Abstract
Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup \(P_{t}\), we give two different stochastic representations of \(dP_{t} f\) for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of \(\frac {1}{2} L\), while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry.
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This work has been supported by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746). The first author supported by project 249980/F20 of the Norwegian Research Council.
Appendix A: Feynman-Kac Formula for Perturbations of Self-Adjoint Operators
Appendix A: Feynman-Kac Formula for Perturbations of Self-Adjoint Operators
1.1 A.1 Essentially Self-Adjoint Operator on Forms
Let M be a manifold with a sub-Riemannian structure (H, gH) with H bracket-generating. Consider the rough sub-Laplacian L = L(∇) relative to some affine connection ∇ on TM. Let g be a complete sub-Riemannian metric taming gH such that ∇g = 0. Assume that
We then have the following statement for operators of the type \(L - \mathcal {C}\) where \(\mathcal {C} \in {\Gamma }(End(T^{*}M))\). To simplify notation, we denote \(\langle \kern .8pt\displaystyle \cdot \kern .8pt, \kern .8pt\displaystyle \cdot \kern .8pt \rangle _{L^{2}(\wedge ^{j} g^{*})}\) as simply 〈 ⋅ , ⋅ 〉 for the rest of this section.
Lemma A.1
Assume that \(\mathcal {C}^{*} = \mathcal {C}\). If \(\mathcal {A} =L- \mathcal {C}\)isbounded from above on compactly supported forms,i.e.if
then\(\mathcal {A}\)is essentially self-adjoint on compactly supported one-forms.
We follow the argument of [40, Section 2]. We begin by introducing the following lemma.
Lemma A.2
[37, Section X.1] Let \(\mathcal {A}\)beany closed, symmetric, densely defined operator on a Hilbert space withdomain \(\text {Dom}(\mathcal {A})\). Assume that \(\mathcal {A}\)isbounded from above by \(\lambda _{0}(\mathcal {A})\)onits domain. Then \(\mathcal {A} = \mathcal {A}^{*}\)ifand only if there are no eigenvectors in the domainof \(\mathcal {A}^{*}\)witheigenvalue \(\lambda > \lambda _{0}(\mathcal {A})\).
Proof of Lemma A.1
Let \(\text {pr}_{H}\) be the orthogonal projection to H. Since \(L = - (\nabla _{\text {pr}_{H}})^{*}(\nabla _{\text {pr}_{H}})\), we have \(- \langle \mathcal {C} \alpha , \alpha \rangle \leq \lambda _{0} \langle \alpha , \alpha \rangle \). Denote the closure of \(\mathcal {A}| {\Gamma }_{c}(T^{*}M)\) by \(\mathcal {A}\) as well. Assume that there exists a one-form α in L2 satisfying \(\mathcal {A}^{*} \alpha = \lambda \alpha \) with \(\lambda > \lambda _{0}\). We know that α is smooth, since L is hypoelliptic. To see the latter, consider any point x ∈ M, and let U be a neighborhood of x such that we can trivialize \(T^{*}M\). Recalling the definition of step from Section 2.1, let r denote the step of H at x. Relative to the trivialization, we have that L equals ΔH along with terms of lower order derivatives in horizontal directions in each component, so by possibly shrinking U, we have that L is maximal hypoelliptic of degree 1/r and hence hypoelliptic on this neighborhood, see [28, Chapter 1] for details. As it is a local property, L is hypoelliptic globally. Let f be an arbitrary function of compact support and write \(d_{H} f = \text {pr}_{H}^{*} df\). Then
Since \((\lambda - \lambda _{0}) \langle f^{2} \alpha ,\alpha \rangle \geq 0\), we have
and hence
Since we assumed that g was complete, there exists a sequence of smooth functions \(f_{j} \uparrow 1\) of compact support satisfying \(\| df_{j}\|_{L^{\infty }(g^{*})} \to 0\). By inserting \(f_{j}\) in (A.1) and taking the limit we obtain \(\| \nabla _{\text {pr}_{H\displaystyle \cdot \kern .8pt}} \alpha \|_{L^{2}(g^{*})}^{2} = - \langle L \alpha , \alpha \rangle = 0\). However, this contradicts our initial hypothesis \(\mathcal {A}^{*} \alpha = \lambda \alpha \) for \(\lambda > \lambda _{0}\). Hence, we obtain our result.
Remark A.3
By replacing the sequence \(f_{j}\) in the proof of Lemma A.1 with (an appropriately smooth approximation of) the sequence found in [41, Theorem 7.3], we can deduce essential self-adjointness of \(L - \mathcal {C}\) just by assuming completeness of \({\sf d}_{g_{H}}\).
1.2 A.2 Stochastic Representation of a Semigroup
Let \((M, H, g_{H})\) be a sub-Riemannian manifold and let g be a complete Riemannian metric taming \(g_{H}\). Define \(L^{2}(T^{*}M)\) as the space of all one-forms in \(L^{2}\) relative to g. Let ∇ be a connection satisfying ∇ g = and \(L^{*} = L\). Relative to L(∇), consider the stochastic flow \(X_{t}(\kern .8pt\displaystyle \cdot \kern .8pt)\) with explosion time τ(⋅). Define \(/\!/_{t}(x)\) as parallel transport along Xt(x) with respect to ∇.
Let \(\mathcal {C}\) be a zero order operator on M, with
Lemma A.4
Assumethat \(L - \mathcal {C}^{s}\)isbounded from above and assumethat \(\mathcal {C}^{a}\)isbounded. For each \(x\), let \(Q_{t}(x) \in End T_{x}^{*}M\)bea continuous process adapted to the filtrationof \(X_{t}(x)\)suchthat for any \(\alpha \in {\Gamma }_{c}(T_{x}^{*}M)\), wehave
where\(\stackrel {\textit {\text {loc.\,m.}}}{=}\)denotes equality modulo differentials of local martingales.
Then there exists a strongly continuous semigroup\(P_{t}^{(1)}\)on\(L^{2}(T^{*}M)\)such that for any\(\alpha \in L^{2}(T^{*}M)\),
and such that\(\lim _{t \downarrow 0} \frac {d}{dt} P_{t}^{(1)} \alpha = (L - \mathcal {C}) \alpha \)for any\(\alpha \in {\Gamma }_{c}(TM)\).
For the proof, we need to consider a special class of Volterra operators. To this end, we follow the arguments of [21, Section III.1]. Let \(\mathfrak {B}\) be a Banach space and let \(\mathcal {L}(\mathfrak {B})\) be the space of all bounded operators on \(\mathfrak {B}\) with the strong operator topology. Consider any strongly continuous semigroup \(\mathbb {R}_{\geq 0} \to \mathcal {L}(\mathfrak {B})\), \(t \mapsto S_{t}\) and let \(\mathcal {A}\colon \mathfrak {B} \to \mathfrak {B}\) be a bounded operator. We define the corresponding Volterra operator \(\mathsf {V}(S; \mathcal {A})\) on continuous functions \(\mathbb {R}_{\geq 0} \to \mathcal {L}(\mathfrak {B})\), \((t, \alpha ) \mapsto F_{t} \alpha \) by
and introduce the operator \(\mathsf {T}(S; \mathcal {A})\) by
The operator \(\mathsf {T}(S; \mathcal {A})\) is well defined, and if \(S_{t}\) has generator \((L, \text {Dom}(L))\) then \(\tilde S_{t} := (\mathsf {T}(S; \mathcal {A}) S)_{t}\) defines a strongly continuous semigroup with generator \((L + \mathcal {A}, \text {Dom}(L))\).
Proof
By Lemma A.1 the operator \(L - \mathcal {C}^{s}\) is essentially self-adjoint. Let \({P^{s}_{t}}\) be the corresponding semigroup on \(L^{2}(T^{*}M)\) with domain \(\text {Dom}^{s} = \text {Dom}(L- \mathcal {C}^{s})\).
Let \(D^{n}\) be an exhausting sequence of \(M\) of relative compact domains, see e.g. [17, Appendix B.1] for construction. Consider the Friedrichs extension \(({\Lambda }^{n}, \text {Dom}({\Lambda }^{n}))\) of \(L- \mathcal {C}^{s}\) restricted to compactly supported forms on \(D^{n}\) and let \(\tilde {P^{n}_{t}}\) be the corresponding semigroup defined by the spectral theorem. Since the operators \({\Lambda }^{n}\) are bounded from above by assumption, the semigroups \(\tilde P^{n}\) are strongly continuous by [21, Chapter II.3 c]. Define \({P_{t}^{s}}\) similarly with respect to the unique self-adjoint extension of \(L-\mathcal {C}^{s}\) restricted to compactly supported forms. Let \(({\Lambda }, \text {Dom}({\Lambda }))\) denote the generator of \({P_{t}^{s}}\) and note that for any compactly supported forms \(\alpha \), we have that \(\tilde {P_{t}^{n}} \alpha \) converge to \({P_{t}^{s}} \alpha \) in \(L^{2}(T^{*}M)\), by e.g. [31, Chapter VIII.3.3]. Define \({P^{n}_{t}} = (\mathsf {T}(\tilde P^{n}; \mathcal {A})\tilde P^{n})_{t}\) and finally \(P_{t}^{(1)} = (\mathsf {T}(P^{s}; \mathcal {C}^{a}) P^{s})_{t}\). These semigroups are strongly continuous with respective generators \(({\Lambda }^{n} + \mathcal {C}^{a}, \text {Dom}({\Lambda }^{n} ))\) and \(({\Lambda } + \mathcal {C}^{a}, \text {Dom}({\Lambda }))\). Furthermore, \({P_{t}^{n}}\alpha \) converge to \(P_{t}^{(1)}\alpha \) in \(L^{2}(TM)\) by [31, Theorem IV.2.23 (c)].
For \(x \in M\), let \(\tau _{n}(x)\) denote the first exist time for \(X_{t}(x)\) of the domain Dn. For any form \(\alpha \) with support in \(D^{k}\), we have that for \(S > 0\) and \(n \geq k\),
is a bounded local martingale, giving us
Taking the limit, and using that \({P^{n}_{t}}\) converges to \(P_{t}^{(1)}\), we obtain
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Grong, E., Thalmaier, A. Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds. Potential Anal 51, 219–254 (2019). https://doi.org/10.1007/s11118-018-9710-x
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DOI: https://doi.org/10.1007/s11118-018-9710-x