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 be a smooth and connected Riemannian manifold. In a local orthonormal frame , one can compute the length of the gradient of a smooth function :
Let us denote by the Riemannian volume measure. We can then consider the pre-Dirichlet form
There exists a unique second order operator such that for every ,
The operator is called the Laplace-Beltrami operator. Locally, we have
where denotes the Levi-Civita connection on .
Definition: A Brownian motion on is a diffusion process with generator , that is for every ,
is a local martingale, where is the lifetime of on .
One can construct Brownian motions by using the theory of Dirichlet forms by using the minimal closed extension of . If the metric is complete (which we always assume for Riemannian metrics in this course), then one can prove that is essentially self-adjoint on and there is a unique closed extension of . Note that even in the complete case 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 , we have
where is a Brownian motion in . 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 and be smooth and connected complete Riemannian manifolds.
Definition: A smooth surjective map is called a Riemannian submersion if its derivative maps are orthogonal projections, i.e. for every , the map is the identity.
Example: (Warped products) Let and be Riemannian manifolds and be a smooth and positive function on . Then the first projection is a Riemannian submersion.
If is a Riemannian submersion and , the set is called a fiber.
For , is called the vertical space at . The orthogonal complement of shall be denoted and will be referred to as the horizontal space at . We have an orthogonal decomposition
and a corresponding splitting of the metric
The vertical distribution 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 , .
If we define its vertical gradient as the projection of its gradient onto the vertical distribution and its horizontal gradient as the projection of the gradient onto the horizontal distribution. We define then the vertical Laplacian as (minus) the generator of the pre-Dirichlet form
where is the Riemannian volume measure on . Similarly, we define the horizontal Laplacian as (minus) the generator of the pre-Dirichlet form
Definition: A horizontal Brownian motion on is a diffusion process with generator , that is for every ,
is a local martingale, where is the lifetime of on .
If is a local orthonormal frame of basic vector fields and a local orthonormal frame of the vertical distribution, then we have
where the adjoints are (formally) understood in . Classically, we have
where is the Levi-Civita connection. As a consequence, we obtain
where denotes the horizontal part of the vector. In a similar way we obviously have
We note that from Hormander’s theorem, the operator 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 is everywhere bracket-generating. The horizontal Laplacian is essentially self-adjoint on the space .
In the sequel, we will often assume that the horizontal distribution 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 from the Brownian motion on . It uses the notion of horizontal lift.
A vector field is said to be projectable if there exists a smooth vector field on such that for every , . In that case, we say that and are -related.
Definition: A vector field on is called basic if it is projectable and horizontal.
If is a smooth vector field on , then there exists a unique basic vector field on which is -related to . This vector is called the horizontal lift of .
A -curve is said to be horizontal if for every ,
Definition: Let be a curve. Let , such that . Then, there exists a unique horizontal curve such that and . The curve is called the horizontal lift of at .
The notion of horizontal lift may easily be extended to Brownian motion paths on by using stochastic calculus (or rough paths theory).
Theorem: Assume that the fibers of the submersion have all zero mean curvature. Let be a Brownian motion on started at . Let such that . The horizontal lift of at is a horizontal Brownian motion.
Proof: Indeed, if is a local orthonormal frame of basic vector fields and a local orthonormal frame of the vertical distribution, let us denote by the vector fields on which are -related to . We have
Therefore, locally solves a stochastic differential equation
where is a Brownian motion in . Since it is easy to check that is -related to , we deduce that locally solves the stochastic differential equation
We now recall that
If the fibers of the submersion have all zero mean curvature, the vector
is always orthogonal to . Thus
and is a horizontal Brownian motion
In this course, we shall mainly be interested in submersion with totally geodesic fibers.
Definition: A Riemannian submersion is said to be totally geodesic if for every , the set is a totally geodesic submanifold of .
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.