## 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.

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