## Stochastic analysis on sub-Riemannian manifolds with transverse symmetries.(English)Zbl 1386.58019

This paper treats a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with transverse symmetries. This representation is closely related to the study of Bochner-Weitzenböck type formulas for sub-Laplacians on one-forms. More precisely, a sub-Riemannian manifold is a smooth manifold $${\mathbb M}$$ equipped with a nonholonomic subbundle $${\mathcal H}$$ $$\subset$$ $$T {\mathbb M}$$ and a fiber inner product $$g_{ {\mathcal H} }$$. The sub-Laplacian $$L$$ is essentially self-adjoint on $$C_0^{\infty}( {\mathbb M} )$$, and $$P_t$$ $$=$$ $$e^{ (1/2)t L}$$ is the semigroup generated by $$\frac{1}{2} L$$. Note that the semigroup $$P_t$$ is stochastically complete. The operator $$\square_{\varepsilon}$$ is defined on one-forms by the formula $\square_{\varepsilon} := - ( \nabla_{ {\mathcal H}} - {\mathfrak T}_{ {\mathcal H}}^{\varepsilon} )^* ( \nabla_{ {\mathcal H}} - {\mathfrak T}_{ {\mathcal H}}^{\varepsilon} ) + \frac{1}{ 2 \varepsilon} J^* J - {\mathcal R}ic_{ {\mathcal H} }, \tag{1}$ where $$\nabla_{ {\mathcal H} }$$ is the horizontal gradient in a local adapted frame of one-form $$\eta$$, $$J^* J$$ is the identity map on the horizontal distribution, $${\mathcal R}ic_{ {\mathcal H}}$$ is the fiberwise symmetric linear map on one-forms, and $${\mathfrak T}_{ {\mathcal H}}^{\varepsilon}$$ is the fiberwise linear map on one-forms. Note that for $$f \in C_0^{\infty}( {\mathbb M} )$$, we have the equality $d P_t f = Q_t^{\varepsilon} d f \tag{2}$ for every $$t \geq 0$$. Then $$\eta_t = Q_t^{\varepsilon} df$$ is the unique solution in $$L^2$$ of the heat equation $\frac{ \partial \eta}{ \partial t} = \frac{1}{2} \square_{\epsilon} \eta \tag{3}$ with initial condition $$\eta_0 = d f$$. Here is the main theorem.

Theorem. Let $$\eta$$ be a smooth and compactly supported one-form. Then for every $$t \geq 0$$ and $$x \in {\mathbb M}$$, $( Q_t^{\varepsilon} \eta )(x) = {\mathbb E}_x [ \tau_t^{\varepsilon} \eta( X_t) ], \tag{4}$ where the process $$\tau_t^{\varepsilon}$$ : $$T_{ X_t}^* {\mathbb M}$$ $$\to$$ $$T_{ X_0}^* {\mathbb M}$$ is the solution of the covariant Stratonovich stochastic differential equation $d [ \tau_t^{\varepsilon} \alpha( X_t)] = \tau_t^{\varepsilon} \left\{ \nabla_{ \circ d X_t} - {\mathfrak T}_{\circ d X_t}^{\varepsilon} + \frac{1}{2} \left( \frac{1}{ 2 \varepsilon} J^* J - {\mathcal R}ic_{ {\mathcal H}} \right) dt \right\} \alpha( X_t), \tag{5}$ with $$\tau_0^{\varepsilon} = Id$$, where $$\alpha$$ is any smooth one-form.
As a consequence, the author proves new hypoelliptic heat semigroup gradient bounds under natural global geometric condition. For other related works, see e.g. [F. Baudoin et al., Math. Ann. 358, No. 3–4, 833–860 (2014; Zbl 1287.53025)] for a sub-Riemannian curvature-dimension inequality and the Poincaré inequality.

### MSC:

 58J65 Diffusion processes and stochastic analysis on manifolds 53C17 Sub-Riemannian geometry 60J60 Diffusion processes

Zbl 1287.53025
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