Qi Feng,Wuchen Li
Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold
degenerate drift–diffusion process,Lyapunov methods,auxiliary Fisher information,sub-Riemannian density manifold,generalized Bochner’s formula,53C17,60D05,58B20
We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.
Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.

Consider the following Stratonovich stochastic differential equation: where is an n-dimensional Brownian motion in , is a matrix-valued function, and is a drift vector field. The convergence analysis of SDE (1) to its invariant distribution lies in the intersection of differential geometry, analysis, the Lie group (subgroup in quantum mechanics), and probability. The convergence analysis also has broad applications in designing fast algorithms in artificial intelligence (AI) and Bayesian sampling/optimization problems. One key question arises: How fast does the probability density function of SDE (1) converge to its invariant distribution? The Gamma calculus, also named Bakry–Émery iterative calculus [1], provides analytical approaches to derive the convergence rate for SDE (1). This lower bound is known as the Ricci curvature lower bound. However, classical studies are limited to the non-degenerate diffusion coefficient matrix a. The classical Gamma calculus is no longer valid when a is a degenerate matrix function; see the generalization of Bakry–Émery calculus in [2]. This paper presents a Lyapunov convergence analysis for the degenerate diffusion process. We selected a class of z-Fisher information as the Lyapunov functional, where z is a matrix function different from matrix a. We derived a generalized Gamma calculus by the dissipation of the Lyapunov functional along the diffusion process. We then derived the generalized Bochner’s formula and obtained the exponential convergence condition. Several concrete examples are presented: gradient-drift–diffusions on the Heisenberg group, the displacement group, and the Martinet sub-Riemannian structure. Our approach extends the classical optimal transport geometry, in particular the second-order calculus of the relative entropy in the density manifold studied in [3,4,5,6]. The generalized Gamma calculus was first introduced by Baudoin–Garofalo [2] for sub-Riemannian manifolds. Related results were studied later in [7,8,9,10,11,12,13,14,15]. The commutative property of the iteration of and (Hypothesis in [2]) was crucial in the previous works. Our algebraic Condition 1 does not have this requirement. We can remove this commutative condition in the weak sense. Thus, our results go beyond the step two-bracket-generating condition. We present algebraic conditions for the existence of the generalized Bochner’s formula. On the other hand, optimal transport on the sub-Riemannian manifold was studied by [16,17,18,19]. An optimal transport metric on a sub-Riemannian manifold was proposed in [18,19]. In this case, the density manifold still forms an infinite-dimensional Riemannian manifold. The Monge–Ampère equation in sub-Riemannian settings was studied in [17]. Our approach is different. We introduced the sub-Riemannian density manifold (SDM) and studied its second-order geometric calculations of relative entropies in the SDM. Using those, we propose a new Gamma z calculus for degenerate stochastic differential equations and established the generalized curvature dimension-type bound. Besides, Refs. [20,21] used the analytical property of optimal transport to formulate the Ricci curvature lower bound in general metric space. Different from [20,21,22], we focused on the geometric calculations in the density manifold introduced by the z direction. Following the second-order geometric calculations in the density manifold, we formulated the new Gamma calculus and the corresponding Ricci curvature tensor for the sub-Riemannian manifold. Besides, our derivation also relates to the entropy methods [23,24]. Using entropy methods, Refs. [25,26] derived the convergence rate for degenerate drift–diffusion processes with constant diffusion coefficients a. Compared to previous works, we applied the entropy method with Gamma calculus and geometric calculations in the density manifold. It derives a generalized Gamma calculus from the dissipation of auxiliary Fisher information. Several concrete examples of convergence conditions are derived in the Lie-group-induced drift–diffusion processes. We organize the paper as follows. We introduce the main result in Section 2. It is an explicit convergence rate condition for the density of degenerate SDEs in the distance. In Section 3, we provide three examples of the proposed convergence analysis, including gradient-drift–diffusions on the Heisenberg group, the displacement group, and the Martinet sub-Riemannian structure. In Section 4, we present the Lyapunov analysis in the sub-Riemannian density manifold. The generalized Gamma calculus and the proof of the generalized Bochner’s formula is presented in Section 5. Some further discussions for other functional inequalities are presented in Section 6.

In this section, we present this paper’s setting and main results.

Consider a Stratonovich SDE: where is an n-dimensional Brownian motion in , is a matrix-valued function, and is a vector field. We refer to [27] (Section 3.13) for the definition of the Stratonovich SDE. According to [28] (Appendix A.7), the SDE (2) can also be written as the following Itô SDE: where We denote as the column vectors of matrix a, and represents We denote as the transpose of matrix a and denote as the row vectors of matrix . In particular, we have , for and . With some abuse of notation, we also denote as the vector fields corresponding to the row vectors , for . We assumed that satisfies the strong Hormander condition (or bracket-generating condition): where represents the Lie bracket between two vector fields. The strong Hörmander condition means that the Lie algebra generated by the vector fields is of full rank at every point (see, e.g., [29] (Section 7.4)). This condition ensures the existence of a smooth probability density function of SDE (2); see the original proofs in [30,31]. For the simplicity of presentation, we assumed the probability density function is strictly positive. Indeed, the positivity of the density follows from the Hörmander condition [32]; for the more technical conditions to show the positivity by using Malliavin calculus, we refer to [33,34] (Theorem 1.4 with H = 1/2). Denote , where is the probability density function of SDE (2). The density function satisfies the Fokker–Planck equation of SDE (2): with a smooth initial condition: In this paper, we assumed that SDE (2) has a unique invariant symmetric measure , where with . Here, solves the equilibrium of Fokker–Planck Equation (6): We studied a particular class of the vector field b for a given invariant distribution . Assumption (Gradient flow formulation): Suppose that b, a, and satisfy the relation: where represents, for , In the Itô formulation, , a, and satisfy for . In this case, we can reformulate Equation (6) as We leave the derivation of Formula (9) in Appendix A. If , then , and is an invariant density function for SDE (2). In Section 4, we demonstrate that Fokker–Planck Equation (6), or its equivalent Formulation (9), forms a “horizontal” gradient flow in the sub-Riemannian density manifold. We designed a Lyapunov functional to study the convergence behavior of this “horizontal” gradient flow (9). Remark 1. Formula (9) can be written as It has a weak formulation that where is a smooth test function. Remark 2 (Non-gradient flow drift). In fact, the proposed method is not limited to the gradient flow assumption of the drift vector field b in (7). See the details in [35].

We now briefly sketch the main results. Denote a sub-elliptic operator as follows: where . Definition 1 (Generalized Gamma z calculus). Consider a smooth matrix function . Denote Gamma one bilinear forms as Define Gamma two bilinear forms as and Here, , are divergence operators defined by for any smooth vector field , and , are vector Gamma one bilinear forms defined as with We next demonstrate that the summation of and can induce the following decomposition and bilinear forms. They are natural extensions of the classical Bakry–Émery calculus in the Riemannian manifold, i.e., non-degenerate matrix function a. Notation 1. For matrix function , we define matrix Q as with . More precisely, for each row (respectively, column) of Q, the row (respectively column) indices of follow (respectively, ). For matrix function , we define matrix P as with . For smooth function , for any and (or , we define vector with components where we denote . We define vector with components We define vector with components We define vector with components We define vector with components We define X as the vectorization of the Hessian matrix of function f: Assumption 1. Assume that there exists vectors such that Definition 2 (Hessian matrix). For smooth function , define a matrix function as where we define the following bilinear forms: and Here, we also denote , such that . The main theorem is presented below, and its proof is postponed to Theorem 3 in Section 5. Theorem 1 (Generalized z Bochner’s formula). If Assumption 1 is satisfied, then the following decomposition holds: where we define We are now ready to prove the convergence property of the degenerate drift–diffusion process (1) and related functional inequalities. Denote the Kullback–Leibler divergence as Denote the -relative Fisher information functional as Theorem 2 (Exponential convergence in the distance). Suppose there exists a constant such that Let be a smooth initial distribution and be the probability density function of (1). Then, ρ converges to the invariant measure π in the sense of In addition, The proof of Theorem 2 is postponed to Proposition (14). Remark 3 (Functional inequalities). Suppose with , then the z-log-Sobolev inequalities hold: for any smooth density function ρ. Remark 4. In the literature [2], the operator is defined by (10), i.e., . In fact, this definition is under the assumption of . This assumption holds true only for the special choice of a and z. In the generalized Gamma z calculus, we introduce a new term (11), which removes the assumption . In fact, in the paper, we show that (11) is exactly the new bilinear form behind the assumption in [2] by considering the weak form. Remark 5. Following [35] (Assumption 1), we know that, for any and , if there exist vectors and , such that the Hessian operator associated with the generator of the SDE and the metric could be represented as Furthermore, we have the following relation: if there exist and as in Assumption 1 such that Assumption 1 is true if Conditions 20 and 21 hold. See the detailed connections in [35] (Remark 11).

In this section, we consider the following degenerate drift–diffusion process: where is a matrix-valued function, for , and is a smooth potential function. We denote the invariant measure of SDE (22) as . We further assumed that The above assumption holds for the later three examples. Remark 6. For , the invariant measure π in the above assumption exists if forms left-invariant structures on unimodular Lie groups. In this case, the sub-Laplacian is the sum of squares of horizontal vector fields and the invariant measure is also symmetric. Stratonovich SDE (22) defines the horizontal Brownian motion on sub-Riemannian structure , and π is the volume form associated with the horizontal Laplacian. In general, if the Lie group structure is not unimodular, the drift . See the related studies about the diffusion process on general manifolds in [36,37,38,39,40,41,42,43]. See the related studies on log-Sobolev inequality in [44,45]. Remark 7. It is also worth mentioning that many sub-Riemannian manifolds are non-compact. Hence, there may not exist a positive constant κ for both classical and directions in the non-compact domain. The non-compactness of the domain brings additional difficulties. To prove the associated inequalities in this case, we need to extend the result derived in [46,47]. This is a direction for future work. Remark 8. It is known that the Heisenberg group is an example of Lie groups in quantum mechanics [48]. In future work, we shall investigate the general convergence analysis of SDEs in Lie groups and their connections with quantum SDEs.

In this subsection, we apply our general theory to the well-known example in sub-Riemannian geometry, which is the Heisenberg group. A related LSI for the horizontal Wiener measure was studied in [46]. Recall briefly that the Heisenberg group admits left-invariant vector fields: . Here, forms an orthonormal basis for the tangent bundle of . In this case, . In particular, X and Y generate the horizontal distribution . To fit into our general theory from the previous section, we take matrices a and z as below: In particular, we have We have the following proposition for Heisenberg group following Theorem 1. Proposition 1. For any smooth function , one has where The proof of Proposition of 1 follows from the proof of Theorem 1 (i.e., Theorem 3) and Lemmas 1–3. The following convergence result follows directly from Theorem 2. Proposition 2. If there exists as shown in Theorem 2, the exponential dissipation result in the distance holds: We next formulate the curvature tensor into a matrix format. Denote and denote as the identity matrix. With a little abuse of notation, there exists a symmetric matrix such that we can represent the tensor as below. which implies that In other words, we need to estimate the smallest eigenvalue of matrix . We next present the formulation of matrix for the Heisenberg group as follows. Corollary 1. The matrix associated with the Heisenberg group has the following form: Proof. The explicit form of matrix follows from the definition in Theorem 1 and the notation in (24) and (25). We have Plugging the explicit representation from Proposition 1 into the above formula and applying matrix symmetrization for the off-diagonal terms, we obtain the desired matrix . □ Next, we present the three key lemmas. Lemma 1. For the Heisenberg group, we have Proof. The proof of this lemma follows from routine computations. Plugging matrices a and z from (23) into Notation 1, we obtain the desired vectors and matrices. We skip the detailed computation here. □ Lemma 2. On , vectors F and G are zero vectors, and we have In particular, we have Lemma 3. By routine computations, we obtain Proof of Lemma 2. We first have By direct computations, we have Completing the squares for the cross terms involving the type of ” and following the reformulation as below: we have The sum of squares terms give , hence and . The remainders generate , which equals . □ We are now left to compute the tensors. Proof of Lemma 3. By direct computation, we have For the four terms above, we have Similar computation applies to the tensor terms and . Since z is a constant matrix, we obtain We now compute the tensor terms involving the drift b. For the drift term in tensor , taking , which means in local coordinates, We now derive the explicit formulas for the above four terms. Summing up the above formulas, we obtain . We now compute the drift tensor term of . By taking , we have We further compute as below by taking advantage of the constant matrix z: The proof is thus completed. □

In this section, we derive the generalized curvature dimension bound for the displacement group, which is one example of three-dimensional solvable Lie groups. We adapted the general setting from [49] below. Denote as the three-dimensional solvable Lie algebra, and denote as the horizontal subspace satisfying Hörmander’s condition, then for a given inner product on H, there exists a canonical basis for , such that forms an orthonormal basis for H and satisfies the following Lie-bracket-generating condition for parameters and : When the parameters and , the Lie algebra has a faithful representation. In particular, it was shown in [49] that the elements of , in local coordinates , correspond to the following left-invariant differential operators: with the following relation: In terms of local coordinates , we have The corresponding Lie group of this special Lie algebra is called the displacement group, denoted as . We chose as the horizontal orthonormal basis for subalgebra H. To fit into the general framework from the previous section, we take with . Our focus here is to derive the curvature tensor in terms of . We then used as the horizontal metric on H. Thus, the sub-Riemannian structure is given by . By direct computations, it is easy to show that, for general smooth function f, . Hence, the classical Gamma z calculus proposed in [2] can not be extended for this case to derive the zLSI. Thus, we need to compute vector G and the tensor term . Following Theorem 1, we have the following z-Bochner’s formula for . Proposition 3. For any smooth function , one has where and In particular, we have The proof of Proposition 3 follows from the proof of Theorem 1 (i.e., Theorem 3) and Lemmas 4–6 below. The following convergence result follows directly from Theorem 2. Proposition 4. If there exists as shown in Theorem 2, the exponential dissipation result in the distance holds: Similarly, we formulated the curvature tensor into a matrix format of . Using the fact , we have the following representation. Corollary 2. The matrix associated with has the following representation: Proof. The derivation for the explicit form of matrix follows from a similar equivalent representation as shown in the proof of Corollary 1 and the explicit bilinear terms derived in Proposition 3. □ Remark 9. By taking as a constant, Proposition 3 reduces to a simple version; in particular, the tensors reduce to be Next, we present the following three key lemmas. Lemma 4. For displacement group , we have . Proof. The proof follows by plugging matrices a and z from (26) into Notation 1. □ Lemma 5. On displacement group , we have In particular, we have Lemma 6. By routine computations, we obtain Proof of Lemma 5. According to Lemma 4 and observing the fact that and , we first have By direct computations, we have Completing the squares for the above terms, we have The first-order terms generate , and the sum of squares terms generate vectors and . We further formulate the above two terms as below: Adding into the term again, we further expand as below: By grouping the bilinear terms of , we obtain □ We are now left to compute the three tensor terms. Proof of Lemma 6. For displacement group , we have and . Recall Theorem 1; we denote , where represents the tensor term involving drift b. We thus have By direct computations, we have For the drift term in tensor , taking , we obtain Plugging into the matrix , we obtain Combining the above computations, we obtain the tensor . Now, we turn to the second tensor , which has the following form: where we denote further that By taking , we further obtain that By direct computations, it is not hard to observe that and Now, we are left to compute the term . Recall that By direct computation, we obtain □

In this part, we apply our result to the Martinet flat sub-Riemannian structure, which satisfies the bracket-generating condition and has a non-equiregular sub-Riemannian structure (see [37]). The sub-Riemannian structure is defined on through the kernel of one-form A global orthonormal basis for the horizontal distribution adapts the following differential operator representation, in local coordinates : The commutative relation gives To apply it in our framework, we take Thus, the sub-Riemannian structure has the form . Proposition 5. In this setting, then Proof. The poof follows from the observation that □ Similar to the previous displacement group case, we have the following identity. Proposition 6. For any smooth function , one has where In particular, we have The proof of Proposition 6 follows from the proof of Theorem 1 (i.e., Theorem 3) and Lemmas 7–9 below. The following convergence results are a direct consequence of Theorem 2. Proposition 7. If there exists as shown in Theorem 2, the exponential dissipation result in the distance holds: Similarly, we summarize the sub-Riemannian Ricci tensor in terms of as follows. Corollary 3. The matrix associated with the Martinet sub-Riemannian structure has the following form: Proof. The proof follows from the similar equivalent matrix formulation as shown in the proof of Corollary 1 and the explicit bilinear forms in Proposition 6. □ Next, we prove the following three key lemmas. Lemma 7. For Martinet sub-Riemannian structure , we have Proof. Plugging matrices a and z from (27) into Notation 1, we complete the proof. □ Lemma 8. For the Martinet sub-Riemannian structure, F and G are zero vectors, and we have In particular, we have Lemma 9. By routine computations, we obtain Proof of Lemma 8. Since F and G are zero vectors, we have By routine computation, we observe that where we use the fact The proof is thus completed. □ We are now left to compute the three tensor terms. Proof of Lemma 9. Similar to the proof of Lemma 6, we have By direct computations, we have For the drift term, we take Plugging into the matrices of , we obtain Combing the above computations, we obtain the tensor . Now, we turn to the second tensor . Since , it is obvious to see that only the drift term of the tensor remains, where we denote By taking , we further obtain that By direct computations, it is not hard to observe that The only non-zero term has the following form: Since matrix is a constant matrix and matrix contains only variable y, it is easy to observe that □

In this section, we illustrate the motivation of this paper, which is to design a matrix condition, whose smallest eigenvalue characterizes the convergence rate of the degenerate SDE. The outline of this section is given below. Consider a density space over the sub-Riemannian manifold. The finite-dimensional sub-Riemannian structure introduces the density space the infinite-dimensional sub-Riemannian structure. We name it the sub-Riemannian density manifold (SDM). We provide the geometric calculations in the SDM. We studied the Fokker–Planck equation as the sub-Riemannian gradient flow in the SDM. We derived the equivalence relation between the second-order calculus of the relative entropy in the SDM and the generalized Gamma z calculus.

Given a finite-dimensional sub-Riemannian manifold with , consider the probability density space: Consider the tangent space at : We introduce the sub-Riemannian structure in probability density space . Definition 3 (sub-Riemannian Wasserstein metric tensor). The sub-Riemannian-Wasserstein metric is defined by Here, , is the metric on , and is the pseudo-inverse of the sub-elliptic operator: For some special choices of a as studied in [19] or forming a positive definite matrix, then is an elliptic operator. In this case, still forms a Riemannian density manifold. In general, given a sub-Riemannian manifold , is only a sub-elliptic operator. Thus, forms an infinite-dimensional sub-Riemannian manifold. We next present the sub-Riemannian calculus in , including both geodesics and the Hessian operator in the tangent bundle. Consider an identification map: Here, . This is the cotangent space in the SDM, and ∼ represents a constant shift relation. Thus, In other words, where the second equality holds by and the last equality holds by the integration by parts. We next derive several basic geometric calculations in the SDM. Proposition 8 (Geodesics in the SDM). The sub-Riemannian geodesics in the cotangent bundle forms Proof. We considered the Lagrangian formulation of geodesics in density. Here, the minimization of the geometric action functional forms where is a density path with fixed boundary points , . Then, the Euler–Lagrange equation in density space forms where is the first variation with respect to and is the first variation with respect to . Here, where the last equality uses the following fact: Denote , then the Euler–Lagrange Equation (30) forms the sub-Riemannian geodesics flow (29). In other words, □ Proposition 9 (Gradient and Hessian operators in the SDM). Given a functional , the gradient operator of in satisfies The Hessian operator of in satisfies where Proof. We first derive the sub-Riemannian gradient operator. We recall the identification map by . Hence, the gradient operator in the SDM satisfies The Hessian operator in the SDM satisfies where satisfies the geodesics Equation (29) with , . Notice the fact that In addition, where the last equality holds by the integration by parts formula. □ We next show the equivalence relation between the Hessian of the relative entropy in the SDM and the classical Gamma two operator. We first demonstrate the relation among , and the gradient operator of the entropy. In particular, we show that the Fokker–Planck equation is a sub-Riemannian gradient flow in the SDM. Denote the KL divergence as Proposition 10 (Gradient flow). The negative gradient operator in forms In addition, the sub-Riemannian gradient flow of in forms the Fokker–Planck equation: Proof. We first derive the sub-Riemannian gradient operator of the entropy and relative entropy. Notice that Thus, where . Following the gradient flow formulation: we finish the derivation of (33). □ We next demonstrate that the Hessian of the relative entropy (KL divergence) is equivalent to the classical Bakry–Émery calculus. Proposition 11 (Hessian of entropy and Bakry–Émery calculus). Given , , then Proof. We first derive the Hessian of in the SDM. Notice the fact that . For simplicity, we denote . By using (31), we have We next rewrite (34) into the iterative Gamma calculus. We first show that where the fourth equality uses the fact that , while the last equality follows the integration by parts. We secondly show that where the second equality applies the fact that , while the last inequality uses the dual-relation between Kolmogorov operators L and in , i.e., Combining the equality of , we prove the result. □ Remark 10. We remark that the above formulations in terms of hold for both Riemannian and sub-Riemannian density manifolds. Here, the major difference is whether matrix function a is full rank or degenerate. In this sense, all formulas derived in this subsection recover the classical Bakry–Émery calculus. However, the classical Hessian operator of the entropy is not enough to study the convergence behavior of degenerate diffusion processes. Briefly, we use a modified Lyapunov functional and derive a tensor for the gradient flow in the SDM. It provides the convergence rate of the degenerate diffusion process.

In this subsection, we introduce the motivation of our new Gamma z calculus from the SDM viewpoint. Consider the SDM gradient flow (33): When a is a degenerate matrix, the classical relative Fisher information may not be the Lyapunov functional. In other words, along the gradient flow, it is possible that . To handle this issue, a new Lyapunov function is considered. It is to add a new direction z into the relative Fisher information functional. Denote and . Construct We next prove the following proposition. Proposition 12. where with the notation and . Proof. For the simplicity of notation, we denote . Notice the fact that From Proposition 11, we have □ We only need to show the following claim. Claim: Proof of Claim. The proof is similar to the ones in Proposition 11. We need to take care of the z direction. Notice that We next estimate (I) and (II) separately. For (I), we notice the fact that Thus, where the last equality holds by integration by parts. For (II), we have Combining (I) and (II), we have Using the notation , we finish the proof. □ We next prove that and in Definition 1 agree with each other in the weak form along the gradient flow. Proposition 13. Denote , then Proof. To prove the proposition, we rewrite as follows. □ Here, we need to prove the following equality. Claim: Proof of Claim. For the simplicity of notation, let and The following property is also used in the proof. For any smooth test function f and , then Notice that , then Here, Notice the fact that Hence, Similarly, by switching a and z, we have Combining the above derivation, we finish the proof. □ Remark 11. From the proof, we can show the following identity: denote , then Therefore, it is clear that, if the commutative assumption holds, the above quantity equals zero. In this case, This means that, under the commutative assumption, the generalized Gamma z calculus agrees with the classical one [2] in the weak sense. With the generalized Gamma z calculus, we are ready to prove the convergence properties and functional inequalities for degenerate drift–diffusion processes. Proposition 14. Suppose with . Denote as the solution of the sub-Riemannian gradient flow (33), then In addition, the z-log-Sobolev inequalities holds: for any smooth density function ρ. Finally, Proof. Here, the proof is very similar to the one in the previous section. Again, consider the sub-Riemannian gradient flow in the SDM. We know that the log-Sobolev inequality relates to the ratio of and . If we cannot estimate a ratio , then We construct the other Lyapunov function: Thus, along the SDM gradient flow (33), we have If , then The convergence result follows directly from Gronwall’s equality. We next prove the z-log-Sobolev inequality. Since then (36) implies the fact that, denoting , then Thus, . Hence, we prove all the results by the fact that implies . In other words, the generalized Gamma z calculus implies the z-log-Sobolev equality (zLSI): We last prove the exponential convergence in the distance. Notice that We apply an inequality between the KL divergence and distance. In other words, This finishes the proof. □ Remark 12. It is worth mentioning that our derivation of the Gamma z calculus is not a direct Hessian operator of the entropy in the SDM. In fact, it combines both the second-order calculus in the SDM and the property of the Hessian operator of the entropy. See similar relations in the mean-field Bakry–Émery calculus [50].

In this section, we introduce the generalized Gamma z calculus. For any smooth functions , the diffusion operator associated with SDE (2) is denoted as where we denote and When , we denote the diffusion operator as We first define the Carré de Champ operator associated with the above second-order diffusion operators. It is easy to check that , , and L share the same : Similarly, we introduce the operator in the direction of below: Next, we define the iterative and for operator L (, respectively) below: Definition 4. We define the generalized Gamma z for operator L below: For matrices and , we denote the divergence operator as and Here, we denote π as the invariant distribution associated with the operator L. Remark 13. In particular, we have the following local coordinates representation. We first present the following key lemmas. Lemma 10. where X, G are defined in Notation 1 and is defined in Definition 2. Lemma 11. where are introduced in Notation 1 and is defined in Definition 2. Lemma 12. where are introduced in Notation 1 and is defined in Definition 2. We then have the following main theorem. In order to distinguish the operators L and , we rewrite Theorem 1 as below, and with some abuse of notation, we denote and . Theorem 3 (z-Bochner’s formula). For smooth function , assume that Assumption 1 holds, then where All the terms are defined in Notation 1 and Definition 2. Proof. By Definition 4 and Formulae (39) and (40), we have We compute the above terms explicitly in the following four steps. Step 1: The term follows from Lemma 11. We are left with the other two terms: and Step 2: The term follows from Lemma 12. We are left to compute the last two terms: and Step 3: Following Lemma 10, which will be proven shortly in the next section, we have where X, G are defined in Notation 1 and is defined in Definition 2. Step 4: Combining the above terms in Lemma 11, in Lemma 12, and , we have Assuming that Assumption 1 is satisfied, we obtain Adding the drift terms from Step 1 and Step 2, we obtain and , which finishes the proof. □

Lemma 13. where X, G are defined in Notation 1 and is defined in Definition 2. Proof. For the first term in the above lemma, we have where is defined in (42). Plugging in (42), we further obtain By further expanding , we obtain Similarly, we obtain where we also obtain Combining all the terms above, we have By direct computations, we separate the above terms into two groups based on “” and “”. We denote as the sum of all “” terms and denote as the sum of all “” terms. Switching indices for the terms in to match , we obtain the following: The first equality follows from the quantities we obtained previously, the second equality from switching ” to ” and ” to ”, and the third equality from switching between ” and ”, ” and ”. Thus, the proof is completed. □

From now on, we keep the following notation: . Furthermore, we fixed the notation for with relation for and Here, we denote Recall that we define Next, we are ready to prove the following lemma. Lemma 14. where are introduced in Notation 1 and is defined in Definition 2. Proof. We plug in the operator into our definition for : Now, we compute the last two terms of the above equation. With , we obtain and It is easy to see We now expand and into local coordinates: and Applying Lemma 15, which will be proven shortly below, we have where Thus, combining with (47) and (48), we have where the last term follows from Lemma 16 below. The proof is thus completed. □ Lemma 15. Here, the local representations for and are given as follows. For , we denote We introduce the following notation (convention) that, for any function F, Proof of Lemma 15. By our definition above, we have where we denote Therefore, we have Next, we compute the following quantity. where we have We continue with our computation as below: From the above, combining (52) and (53), we further obtain Recall that we denote to emphasize the transpose of the matrix a and : and Subtracting the above two terms, we have Now, we eventually obtain the the following step: Thus, the proof is completed. □ Below, we further investigate the extra term explicitly in the above Lemma 15. Lemma 16. Recall that matrix Q and vectors X, C, and D are defined in Notation 1. Proof. We expand the two terms in Lemma 16. The first term reads as The second term reads as Next, we expand and completely and obtain the following: Additionally, Our next step is to complete the squares for all the above terms. Look at the term first. The terms , , and play the role of crossing terms inside the complete squares. In particular, for convenience, we change the index inside the sum of and , switching for and switching for . Then, we obtain the following. We denote The above Equality (55) can be represented in the following matrix form: where Q and X are defined in (12) and (19). Now, we can represent term as . Next, we want to represent and in the following form in terms of vector X: where C is defined in (14). Similarly, we can represent by X: where D is defined in (1). Summingover the above terms, we have the following quadratic form: Taking into account the fact that and , we have which completes the proof. □

Lemma 17. where is defined in Definition 2. Proof. The proof follows directly from Lemmas 18 and 19. □ Lemma 18. Proof. Step 1: We first define , then we have By our definition above, we directly obtain where we denote We have Next, we compute the following quantity. From Lemma 15, we have We continue with our computation as below: From the above, combining (58) and (59), we further obtain Recall here that we denote to emphasize the transpose of the matrix a and : Subtracting the above two terms, we obtain the following: Now, we eventually end up with the following formula: Step 2: Computation of . Now, we compute the last two terms of the above equation, with : It is easy to see that We now expand and into local coordinates: Combining the above two steps, we thus obtain □ Lemma 19. Proof. We expand the two terms in Lemma 19. Next, we expand and completely and obtain the following: Our next step is to complete the squares for all the above terms. We look at term first. The terms , , and play the role of crossing terms inside the complete squares. In particular, for convenience, we changed the index inside the sum of and , switched for , and switched for , then we obtain the following. We denote The above Equality (63) can be represented in the following matrix form: where P and X are defined in (13) and (19). Now, we can represent term as . Next, we want to represent and in the following form in terms of vector X: where F is defined in (16). Similarly, we can represent by X: where E is defined in (17). We thus have the following form: Taking into account the fact that and , we have which completes the proof. □

In this section, we apply the generalized Gamma calculus to study the entropic inequality for the semi-group associated with the drift–diffusion process. With a little abuse of notation, we denote the generator of the semi-group as instead of L, and we denote as the corresponding diffusion process. Definition 5. We define the semigroup , where L is invariant with respect to the invariant measure . We denote , and where the infinitesimal generator of this process is , and we denote as the product of the transition kernel and the volume measure π. Remark 14. Following the standard treatment as in [2] (Section 5), whenever we consider the differentiating operation on , we shall always consider first with , for . Then, we take the limit as . Throughout this section, we directly use instead of for convenience. Remark 15. In the standard sub-Riemannian setting, the semi-groups are in general defined with respect to the invariant measure . In this paper, we formulate the semi-group and the transition kernel with respect to the Lebesgue measure . Following the framework in [2], we also need the following assumption, which is necessary to rigorously justify the computations on functionals of the heat semigroup. Assumption 2. The semigroup is stochastically complete, that is, for , and for any and with compact support, we assume that We believe that the above Assumption 2 should follow from the the assumption if we assume the appropriate lower bound . We leave this for further studies. Related gradient estimates are presented in order below. For the infinitesimal generator associated with linear semi-group , we have the following property. Proposition 15. For all smooth function f, we have: ; For all functions , the map is continuous from to ; For all one has ; , . Next, we present the entropic inequality under Assumption 1. We follow closely the framework introduced in [2] and define the following two functionals: Lemma 20. We have the following relation: Proof. Denote , and we have the following relation: By direct computation, one obtains where we have ; thus, (66) is proven. Similarly, we obtain the following for : The proof then follows. □ Now, we are ready to present the following important lemma, which prepares us to prove the new entropy inequality without the assumption: Lemma 21. For any , we denote as the transition kernel of diffusion process starting at x defined in Definition 5, and the following equality is satisfied: Here, we denote and Proof. We first expand in the following integral form. We skip for simplicity. Take . Claim 1: Recall that we denote and . Use the following identity: We then obtain Similarly, the other equality is satisfied. Claim 2: First, observe that Similarly, one obtains For the next term, one obtains where the last equality follows from the integration by parts for the first term and the direct expansion of the divergence for the second term. Similarly, we obtain Observing, by integration by parts, we obtain Combining the above formulas, the proof is completed. □ With the above lemma in hand, we are ready to prove the following entropic inequality. We first define the following energy form: Recall that we define Theorem 4. Denote ; if the following condition is satisfied: we then conclude where depends on the estimate of the transition kernel associated with semi-group (see Definition 5). Remark 16. Based on Theorem 3, we can also prove the above theorem for operator with the drift term involved. Since the proof is similar, we skip the proof here. Proof. Take . Let be the diffusion Markov process with semigroup . (Similar proofs can be found in [2] (Proposition 4.5).) Let smooth function be such that, for every , and . We have for every where is a local martingale. Let be an increasing sequence of stopping times such that, almost surely, and is a martingale. We obtain By using the dominated convergence theorem, we obtain Applying the above equality to , we obtain We now look at the term with : By using the above Lemma 21, let , and we obtain Applying Theorem 3 here with as the transition kernel function, we obtain a time-dependent version of Theorem 3. Assume that the following bound is satisfied where the bound depends on kernel : We then conclude with the following bound: Plugging into the time integral , the proof follows. □ Remark 17. We prove the entropic inequality Theorem 4 in this section without the the assumption: . A similar entropic inequality under the assumption was first proven in [2] (Proposition 4.5 and Theorem 5.2). With this new inequality Theorem 4 in hand, similar gradient estimates and other inequalities from [2] follow. We leave them for future studies. Proposition 4.5 in [2] is based on a pointwise estimate given the commutative assumption of and . We removed the commutative assumption, and our estimate is in a weak form, which is presented in the above Lemma 21.