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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

Citations:

Zbl 1287.53025
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