectable if there exists a smooth vector field $\overline{X}$ on $\mathbb B$ such that for every $x \in \mathbb M$, $T_x \pi ( X(x))= \overline {X} (\pi (x))$. In that case, we say that $X$ and $\overline{X}$ are $\pi$-related.

Definition: A vector field $X$ on $\mathbb M$ is called basic if it is projectable and horizontal.

If $\overline{X}$ is a smooth vector field on $\mathbb B$, then there exists a unique basic vector field $X$ on $\mathbb M$ which is $\pi$-related to $\overline{X}$. This vector is called the horizontal lift of $\overline{X}$.

A $C^1$-curve $\gamma: [0,+\infty) \to \mathbb M$ is said to be horizontal if for every $t \ge 0$,

$\gamma'(t) \in \mathcal{H}_{\gamma(t)}.$

Definition: Let $\bar{\gamma}: [0,+\infty) \to \mathbb{B}$ be a $C^1$ curve. Let $x \in \mathbb{M}$, such that $\pi(x)=\gamma(0)$. Then, there exists a unique $C^1$ horizontal curve $\gamma: [0,+\infty) \to \mathbb M$ such that $\gamma (0)=x$ and $\pi (\gamma(t))=\gamma(t)$. The curve $\gamma$ is called the horizontal lift of $\bar{\gamma}$ at $x$.

The notion of horizontal lift may easily be extended to Brownian motion paths on $\mathbb{B}$ by using stochastic calculus (or rough paths theory).

Theorem: Assume that the fibers of the submersion $\pi$ have all zero mean curvature. Let $(B_t)_{t \ge 0}$ be a Brownian motion on $\mathbb{B}$ started at $b \in \mathbb{B}$. Let $x \in \mathbb{M}$ such that $\pi (x)=b$. The horizontal lift $(X_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ at $x$ is a horizontal Brownian motion.

Proof: Indeed, if $X_1,\cdots,X_n$ is a local orthonormal frame of basic vector fields and $Z_1,\cdots,Z_m$ a local orthonormal frame of the vertical distribution, let us denote by $\overline{X}_1,\cdots,\overline{X}_n$ the vector fields on $\mathbb B$ which are $\pi$-related to $X_1,\cdots,X_n$ . We have

$\Delta_{\mathbb B}=\sum_{i=1}^n \overline{X}_i^2 -\sum_{i=1}^n D_{\overline{X}_i}\overline{X}_i.$
Therefore, $(B_t)_{t \ge 0}$ locally solves a stochastic differential equation

$dB_t=\sum_{i=1}^n \overline{X}_i(B_t) \circ dB^i_t -\frac{1}{2} \sum_{i=1}^nD_{\overline{X}_i} \overline{X}_i (B_t) dt,$
where $B^1,...,B^n$ is a Brownian motion in $\mathbb{R}^n$. Since it is easy to check that $D_{\overline{X}_i}\overline{X}_i$ is $\pi$-related to $(D_{X_i}X_i)_{\mathcal{H}}$, we deduce that $(X_t)_{t \ge 0}$ locally solves the stochastic differential equation

$dX_t=\sum_{i=1}^n X_i(X_t) \circ dB^i_t -\frac{1}{2} \sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}} (X_t) dt.$
We now recall that

$\Delta_{\mathcal{H}}=\sum_{i=1}^n X_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}} -\sum_{i=1}^m (D_{Z_i}Z_i)_{\mathcal{H}},$

If the fibers of the submersion $\pi$ have all zero mean curvature, the vector

$T=\sum_{i=1}^m D_{Z_i}Z_i$
is always orthogonal to $\mathcal{H}$. Thus

$\Delta_{\mathcal{H}}=\sum_{i=1}^n X_i^2 -\sum_{i=1}^n (D_{X_i}X_i)_{\mathcal{H}} ,$
and $(X_t)_{t \ge 0}$ is a horizontal Brownian motion $\square$

In this course, we shall mainly be interested in submersion with totally geodesic fibers.

Definition: A Riemannian submersion $\pi: (\mathbb M , g)\to (\mathbb B,j)$ is said to be totally geodesic if for every $b \in \mathbb B$, the set $\pi^{-1}(\{ b \})$ is a totally geodesic submanifold of $\mathbb M$.

Observe that for totally geodesic submersions, the mean curvature of the fibers are zero, and thus the horizontal Brownian motion may be constructed as a lift of the Brownian motion on the base space.

## Lecture 2. Horizontal Brownian motion on the Heisenberg group

We now study in more details the geometric structure behind the diffusion underlying the Levy area process
$S_t=\int_0^t B^1_s dB^2_s-B^2_s dB^1_s,$
where $B_t=(B^1_t,B^2_t)$, $t \ge 0$, is a two dimensional Brownian motion started at 0. Let us recall that if we consider the 3-dimensional process

$X_t=(B^1_t,B^2_t,S_t),$
then $X_t$ is a Markov process with generator

$L=\frac{1}{2}( X^2+Y^2)$
where $X,Y$ are the following vector fields

$X=\frac{\partial}{\partial x}-y \frac{\partial}{\partial z}$

$Y=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z}.$
Denote $Z$ the vector field

$Z=\frac{\partial}{\partial z}$
We have then Lie brackets commutation relations
$[X,Y]=2Z, [X,Z]=[Y,Z]=0.$
As a consequence, $X,Y,Z$ generate a 3-dimensional nilpotent Lie algebra of (complete) vector fields. This is the Lie algebra of the Heisenberg group

$\mathbb{H}^3=\{ (x,y,z) \in \mathbb{R}^3 \}$
where the non-commutative group law is given by
$(x_1,y_1,z_1) \star (x_2,y_2,z_2) =(x_1+x_2,y_2+y_2,z_1+z_2+x_1y_2-x_2y_1).$

For $s \le t$, one has

$X_s^{-1} \star X_t =(-B^1_s,-B^2_s,-S_s) \star (B^1_t,B^2_t,S_t) =(B^1_t-B^1_s,B^2_t-B^2_s,S_t-S_s-B^1_sB^2_t+B^2_sB^1_t)$
Observe now that

$S_t-S_s-B^1_sB^2_t+B^2_sB^1_t=\int_s^t (B^1_u-B^1_s)dB^2_u-(B^2_u-B^2_s)dB^1_u.$
Therefore, $X_s^{-1} \star X_t$ is independent from $\sigma(X_u, u \le s)$ and distributed as $X_{t-s}$. It is therefore natural to call $(X_t)_{t\ge 0}$ a Brownian motion in the Heisenberg group. Observe that $X,Y,Z$ form a basis of the Lie algebra, but that the generator of  $X_t$ only involves $X$ and $Y$. Thus, the direction $Z$ is missing. Calling $\mathbf{span}(X,Y)$ the set of horizontal directions, we then refer to $X_t$ as a horizontal Brownian motion.

This construction is easily extended to higher dimensions.

Let $Z_t =B_t +i \beta_t$ be a Brownian motion in $\mathbb{C}^n$ started at 0. This means that $B$ and $\beta$ are two independent Brownian motions in $\mathbb{R}^n$. We can then consider the one-form

$\alpha=\sum_{i=1}^n x_i dy_i -y_i dx_i$
and the generalized Levy area

$S_t =\int_{Z[0,t]} \alpha =\sum_{i=1}^n \int_0^t B^i_s d\beta^i_s -\beta^i_sdB_s^i$
The process

$X_t=(Z_t,S_t)$
is then a diffusion process in $\mathbb{C}^n \times \mathbb{R}$ with generator

$L=\frac{1}{2} \sum_{i=1}^n X_i^2+Y_i^2$
where

$X_i=\frac{\partial}{\partial x_i} -y_i \frac{\partial}{\partial z}$
$Y_i=\frac{\partial}{\partial y_i} +x_i \frac{\partial}{\partial z},$
$z$ being the last coordinate in $\mathbb{C}^n \times \mathbb{R}$. If we denote $Z= \frac{\partial}{\partial z}$, we have the following Lie brackets relations

$[X_i,X_j]=[Y_i,Y_j]=[X_i,Z]=[Y_j,Z]=0$
and

$[X_i,Y_j]=2 \delta_{ij} Z.$
In particular, the Hormander’s condition is satisfied. We also see that $X_i,Y_j,Z$ generate the Lie algebra of the $n$-dimensional Heisenberg group

$\mathbb{H}^{2n+1}=\{ (x,y,z) \in \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R} \}$
where the group law is given by

$(x_1,y_1,z_1) \star (x_2,y_2,z_2) =(x_1+x_2,y_2+y_2,z_1+z_2+\sum_{i=1}^n x^i_1y^i_2-x^i_2y^i_1).$
As before, we can then interpret $(X_t)_{t \ge 0}$ as a horizontal Brownian motion on $\mathbb{H}^{2n+1}$.

The group structure is specific to the particular choice of the one-form $\alpha$. If one wants to study more general situations, one has to use some Riemannian geometry.

Consider for instance a general smooth one-form

$\alpha=\sum_{i=1}^n \alpha^i(x)dx_i$
on $\mathbb{R}^n$ and let $(B_t)_{t \ge 0}$ be a $n$-dimensional Brownian motion. We have

$\int_{B[0,t]} \alpha =\sum_{i=1}^n \int_0^t \alpha^i(B_s) \circ dB^i_s,$
where the stochastic integrals have to be understood in the Stratonovitch sense. The process

$X_t=\left(B_t, \int_{B[0,t]} \alpha \right)$
is a diffusion process in $\mathbb{R}^n \times \mathbb{R}$ with generator

$L=\frac{1}{2} \sum_{i=1} X_i^2,$
where

$X_i=\frac{\partial}{\partial x_i} +\alpha^i(x) \frac{\partial}{\partial z} .$
We have

$[X_i,X_j]=\left( \frac{\partial \alpha^j}{\partial x_i}-\frac{\partial \alpha^i}{\partial x_j} \right) \frac{\partial}{\partial z}.$
Therefore, if the two-form $d\alpha$ is never 0, then the Hormander’s condition is satisfied.

We would like to call $(X_t)_{t \ge 0}$ the horizontal Brownian motion of some relevant geometric structure…

## Lecture 1. The Paul Levy’s stochastic area formula

When studying functionals of a Brownian motion, it may be useful to embed this functional into a larger dimensional Markov process.

Consider the case of the Levy area
$S_t=\int_0^t B^1_s dB^2_s-B^2_s dB^1_s,$
where $B_t=(B^1_t,B^2_t)$, $t \ge 0$, is a two dimensional Brownian motion started at 0. We can write
$S_t=\int_{B[0,t]} \alpha$
where $\alpha=xdy-ydx$. Since $d\alpha =2 dx \wedge dy$, we interpret $S_t$ as (two times) the algebraic area swept out in the plane by the Brownian curve up to time $t$. The process $(S_t)_{t \ge 0}$ is not a Markov process in its own natural filtration. However, if we consider the 3-dimensional process
$X_t=(B^1_t,B^2_t,S_t),$
then $X_t$ is solution of a stochastic differential equation
$dX^1_t =dB^1_t$

$dX^2_t =dB^2_t$

$dX^3_t =-X^2_t dB^1_t+X^1_t dB^2_t$

As a consequence $X_t$ is a Markov process with generator

$L=\frac{1}{2} (X^2+Y^2)$
where $X,Y$ are the following vector fields
$X=\frac{\partial}{\partial x}-y \frac{\partial}{\partial z}$
$Y=\frac{\partial}{\partial y}+x \frac{\partial}{\partial z}.$
Observe that the Lie bracket

$[X,Y]=XY-YX =2 \frac{\partial}{\partial z}.$
Thus, for every $x \in \mathbb{R}^3$, $(X(x),Y(x), [X,Y](x))$ is a basis of $\mathbb{R}^3$. From the celebrated Hormander’s theorem, this implies that for every $t > 0$ the random variable $X_t$ has a smooth density with respect to the Lebesgue measure of $\mathbb{R}^3$. In particular $S_t$ also has smooth density whenever $t>0$. We are interested in an expression for this density. The first idea is to reduce the complexity of the random variable $X_t$ by making use of symmetries.

Lemma: Let $r_t=\| B_t \|=\sqrt{ (B^1_t)^2 +(B^2_t)^2 }$, $t \ge 0$. Then, the couple

$(r_t , S_t)_{t \ge 0}$
is a Markov process with generator

$\mathcal{L}=\frac{1}{2r} \frac{\partial}{\partial r}+\frac{1}{2} \frac{\partial^2}{\partial r^2}+\frac{1}{2}r^2 \frac{\partial^2}{\partial s^2}$

Proof:
From Ito’s formula, we have

$dr_t =\frac{dt}{2 r_t}+\frac{B^1_t dB^1_t+B^2_t dB^2_t}{ \sqrt{ (B^1_t)^2 +(B^2_t)^2 }}$

$dS_t=r_t \frac{B^1_t dB^2_t-B^2_t dB^1_t}{ \sqrt{ (B^1_t)^2 +(B^2_t)^2 }}.$

Since the two processes

$\beta_t=\int_0^t \frac{B^1_s dB^1_s+B^2_s dB^2_s}{ \sqrt{ (B^1_s)^2 +(B^2_s)^2 }}$

$\gamma_t=\int_0^t \frac{B^1_s dB^2_s-B^2_s dB^1_s}{ \sqrt{ (B^1_s)^2 +(B^2_s)^2 }},$

are two independent Brownian motions, the conclusion easily follows. $\square$

We are now ready to prove the celebrated Levy’s area formula.

Theorem: For $t>0$ and $x \in \mathbb{R}^2$, and $\lambda >0$

$\mathbb{E}\left( e^{i\lambda S_t} | B_t=x\right)=\frac{\lambda t}{\sinh \lambda t} e^{-\frac{\|x\|^2}{2t}(\lambda t \coth \lambda t -1) }.$

Proof:
First, we observe that by rotational symmetry of the Brownian motion $(B_t)_{t \ge 0}$, we have

$\mathbb{E}\left( e^{i\lambda S_t} | B_t=x\right)=\mathbb{E}\left( e^{i\lambda S_t} | \| B_t \| =\| x \| \right).$
Then, according to the previous lemma,

$\mathbb{E}\left( e^{i\lambda S_t} | \| B_t \| =\| x \| \right)=\mathbb{E}\left( e^{i\lambda \gamma_{\int_0^t r_s^2 ds}} | r_t =\| x \| \right),$
where $\gamma_t$ is a Brownian motion independent from $r$. We deduce

$\mathbb{E}\left( e^{i\lambda S_t} | B_t=x\right)=\mathbb{E}\left( e^{-\frac{\lambda^2}{2} \int_0^t r_s^2 ds} | r_t =\| x \| \right),$
As we have seen, $r_t$ solves a stochastic differential equation

$dr_t =\frac{dt}{2 r_t}+d\beta_t,$
where $\beta$ is a one-dimensional Brownian motion.
One considers then the new probability

$\mathbb{P}_{/ \mathcal{F}_t}^\lambda = \exp \left(- \lambda \int_0^t r_s d\beta_s -\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right)\mathbb{P}_{/ \mathcal{F}_t},$
where $\mathcal{F}$ is the natural filtration of $\beta$. Observe that

$\int_0^t r_s d\beta_s =\int_0^t r_s dr_s -\frac{t}{2} =\frac{1}{2} r_t^2 -t$

Therefore

$\exp \left( -\lambda \int_0^t r_s d\beta_s -\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right)=e^{\lambda t} \exp \left( -\frac{\lambda}{2} r_t^2 -\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right)$

In particular, one deduces that

$\exp \left( -\lambda \int_0^t r_s d\beta_s -\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right) \le e^{\lambda t},$
which proves that $\exp \left( -\lambda \int_0^t r_s d\beta_s -\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right)$ is a martingale. By using this change of probability, if $f$ is a bounded and Borel function, we have

$\mathbb{E}\left( f(r_t) e^{-\frac{\lambda^2}{2} \int_0^t r_s^2 ds} \right) =e^{-\lambda t} \mathbb{E}^\lambda \left(f(r_t) \exp \left( \frac{\lambda}{2} r_t^2 +\frac{\lambda^2}{2} \int_0^t r_s^2 ds \right) e^{-\frac{\lambda^2}{2} \int_0^t r_s^2 ds} \right) =e^{-\lambda t} \mathbb{E}^\lambda \left(f(r_t) \exp \left( \frac{\lambda}{2} r_t^2 \right) \right).$
Putting things together, we are thus let with the computation of the distribution of $r_t$ under the probability $\mathbb{P}^\lambda$. From Girsanov’s theorem, the process

$\beta^\lambda_t=\beta_t +\lambda \int_0^t r_s ds$

is a Brownian motion under the probability $\mathbb{P}^\lambda$. Thus

$dr_t =\left( \frac{1}{2 r_t} -\lambda r_t \right) dt+d\beta^\lambda_t.$

In law, this is the stochastic differential equation solved by $\| Y_t \|$ where

$dY_t =-\lambda Y_t dt +dB^\lambda_t, \quad Y_0=0.$

We deduce that $r_t$ is distributed as $\| Y_t \|$, the norm of a two-dimensional Ornstein Uhlenbeck process with parameter $-\lambda$. Since $Y_t$ is a Gaussian random variable with mean 0 and variance $\frac{1-e^{-2\lambda t}}{2\lambda} \mathbf{Id}$, the conclusion follows from standard computations about the Gaussian distribution. $\square$

This formula is due to Paul Levy who originally used a series expansion of the Brownian motion. The proof we present here is due to Marc Yor.

The Levy’s area formula has several interesting consequences. First, when $x=0$, we deduce that

$\mathbb{E}\left( e^{i\lambda S_t} | B_t=0\right)=\frac{\lambda t}{\sinh \lambda t}.$

This gives a formula for the characteristic function of the algebraic stochastic area within the Brownian loop with length $t$. Inverting this Fourier transform yields

$\mathbb{P} \left( S_t \in ds | B_t=0 \right)=\frac{\pi}{2t} \frac{1}{\cosh^2 \left( \frac{\pi s}{t}\right)} ds.$

Next, integrating the Levy’s area formula with respect to the distribution of $B_t$ yields the characteristic function of $S_t$:

$\mathbb{E}\left( e^{i\lambda S_t} \right)=\frac{1}{\cosh (\lambda t)}$

Unfortunately, this Fourier transform may not be easily inverted. However, one may deduce from it the following formula (due to Biane-Yor): For $\alpha>0$,

$\mathbb{E} (|S_t|^\alpha)=\frac{2^{\alpha+2} \Gamma(1+\alpha)}{\pi^{1+\alpha}} L(1+\alpha) t^\alpha,$

where $L(s)=\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$ is the Dirichlet function. This provides an unexpected and fascinating connection with the Riemann zeta function.