Molecules as metric measure spaces with Kato-bounded Ricci curvature

SetΨ := log(Ψ̃), with Ψ̃> 0 the ground state of an arbitrary molecule with n electrons in the infinite mass limit (neglecting spin/statistics). Let Σ ⊂ R3n be the set of singularities of the underlying Coulomb potential. We show that the metric measure space M given by R3n with its Euclidean distance and the measure μ(dx) = e−2Ψ(x)dx has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function x 7→ |x −Σ|−1, which we show to be an element of the Kato class induced by M . In addition, it is shown that M is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for Ψ̃ by Fournais/Sørensen, as well as Aizenman/Simon’s Harnack inequality for Schrödinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class. Manuscript received 11th September 2019, revised 30th April 2020, accepted 19th May 2020. 1. Metric measure spaces Ever since the pioneering papers by Sturm [25,26] and Lott/Villani [19], which based on the earlier results from [9, 21, 22], metric measure spaces with a Ricci curvature which is bounded below by a constant have been examined in great detail and revealed many deep geometric and analytic results and in particular stability properties. Several equivalent definitions of such a lower bound have been given in the last years, which are typically in the spirit of a (possibly rather technical) convexity assumption on a certain nonlinear functional (like for example the convexity of the entropy functional on Wasserstein space). In the situation of a weighted Riemannian manifold, which is a pair given by Riemannian manifold M and a function Φ : M → R, one canonically ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 596 Batu Güneysu and Max von Renesse gets a metric measure space by taking the geodesic distance on M and the weighted Riemannian volume measure e−2Φ(x)vol(dx). In this case, the above convexity assumptions turn out (in their simplest dimension free form) to be equivalent to the lower boundedness of the Bakry-Emery Ricci curvature [5] RicΦ := Ric+2∇2Φ. (1) The reader may find some of the central results on the geometry and analysis of abstract metric measures with lower bounded Ricci curvature in [2–4,7,8,10,12,17,20] and the references therein. The point we want to make in this note is, that, on the other hand, there exist very natural metric measure spaces whose Ricci curvature is not even locally bounded from below or above by a constant, but by a function which lies in the Kato class of the underlying metric measure space. We believe that our main result, Theorem 7 below, suggests to define and investigate systematically metric measure spaces having a Ricci curvature which is bounded from below and/or above by a Kato function. As we will exclusively work in the setting of metric measure spaces that arise from perturbing the Lebesgue measure of the standard Euclidean metric measure space, we start by briefly recalling the notions from metric measure spaces that will be relevant for us in this special class. In the sequel, we denote with ( · , · ) the Euclidean scalar product and with | · | the associated norm. With C 0,1(Rm) the space of all globally Lipschitz functions on Rm , that is, the space of all f :Rm →C such that sup x 6=y | f (x)− f (y)| |x − y |−1 <∞, we understand C 0,1 loc(R m) to be the space of all f :Rm →C with φ f ∈C 0,1(Rm) for all φ ∈C∞ c (Rm). Assume we are given a real-valued function Φ ∈ C 0,1 loc(R) with ∆Φ ∈ Lloc(R). Note that, in particular, ∇Φ ∈ Lloc(R ,Rm) by Rademacher’s theorem. Definition 1. The metric measure space MΦ is defined by MΦ := (R , | ·− · |,μΦ), where μΦ denotes the measure μΦ(dx) := e−2Φdx. In view of (1), we define the Bakry-Emery Ricci tensor RicΦ of MΦ by RicΦ := 2∇2Φ ∈ Lloc(R ,Matm×m(R) ) . Note that the asserted local square integrability follows from Φ,∆Φ ∈ Lloc(R) and the CalderonZygmund inequality. The Cheeger energy form QΦ of MΦ in the Hilbert space LΦ(R m) is given by QΦ( f ) = 1 2 ∫ |∇ f |dμΦ with domain of definition W 1,2 Φ (Rm) given by all f ∈ LΦ(R) with ∇ f ∈ LΦ(R ,Cm), where for q ∈ [1,∞) the Banach space L Φ (Rm) is defined to be space of the equivalence classes1 Borel functions f on Rm with ∫ | f |q dμΦ <∞, and likewise for vector-valued functions. The weighted Laplacian ∆Φ on MΦ is given by ∆Φ =∆−2(∇Φ, · ) and its domain of definition is W 2,2 Φ (Rm), the space of all f ∈ LΦ(R) with ∆Φ f ∈ LΦ(R), and an integration by parts shows that −∆Φ/2 ≥ 0 is the self-adjoint operator induced by QΦ. As in 1with respect to dx or equivalently dμΦ; note that L ∞ Φ (Rm ) = L∞(Rm ). C. R. Mathématique, 2020, 358, n 5, 595-602 Batu Güneysu and Max von Renesse 597 the proof of Theorem 11.5 in [13] one finds that the operator −∆Φ/2 is essentially self-adjoint on C∞ c (Rm). The space-time function (t , x) 7→ e t 2∆Φ f (x) has a jointly continuous version for all f ∈ LΦ(R), and there exists a uniquely determined a jointly continuous map (0,∞)×Rm ×Rm 3 (t , x, y) 7−→ e t 2∆Φ (x, y) ∈ [0,∞) which for all f ∈ LΦ(R), t > 0, x ∈Rm satisfies e t 2∆Φ f (x) = ∫ e t 2∆Φ (x, y) f (y)dμΦ(y). (2) This integral kernel has the following properties for all s, t > 0, x, y ∈Rm : e t 2∆Φ (y, x) = e t 2∆Φ (x, y), e t+s 2 ∆Φ (x, y) = ∫ e t 2∆Φ (x, y)e s 2∆Φ (y, z)dμΦ(z), ∫ e s 2∆Φ (x, y)dμΦ(y) ≤ 1. In particular, one can extend the definition e t 2∆Φ f (x) to f ∈ ⋃q∈[1,∞] L Φ(R) or all nonnegative Borel f ’s using formula (2). Definition 2. The Kato class KΦ(R) of MΦ is defined by all Borel functions v on Rm satisfying lim t→0 sup x∈Rm ∫ t 0 ∫ e s∆Φ (x, y)|v(y)|dμΦ(y)ds = lim t→0 sup x∈Rm ∫ t 0 e s∆Φ |v(x)|ds = 0. In view of ∫ e s 2∆Φ (x, y)dμΦ(y) ≤ 1 for all s > 0, x ∈Rm , one trivially has L∞(Rm) ⊂KΦ(R), while inclusions of the type L Φ(R) ⊂KΦ(R) will in general depend on the geometry induced byΦ. We refer the reader to [1,14,18,23,24] for several abstract and Riemannian results on the Kato class. Remark 3. A simple result [1, 23] in this context forΦ= 0 is that L (R) ⊂K (R) for all q > m/2 if m ≥ 3, and that if v ∈K (Rm) and if T :Rm →Rm is a surjective linear map, then v ◦T ∈K (Rm ). Let (Ω,F∗, X , (P x Φ)x∈Rm ) be the canonical ∆Φ/2-diffusion, which in general lives on the space of continuous pathsΩwith values in one-point compactification ofRm . In particular, F∗ denotes the canonical filtration ofΩwhich is generated by the coordinate process X t (ω) =ω(t ). For every Borel set A ⊂Rm , t > 0, x ∈Rm one has the defining relation P x Φ{A, t < ζ} = ∫ A e t 2∆Φ (x, y)dμΦ(y), with ζ := inf{t ≥ 0 : X t =∞} :Ω−→ [0,∞] the canonical explosion time. Definition 4. The metric measure space MΦ is called stochastically complete, if one has P x Φ{t < ζ} = 1 for all t > 0, x ∈Rm . Note that forΦ= 0 the above diffusion is nothing but Euclidean Brownian motion. In particular, P x Φ is obtained as the law of the solution X (x) of the Ito equation dX t (x) =−∇Φ(XΦ t (x))dt +dWt , X 0 (x) = x, (3) where W is a Euclidean Brownian motion. C. R. Mathématique, 2020, 358, n 5, 595-602 598 Batu Güneysu and Max von Renesse 2. The metric measure space of a molecule Given n,m ∈N, R = (R1, . . . ,Rm) ∈R3m , Z = (Z1, . . . , Zm) ∈N3m , and the potential V :R3n →R V (x1, . . . ,xn) :=− m ∑ j=1 n ∑ i=1 Z j |xi −R j | + ∑ 1≤i< j≤n 1 |xi −x j | , consider the Hamilton operator −∆/2+V in L2(R3n) of a molecule with n electrons and m nuclei, where in the infinite mass limit the j -th nucleus is considered to be fixed in R j . In addition, we have ignored spin/statistics and we have set the elementary charge equal to 1. This operator is essentially self-adjoint on C∞ c (R3n) [16] and its domain of definition is W 2,2(R3n). Let λ> 0 be the corresponding ground state energy and 0 < Ψ̃ ∈W 2,2(R3n) the ground state, so (−∆/2+V )Ψ̃=λΨ̃. We know from [16] that Ψ̃ ∈C 0,1(R3n). In fact, there exists a constant C > 0 such that |∇Ψ̃| ≤C on R \Σ, where Σ := { x ∈R3n : m ∏ j=1 n ∏ i=1 |xi −R j | ∏ 1≤i< j≤N |xi −x j | = 0 } is the set of singularities of the underlying Coulomb potential, a closed set of measure zero. Of course, by local elliptic regularity, Ψ̃ ∈C∞(R3n \Σ). Definition 5. WithΨ := log(Ψ̃), the metric measure space MΨ = (R , | ·− · |,μΨ), is called a molecular metric measure space. Remarks 6. (1) ClearlyΨ ∈C 0,1 loc(R) and it follows from the formula ∆Ψ= (1/Ψ̃)∆Ψ̃− (1/Ψ̃) 3n ∑ j=1 (∂i Ψ̃) 2 in combination with |∂i Ψ̃| ≤C , the continuity of Ψ̃ and∆Ψ̃ ∈ L2(R3n) that∆Ψ ∈ Lloc(R), as required for the theory from the previous section. (2) The operator −∆/2+V in L2(R3n) is unitarily equivalent to the operator −∆Ψ/2+λ in LΨ(R 3n) via LΨ(R ) −→ L(R), UΨ f (x) = e f (x). Indeed, as both operators are essentially self-adjoint on C∞ c (R3m) in their respective Hilbert spaces, it suffices to check UΨ (−∆Ψ/2+λ) f = (−∆/2+V )UΨ f for all f ∈C∞ c (R), which is a standard calculation. (3) It is well-known [1,23] that V ∈K (R3n) (in fact, this is a simple consequence of Remark 3). Here comes our main result: C. R. Mathématique, 2020, 358, n 5, 595-602 Batu Güneysu and Max von Renesse 599 Theorem 7. (a) There exist constants A1, A2 > 0 such that ∣∣RicΨ(x)v∣∣≤ (A1 + A2|x −Σ|−1)|v |, for all (x, v) ∈ (R \Σ)×R3n . (b) One has | ·−Σ|−1 ∈KΨ(R). (c) MΨ is stochastically complete. Proof. Before we come to the actual proof of the statements, we record some central results for atomic Schrödinger operators: (i) As V ∈K (R3m), Aizenman/Simon’s Harnack inequality [1] for positive eigenfunctions of Schrödinger operators with Kato potentials states the existence of a constant C such that sup {y∈R3n :|y−x|<1/2} Ψ̃(y) ≤C inf {y∈R3n |y−x|<1/2} Ψ̃(y) for all x ∈R3n . (ii) Recently established estimates by Fournais/Sørensen (see also [15] for |α| = 1) state that for every multi-index α there exists [11] a constant cα with |∂αΨ̃(x)| ≤ cαmin(1, |x −Σ|)1−|α| sup {y∈R3n :|y−x|<1/2} Ψ̃(y) for all x ∈R3n \Σ. (iii) As a consequence of (i), (ii) we get that for every multi-index α one has |∂αΨ̃(x)| Ψ̃(x) ≤ cαmin(1, |x −Σ|)1−|α| sup{y∈R3n :|y−x|<1/2} Ψ̃(y) inf{y∈R3n :|y−x|<1/2} Ψ̃(y) ≤ c ′ αC min(1, |x −Σ|)1−α, for all x ∈R3n \Σ. (iv) For all q ∈ [1,∞) the process


Metric measure spaces
Ever since the pioneering papers by Sturm [25,26] and Lott/Villani [19], which based on the earlier results from [9,21,22], metric measure spaces with a Ricci curvature which is bounded below by a constant have been examined in great detail and revealed many deep geometric and analytic results and in particular stability properties. Several equivalent definitions of such a lower bound have been given in the last years, which are typically in the spirit of a (possibly rather technical) convexity assumption on a certain nonlinear functional (like for example the convexity of the entropy functional on Wasserstein space). In the situation of a weighted Riemannian manifold, which is a pair given by Riemannian manifold M and a function Φ : M → R, one canonically gets a metric measure space by taking the geodesic distance on M and the weighted Riemannian volume measure e −2Φ(x) vol(dx). In this case, the above convexity assumptions turn out (in their simplest dimension free form) to be equivalent to the lower boundedness of the Bakry-Emery Ricci curvature [5] Ric Φ := Ric + 2∇ 2 Φ. (1) The reader may find some of the central results on the geometry and analysis of abstract metric measures with lower bounded Ricci curvature in [2][3][4]7,8,10,12,17,20] and the references therein. The point we want to make in this note is, that, on the other hand, there exist very natural metric measure spaces whose Ricci curvature is not even locally bounded from below or above by a constant, but by a function which lies in the Kato class of the underlying metric measure space. We believe that our main result, Theorem 7 below, suggests to define and investigate systematically metric measure spaces having a Ricci curvature which is bounded from below and/or above by a Kato function.
As we will exclusively work in the setting of metric measure spaces that arise from perturbing the Lebesgue measure of the standard Euclidean metric measure space, we start by briefly recalling the notions from metric measure spaces that will be relevant for us in this special class.
In the sequel, we denote with ( · , · ) the Euclidean scalar product and with | · | the associated norm. With C 0,1 (R m ) the space of all globally Lipschitz functions on R m , that is, the space of all f : R m → C such that sup . Assume we are given a real-valued function Φ ∈ C 0,1 loc (R m ) with ∆Φ ∈ L 2 loc (R m ). Note that, in particular, ∇Φ ∈ L ∞ loc (R m , R m ) by Rademacher's theorem.
In view of (1), we define the Bakry-Emery Ricci tensor Ric Φ of M Φ by Ric Φ := 2∇ 2 Φ ∈ L 2 loc (R m , Mat m×m (R) . Note that the asserted local square integrability follows from Φ, ∆Φ ∈ L 2 loc (R m ) and the Calderon- and likewise for vector-valued functions. The weighted Laplacian ∆ Φ on M Φ is given by and its domain of definition is and an integration by parts shows that −∆ Φ /2 ≥ 0 is the self-adjoint operator induced by Q Φ . As in the proof of Theorem 11.5 in [13] one finds that the operator −∆ Φ /2 is essentially self-adjoint , and there exists a uniquely determined a jointly continuous map This integral kernel has the following properties for all s, t > 0, x, y ∈ R m : In particular, one can extend the definition e or all nonnegative Borel f 's using formula (2).
will in general depend on the geometry induced by Φ. We refer the reader to [1,14,18,23,24] for several abstract and Riemannian results on the Kato class.

Remark 3.
A simple result [1,23] in this context for Φ = 0 is that Let (Ω, F * , X , (P x Φ ) x∈R m ) be the canonical ∆ Φ /2-diffusion, which in general lives on the space of continuous paths Ω with values in one-point compactification of R m . In particular, F * denotes the canonical filtration of Ω which is generated by the coordinate process X t (ω) = ω(t ). For every the canonical explosion time.
Note that for Φ = 0 the above diffusion is nothing but Euclidean Brownian motion. In particular, P x Φ is obtained as the law of the solution X Φ (x) of the Ito equation where W is a Euclidean Brownian motion.

The metric measure space of a molecule
Given n, m ∈ N, R = (R 1 , . . . , R m ) ∈ R 3m , Z = (Z 1 , . . . , Z m ) ∈ N 3m , and the potential V : consider the Hamilton operator −∆/2+V in L 2 (R 3n ) of a molecule with n electrons and m nuclei, where in the infinite mass limit the j -th nucleus is considered to be fixed in R j . In addition, we have ignored spin/statistics and we have set the elementary charge equal to 1. This operator is essentially self-adjoint on C ∞ c (R 3n ) [16] and its domain of definition is W 2,2 (R 3n ). Let λ > 0 be the corresponding ground state energy and 0 < Ψ ∈ W 2,2 (R 3n ) the ground state, so We know from [16] that Ψ ∈ C 0,1 (R 3n ). In fact, there exists a constant C > 0 such that is the set of singularities of the underlying Coulomb potential, a closed set of measure zero. Of course, by local elliptic regularity, Ψ ∈ C ∞ (R 3n \ Σ).

Definition 5.
With Ψ := log( Ψ), the metric measure space is called a molecular metric measure space.
(2) The operator −∆/2 + V in L 2 (R 3n ) is unitarily equivalent to the operator −∆ Ψ /2 + λ in . Indeed, as both operators are essentially self-adjoint on C ∞ c (R 3m ) in their respective Hilbert spaces, it suffices to check which is a standard calculation. (3) It is well-known [1,23] that V ∈ K (R 3n ) (in fact, this is a simple consequence of Remark 3).
Here comes our main result: (a) There exist constants A 1 , A 2 > 0 such that Proof. Before we come to the actual proof of the statements, we record some central results for atomic Schrödinger operators: (ii) Recently established estimates by Fournais/Sørensen (see also [15] for |α| = 1) state that for every multi-index α there exists [11] a constant c α with (iii) As a consequence of (i), (ii) we get that for every multi-index α one has Indeed, by Novikov's theorem it suffices to show that for all t > 0 one has which trivially follows from by inequality (iii). (v) Girsanov's theorem and uniqueness in law for the solution of the corresponding SDE imply that for all t > 0, x ∈ R 3n we have (a). One calculates In view of the above inequality (iii), the absolute value of the first summand can be controlled by ≤ C | · −Σ| −1 , while the absolute value of the second summand can be estimated by a constant, again using (iii) above.
(b). Using (v), for all q ∈ (1, ∞) with q * ∈ (1, ∞) its Hölder dual, and all t ≥ s ≥ 0, and all x, e s 2 ∆ Ψ (x, y)|y − Σ| −1 dµ(y) where we have estimated as follows: using (5). By (iv) we have and by Jenen's inequality, for all t ≤ 1, In in order to estimate this, we record that for all y = (y 1 , . . . , y n ) ∈ R 3n one has and the latter function of y is in K (R 3n ) for some q > 1 by Remark 3. Indeed, fix 1 < q < 6/5. Then for all R ∈ R 3 the function Thus for some q > 1 we arrive at as t → 0+, which completes the proof of (b).

Remark 8.
Based on the results from [27] (see also [6]), one can prove that the global Bismut-Elworthy-Li derivative formula holds for the gradient of e t 2 ∆ Ψ f , f ∈ L ∞ (R 3n ), where Ψ is an appropriately chosen mollification of Ψ. Then, taking → 0+, using an appropriate form of Mosco convergence, one can obtain the following Lipschitz smoothing property: The (partially technical) details will be given elsewhere.
As a corollary to the stochastic completeness of M Ψ and the Feynman-Kac formula we get the following seemingly completely new formula: Corollary 9. For all t > 0, x ∈ R 3n one has E x e − t 0 V (X s )ds / Ψ(X t ) = e −t λ / Ψ(x).
Proof. Let (K n ) be a compact exhaustion of R 3n . Then for f n := 1 K n e −Ψ ∈ L 2 (R 3n ) we have e −t λ e −Ψ(x) e t ∆ Ψ /2 (e Ψ f n )(x) = e −t (−∆/2+V ) f n (x) = E x e − t 0 V (X s )ds f n (X t ) , where the first equality follows from the unitary equivalence from Remark 6 and the second from the Feynman-Kac formula [23]. Taking n → ∞ the RHS of (7) tends to the LHS of (6) by monotone convergence. The LHS of (7) is equal to e −t λ e −Ψ(x) e t ∆ Ψ /2 (e Ψ f n )(x) = e −t λ Ψ(x) K n e t ∆ Ψ /2 (x, y)dµ Ψ (y), which tends to e −t λ / Ψ(x) by monotone convergence and the stochastic completeness of M Ψ .