Derivative and divergence formulae for diffusion semigroups (Q2414141)

This paper treats the equivalence between probabilistic formulae for \(P_T( V(f))\) and shift-Harnack inequalities, where \(P_T\) is a Markov operator associated to a nondegenerate diffusion \(X_t\) on a smooth finite-dimensional manifold \(M\), and \(V\) is a bounded smooth vector field with bounded \(div\, V\), and \(f\) is a bounded smooth function. According to \textit{F.-Y. Wang} [ibid. 42, No. 3, 994--1019 (2014; Zbl 1305.60042)], for \(\delta_e \in (0,1)\) and \(\beta_e \in C( ( \delta_e, \infty) \times E ; [0, \infty))\) with a Banach space \(E\), \(e \in E\) and a Markov operator \(P\), the derivative-entropy estimate \[ \vert P( \nabla_e f ) \vert \leqslant \delta \{ P(f \log f) - (Pf) \log Pf \} + \beta_e( \delta, \cdot) Pf \tag{1} \] holds for any \(\delta \geq \delta_e\) and positive \(f \in C_b^1(E)\) if and only if the inequality \[ ( P f)^p \leqslant \{ P( f^p ( re + \cdot) ) \} \times \exp \left\{ \int_0^1 \frac{pr}{ 1 +(p-1) s} \beta_e \left( \frac{p-1}{ r + r(p-1)s}, \cdot + sre \right) ds \right\} \tag{2} \] holds for any \(p \geq 1/( 1- r \delta_e)\), \(r \in (0, 1 / \delta_e)\) and positive \(f \in B_b(E)\); furthermore, if \(C \geq 0\) is a constant then the \(L^2\)-derivative inequality \[ \vert P( \nabla_e f) \vert^2 \leqslant C \cdot Pf^2 \tag{3} \] holds for any nonnegative \(f \in C_b^1(E)\) if and only if the inequality \[ Pf \leqslant P( f( \alpha e + \cdot )) + \vert \alpha \vert \sqrt{ C\cdot P f^2 } \tag{4} \] holds for any \(\alpha \in \mathbb{R}\) and nonnegative \(f \in B_b(E)\). The purpose of this article is to find probabilistic formulae for \(P_T( V(t))\) from which such estimates can be derived. Let \(M\) be a Riemannian manifold and \(Z\) a smooth vector field on \(M\). Let \(X = X(x)\) be a nonexplosive diffusion to \(\Delta + Z\) on \(M\), starting at \(X_0(x) =x\). For \(T > 0\), \(h_t(\omega)\) \(\in\) \(L^{1,2}( [0, T]; \mathbb{R})\) is an adapted process satisfying \(h_o=0\) and \(h_T=1\). Assume that \[ div \, \alpha_{T-t} h_t - \frac{1}{2} \alpha_{ T- t} \left( //_t \Theta_t \int_s^t ( \dot{h}_s - (div \, Z) (X_s(x)) h_s ) \Theta_s^{-1} //_s^{-1} d B_s \right) \tag{5} \] is a true martingale, where a bounded 1-form \(\alpha\) satisfies \[ \frac{d}{dt} \alpha_t = ( \square + \nabla_Z - Ric_{-Z}^* + div \, Z ) \alpha_t \tag{6} \] with \(\alpha_0= \alpha\), where \(div \, Z\) acts fibrewise as a multiplication operator, and \(Ric_{-Z}^*\) is the adjoint to \(Ric_{-Z}\) acting as endomorphism of \(T_x M\). \(//_t\) denotes the stochastic parallel transport associated to \(X_t(x)\), whose antidevelopment to \(T_x M\) has martingale part \(B\). Here is the first main result: if \(f\) is a bounded smooth function and \(V\) is a bounded smooth vector field with bounded \(div \, V\), then \[ \begin{multlined} P_T( V(t))(x) = - \mathbb{E} [ f( X_T(x)) (div \, V) (X_T(x)) ] \\ + \frac{1}{2} \mathbb{E} \left[ f(X_T(x)) \langle V(X_T(x)), //_T \Theta_T \int_0^T ( \dot{h}_T - (div \, Z) ( X_t(x) h_t) \Theta_t^{-1} d B_t \rangle \right] \end{multlined} \tag{7} \] is obtained, from which the following estimate (shift-Harnack inequality) is derived: \[ ( P_t f)^p \leqslant ( P_t( f^p \circ F_1)) \exp \left\{ \int_0^1 \frac{p}{ \beta(s)} \alpha_1 \left( \frac{ \beta'(s)}{\beta(s)}, t, V_s \right) ds \right\}. \tag{8} \] The other formula which the authors found is the following one: \[ \begin{multlined} P_T( V(f)) = - \sum_{i=1}^m \mathbb{E} [ f( X_T ) A_i \langle V, A_i \rangle (X_T) ] \\ + \frac{1}{2} \mathbb{E} \left[ f( X_T) \left\langle V, \Xi_T \int_0^T \Xi_t^{-1} \left( ( \dot{h}_t - ( trace \, \hat{\nabla} A_0 ) ( X_t) h_t ) A( X_T) d B_t + 2 h_t A_0^A dt \right) \right\rangle \right], \end{multlined}\tag{9} \] where \(A_0\) is a smooth vector field and \(A : M \times \mathbb{R}^m\) \(\ni (x,e)\) \(\mapsto\) \(A(x)e \in TM\) is a smooth bundle map over \(M\). Note that the vector field \(A_0^A\) appears to depend on the Levi-Civita connection via the sum of the vector fields \(\nabla_{A_i} A_i\). Note that \(V\) is a bounded smooth vector field with bounded \(\sum_{i=1}^m A_i \langle V, A_i \rangle\). Assume that the Stratonovich stochastic differential equation \(d X_t\) \(=\) \(A_0(X_t) dt\) \(+\) \(A( X_t) \circ d B_t\) is complete and \(\alpha_{T-t}(\Xi _t)\) and the term \[ \begin{multlined} \hat{\delta} \alpha_{T -t} h_t - \int_0^t h_s \alpha_{T-s} ( A_0^A) ds \\ + \frac{1}{2} \alpha_{T-t} \left\{ \Xi_t(x) \int_0^t ( \dot{h}_s - ( trace \hat{\nabla} A_0) ( X_s(x)) h_s ) \Xi_s(x)^{-1} A( X_s(x)) d B_s \right\} \end{multlined} \tag{10} \] is a local martingale, where a differential operator \(\hat{\delta}\) is given by \[ \hat{\delta} \phi := - \sum_{i=1}^m \iota_{ A_i} L_{A_i} \phi \tag{11} \] with \(L_{A_i} \phi = \iota_{A_i} d \phi + d ( \iota_{ A_i} \phi)\), while the codifferential \(\delta\) satisfies \(\delta \phi\) \(=\) \(- \sum_{i=1}^m\) \(( \nabla_{A_i} \phi)(A_i)\) for a 1-form \(\phi\). For other related works, see [\textit{B. K. Driver} and the first author, J. Funct. Anal. 183, No. 1, 42--108 (2001; Zbl 0983.58018); the first author and \textit{F.-Y. Wang}, Bull. Sci. Math. 135, No. 6--7, 816--843 (2011; Zbl 1242.58020)]; and see also [\textit{A. B. Cruzeiro} and \textit{X. Zhang}, Potential Anal. 25, No. 2, 121--130 (2006; Zbl 1106.58015)] for Bismut type formulas for vector bundles, and [\textit{F.-Y. Wang}, Ann. Probab. 42, No. 3, 994--1019 (2014; Zbl 1305.60042)] for shift Harnack inequality for stochastic equations.