A comparison principle based on couplings of partial integro-differential operators
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Della Corte, Serena; Fuchs, Fabian; Kraaij, Richard; Nendel, Max Working Paper A comparison principle based on couplings of partial integro-differential operators Center for Mathematical Economics Working Papers, No. 696 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Della Corte, Serena; Fuchs, Fabian; Kraaij, Richard; Nendel, Max (2024) : A comparison principle based on couplings of partial integro-differential operators, Center for Mathematical Economics Working Papers, No. 696, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29941902 This Version is available at: https://hdl.handle.net/10419/306536 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
696 October 2024 A COMPARISON PRINCIPLE BASED ON COUPLINGS OF PARTIAL INTEGRO-DIFFERENTIAL OPERATORS Serena Della Corte, Fabian Fuchs, Richard C. Kraaij, and Max Nendel Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en
A COMPARISON PRINCIPLE BASED ON COUPLINGS OF PARTIAL INTEGRO-DIFFERENTIAL OPERATORS SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Abstract. This paper is concerned with a comparison principle for viscosity solutions to Hamilton–Jacobi (HJ), –Bellman (HJB), and –Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii–Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes. Keywords: Comparison principle, viscosity solution, Hamilton–Jacobi-Bellman–Isaacs equation, coupling of operators, Lyapunov function, Jensen perturbation, mixed topology. MSC 2020 classification: Primary 35J60; 35D40; 45K05; Secondary 49L25; 49Q22. 1. Introduction In this work, we provide a new perspective on comparison principles for viscosity solutions to the Hamilton–Jacobi equation f−λHf =h, λ > 0, h ∈Cb(Rq),(1.1) for Hamiltonians Hof the type Hf(x) = ⟨b(x),∇f(x)⟩+1 2Tr ΣΣT(x)D2f(x) +Zf(x+z)−f(x)−χB1(0)(z)⟨z,∇f(x)⟩µx(dz) + H(∇f(x)) (1.2) and, more generally, for those in Bellman and Isaacs form Hf(x) = sup θ∈Θ{Hθf(x)−I(x, θ)}and Hf(x) = sup θ1∈Θ1 inf θ2∈Θ2{Hθ1,θ2f(x)−I(x, θ1, θ2)}, with Hθand Hθ1,θ2as in (1.2) but with θand (θ1, θ2)dependent coefficients, respectively, and an appropriate cost functional I. Motivated by convex Hamiltonians, for which no unique classical or weak solutions exist in general, [17] introduced the notion of viscosity solutions. The seminal works [13], [15], [26], [27], [31] explore this framework for first-order equations. Date: October 28, 2024. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1283/2 2021 – 317210226 and by The Netherlands Organisation for Scientific Research (NWO), grant number 613.009.148. 1 arXiv:2410.19566v1 [math.AP] 25 Oct 2024
2 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Most modern comparison proofs for operators containing second-order terms are based on results of [28], [29]. Using then recent advances for generalized differentials, [14] provided what is nowadays known as the Crandall–Ishii Lemma. An overview over uniqueness results for viscosity solutions to degenerate elliptic equations is given in the User’s Guide [16]. The treatment of non-local operators was initially motivated by problems in optimal control theory; see [2], [3], [34] for early examples with non-local operators. The work [5] gives a non-local version of the Crandall–Ishii Lemma by adapting the original procedure in [16], and [23] extends these results to unbounded solutions. We also refer to [20] for an overview of the Hilbertian setting, [8] for comparison principles for convex monotone semigroups on spaces of continuous functions, to [18] for the classical well-posedness of convex Cauchy problems on Lp, to [25] for a comparison principle in the framework of G-Lévy processes, and to [7] for a comparison principle for HJB equations on the set of probability measures. Our approach and our main results, Theorem 3.1 and Corollary 3.2, innovate upon classical comparison principles in the following three ways: (1) We reinterpret the classical doubling-of-variables method in the context of secondorder equations by casting the Crandall–Ishii Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as generators of Lévy processes, in the same framework as secondorder operators, paving the way for stability results. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to both continuous and discrete operators. We point out that [15] also discusses a coupling point of view, but only for first order operators. (3) We strengthen the typical comparison principle using Lyapunov functionals from a sup-norm contractivity result to what we call the strict comparison principle, cf. Definition 2.4, which encodes continuity in the strict or sometimes also called mixed topology, cf. [9], [33]. The results are illustrated in various examples in Section 4. To introduce the first two innovations, we heuristically trace back the classical doubling-of-variables procedure used to obtain comparison principles for first and second-order equations. For the sake of exposition, we focus on the θ-independent case. Given a subsolution uand supersolution vto an equation of type (1.1) and, for α > 1, optimizers (xα, yα)to u(xα)−v(yα)−α 2d2(xα, yα) = sup x,y∈Rqnu(x)−v(y)−α 2d2(x, y)o,(1.3) one estimates sup x∈Rq u(x)−v(x)≤h(xα)−h(yα)+λhHα 2d2(·, yα)(xα)−H−α 2d2(xα,·)(yα)i. Consequently, comparison then holds, if lim inf α→∞ Hα 2d2(·, yα)(xα)−H−α 2d2(xα,·)(yα)≤0.(1.4) The estimate (1.4), then translates into explicit conditions on H. When His, for example, of the form Hf(x) = ⟨b(x),∇f(x)⟩+1 2|∇f(x)|2,
A COMPARISON PRINCIPLE BASED ON COUPLINGS 3 the estimate (1.4) translates into Hα 2d2(·, yα)(xα)−H−α 2d2(xα,·)(yα) =⟨b(xα), α(xα−yα)⟩+α2 2d2(xα, yα)−⟨b(yα), α(xα−yα)⟩+α2 2d2(xα, yα) ≤ ⟨b(xα)−b(yα), α(xα−yα)⟩, which goes to 0for α→ ∞, if bis one-sided Lipschitz. For second order operators, however, the same strategy fails since, considering, for example, the Laplacian Hf(x) = 1 2∆f(x) = 1 2Tr D2f(x), we get Hα 2d2(·, yα)(xα)−H−α 2d2(xα,·)(yα)=2α, which diverges as α→ ∞. The works [28], [29] use the key insight that, while the first order-viscosity solution method explores the sequences of optimizers of (1.3) separately (fix yαand vary xfor the subsolution part and vice versa), for second order equations, one needs to treat the two sequences jointly. This insight was later formalized in [14] and as Theorem 3.2 in the User’s Guide [16], now known as the Crandall–Ishii Lemma. The lemma states for equations of type Hf(x) = 1 2Tr D2f(x)that, given Xα=D2u(xα)and Yα=D2v(yα) or their appropriate generalizations, we have the estimate Xα0 0−Yα≤3α 1 − 1 − 1 1 . Conjugating the matrices with C:=1 √2 1 1 1 1 ,(1.5) i.e. essentially using Cto couple the subsolution and supersolution problems, we arrive at the desired estimate 1 2Tr(Xα)−1 2Tr(Yα) = 1 4Tr Xα−YαXα−Yα Xα−YαXα−Yα≤0.(1.6) We now briefly describe the three innovations (1)–(3). Innovation 1: A test function framework. Examining the proof of the Crandall– Ishii Lemma, we can interpret the procedure as the construction of two test functions ϕα, ψα∈C2(Rq)that are squeezed between uand von one-hand and α 2d2on the other. To be more precise, we find ϕα, ψα∈C2(Rq)such that u(xα)−ϕα(xα) = sup x∈Rq{u(x)−ϕα(x)}and v(yα)−ψα(yα) = inf y∈Rq{v(y)−ψα(y)}, and ϕα(xα)−ψα(yα)−α 2d2(xα, yα) = sup x,y∈Rqnu(x)−v(y)−α 2d2(x, y)o.(1.7) As before, comparison now follows from the estimate lim inf α→∞ Hϕα(xα)−Hψα(yα)≤0. For the Laplacian Hf(x) = 1 2Tr D2f(x), this translates to Hϕα(xα)−Hψα(yα) = 1 2Tr(D2ϕα(xα)) −1 2Tr(D2ψα(yα)).(1.8) At this point in proofs using the Crandall–Ishii Lemma, the estimate (1.6) is performed by conjugation with the matrix Cin (1.5). We formalize this step by adapting the
4 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL probabilistic notion of couplings, cf. [10], [30], [37], and identify the choice of the matrix Cin (1.5) with the synchronous coupling (also called co-monotone coupling). Innovation 2: The coupling approach. Indeed, given two Brownian motions starting in xand y, one can construct a coupling of the two by considering (X(t), Y (t)) = (x+B(t), y +B(t)),(1.9) where B(t)is a standard Brownian motion. The generator of the coupled process (1.9) is given by b Hg(x, y) := 1 2(∂x+∂y)2g(x, y) = 1 2Tr 1 1 1 1 D2g(x, y)=1 2Tr CD2g(x, y)CT, where we recover the matrix Cof (1.5). Note that b His indeed a coupling: For f1, f2∈ Cb(Rq)and (f1⊕f2)(x, y) := f1(x) + f2(y), we have b H(f1⊕f2)(x, y) = Hf1(x) + Hf2(y). Using the coupling b H, we can now rewrite (1.8) as Hϕα(xα)−Hψα(yα) = b H(ϕα⊕−ψα) (xα, yα) ≤b Hα 2d2(xα, yα)=0,(1.10) where the first equality follows by the definition of a coupling, the inequality is based on the positive maximum principle with the optimizers from equation (1.7), and the final equality is due to the fact that the synchronous coupling controls distance growth. A similar strategy can be used to treat a discretized version of the Brownian Motion by considering the generator Hf(x) = 1 2[f(x+ 1) −f(x)] + 1 2[f(x−1) −f(x)] of a random walk: We synchronously couple the random walk with itself using the operator b Hf(x, y) = 1 2[f(x+ 1, y + 1) −f(x, y)] + 1 2[f(x−1, y −1) −f(x, y)] . The argument in (1.10) then works for the random walk exactly as it did for the Brownian motion. This coupling approach is one of the main contributions of this paper, allowing for a unifying framework to show comparison for Hamilton–Jacobi equations with Hamiltonians of type (1.2) and their Bellman and Isaacs versions, cf. Theorem 3.1 and Corollary 3.2. Innovation 3: The strict comparison principle. Our third innovation is on the final estimate that is obtained as the comparison principle. For a subsolution uto f−λHf =h1 and a supersolution vto f−λHf =h2 the comparison principle amounts to establishing that sup x∈Rq u(x)−v(x)≤sup x∈Rq h1(x)−h2(x). The comparison principle, once established, thus implies sup-norm contractivity for the solution map R(λ) : Cb(Rq)→Cb(Rq), where R(λ)his the unique viscosity solution for the Hamilton–Jacobi equation (1.1). It is well-known from examples, cf. [4], [11], [22], [39], that the map R(λ)htakes the form of an exponentially discounted Markovian control problem. If the dynamics admits a Lyapunov function V, having compact sublevel sets and satisfying HV ≤c, then the controlled Markov processes satisfy tightness properties. More precisely, if the controlled process starts in a compact set K, one can find, for any time horizon T > 0
A COMPARISON PRINCIPLE BASED ON COUPLINGS 5 and ε > 0, a compact set b K⊇K, given in terms of the sublevel sets of Vsuch that, with probability 1−ε, the process remains in b Kup to time T. Rewriting this in terms of an estimate on the solution map R(λ), we then find sup x∈K R(λ)h1(x)−R(λ)h2(x)≤ε||h1−h2||+ sup x∈ b K h1(x)−h2(x).(1.11) Estimates of this type are indeed characterized by the strict topology, as was first established for linear functionals in [33, Theorem 5.1] and for convex, monotone functionals in [32, Corollary 2.10]. Note that in this paper, we do not establish convexity of h7→ R(λ)h, but want to point out that given a convex H, convexity of R(λ)his to be expected by performing a comparison principle in terms of three variables using variants of the, e.g., three dimensional Theorem 3.2 of [16], see also the domination principle of Theorem 2.22 and Corollary 2.26 of [23]. We leave this for future work. Building upon the notion of Lyapunov functions, we will show that we can directly establish a variant of (1.11) for a subsolution uand a supersolution v. Given its motivation, we will call this estimate the strict comparison principle, see Definition 2.4 and the main result, Theorem 3.1, below. Organization of the paper. The rest of the paper is organized as follows: Section 2 introduces the notation and definitions. Section 3introduces the framework by stating the necessary assumptions and formalizing the main results. In Section 4, we show how to apply our framework to operators of the form (1.2). Section 5contains the construction of the required optimizing points and test functions. Finally, Section 6 contains the proof of the main theorems. 2. Preliminaries and general setting 2.1. Notation and Preliminaries. Throughout the paper, let q∈Nand E=Rq. We write C(E)for the set of all real-valued continuous functions on E, where Eis endowed with the topology induced by the Euclidean distance don Rq. Let C(E)and Cb(E)be the set of continuous and bounded continuous functions. For k∈N, let Ck(E)denote the space of all real-valued functions on Ethat are ktimes continuously differentiable. Let Ck b(E)the set of all functions in Ck(E)with bounded derivatives up to order k. We denote the space of all smooth functions that are constant outside of a compact set by C∞ c(E). We write Cu(E)and Cl(E)for the set of continuous functions on Ethat are uniformly bounded from above and below, respectively. Moreover, we write C+(E) := {f∈C(E)|fhas compact sub-level sets}, C−(E) := {f∈C(E)|fhas compact super-level sets}, Cc(E) := {f∈C(E)|fis constant outside of a compact set}. We furthermore define the following intersections: C2 c(E) = Cc(E)∩C2(E), C2 +(E):=C+(E)∩C2(E), C2 −(E):=C−(E)∩C2(E). For a, b ∈R, we write a∨b:= max{a, b}and a∧b:= min{a, b}. We denote the supremum norm by ||·||, that is ||f|| = sup x∈E|f(x)|, for f∈Cb(E), while, for u∈C(E), we use the notation ⌈u⌉:= sup x∈E u(x),⌊u⌋:= inf x∈Eu(x)
6 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL for a supremum or infimum over the entire space and ⌈u⌉C:= sup x∈C u(x),⌊u⌋C:= inf x∈Cu(x) for a supremum or infimum over a subset C⊆E. We say that a function ω: [0,∞)→[0,∞)is a modulus of continuity, if ωis upper semi-continuous with ω(0) = 0. We say that a function f∈C(E)admits a modulus of continuity, if, for every compact K⊆E, there exists a modulus of continuity ωK: [0,∞)→[0,∞)such that, for all x, y ∈K, we have |f(x)−f(y)| ≤ ωK(d(x, y)). A function ϕ:E→Ris called semi-convex with constant κ∈Rif for any x0∈E the map x7→ ϕ(x) + κ 2d2(x, x0) is convex. Moreover, ϕis called semi-concave with constant κ∈Rif −ϕis semi-convex with constant −κ. We say that a function f∈C(E, Rq)is one-sided Lipschitz if, for all x, y ∈Eand some constant C∈R, we have ⟨x−y, f(x)−f(y)⟩ ≤ Cd2(x, y). For any z∈E, let sz:E→Rqbe the shift map sz(x) = x−z. For any z1, z2∈E, let dz1,z2(x, y):=d(sz1(x), sz2(y)) . Let f1, f2∈C(E). Then, we define the direct sum f1⊕f2, f1⊖f2∈C(E×E)as (f1⊕f2)(x1, x2):=f1(x1) + f2(x2)and (f1⊖f2)(x1, x2):=f1(x1)−f2(x2) for all x1, x2∈E. For two sets of functions F1, F2⊆C(E), we define F1⊕F2:={f1⊕f2|f1∈F1, f2∈F2}and F1⊖F2:={f1⊖f2|f1∈F1, f2∈F2}. 2.2. Operator notions. We consider operators H⊆C(E)×C(E), where we identify Hby its graph. As usual, the domain of His given by D(H):={f∈C(E)|∃g∈C(E): (f, g)∈H}. Let H1, H2⊆C(E)×C(E). We define H1+H2:={(f, g1+g2)|(f, g1)∈H1,(f, g2)∈H2}, which is an operator with domain D(H1+H2):=D(H1)∩D(H2). We say that His linear on its domain if, for any f, g ∈ D(H)and a∈Rsuch that af +g∈ D(H), we have H(af +g) = aHf +Hg. We will prove the comparison principle for the equation in terms of Hby relating it to two equations in terms of two restrictions of H. To do so, we will need to be able to construct test functions in the domain of Hfrom functions in the domain of the restrictions. In particular, we will need the following notion. Definition 2.1 (Sequential Denseness).Let D ⊆ Cb(E),D+⊆C+(E), and D−⊆ C−(E).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 7 •We say that Dis upward sequentially dense in D+if, for any f†∈ D+and constant a∈R, there exists a function f†,a ∈ D such that (f†,a(x) = f†(x)if f†(x)≤a, a < f†,a(x)≤f†(x)if f†(x)> a. •We say that Dis downward sequentially dense in D−if, for any f‡∈ D−and constant a∈R, there exists a function f‡,a ∈ D such that (f‡,a(x) = f†(x)if f‡(x)≥a, a > f‡,a(x)≥f‡(x)if f‡(x)< a. 2.3. Viscosity solutions. For λ > 0, consider h1∈Cl(E)and h2∈Cu(E)and two operators H1⊆Cl(E)×C(E)and H2⊆Cu(E)×C(E). We study the pair of equations f−λH1f≤h1,(2.1) f−λH2f≥h2.(2.2) The notion of viscosity solution is built upon the maximum principle. Definition 2.2 (Maximum principle).We say that an operator H⊆C(E)×C(E) satisfies the maximum principle if, for all f1, f2∈ D(H)and x0∈Ewith f1(x0)−f2(x0) = sup x∈E{f1(x)−f2(x)}, we have Hf1(x0)≤Hf2(x0) and, analogously, for all f1, f2∈ D(H)and x0∈Ewith f1(x0)−f2(x0) = inf x∈E{f1(x)−f2(x)}, we have Hf1(x0)≥Hf2(x0). Observe that every operator H⊆C(E)×C(E)that satisfies the maximum principle is single-valued, i.e., for all f∈ D(H), #{g∈C(E)|(f, g)∈H}= 1. Definition 2.3 (Viscosity suband supersolutions).Let H1⊆Cl(E)×C(E)and H2⊆Cu(E)×C(E)be two operators with domains D(H1)and D(H2), respectively. Moreover, let λ > 0,h1∈Cl(E), and h2∈Cu(E). (a) A bounded, upper semicontinuous function u:E→Ris called a (viscosity) subsolution to (2.1) if, for all (f, g)∈H1, there exists a sequence (xn)n∈N⊆E such that lim n→∞ u(xn)−f(xn) = sup x∈E u(x)−f(x), lim sup n→∞ u(xn)−λg(xn)−h1(xn)≤0. (b) A bounded, lower semicontinuous function v:E→Ris called a (viscosity) supersolution to (2.2) if, for all (f, g)∈H2, there exists a sequence (xn)n∈N⊆E such that lim n→∞ v(xn)−f(xn) = inf x∈Ev(x)−f(x), lim inf n→∞ v(xn)−λg(xn)−h2(xn)≥0.
14 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL its domain and compatible with V,{ζz,p}z∈E,p∈Rq, and {ζz}z∈Eand with associated controlled growth coupling b A:= b A1+b A2. Remark 4.3. Let B1,B2⊆C(E)×C(E)be compatible with V,{ζz,p}z∈E,p∈Rq, and {ζz}z∈Eand convex semi-monotone operators. Then B:= B1+B2is compatible with V,{ζz,p}z∈E,p∈Rq, and {ζz}z∈Eand convex semi-monotone operator. The rest of this section is organized as follows: •In Section 4.1, we consider drift terms and convex first-order Hamiltonians; •In Section 4.2, we consider diffusion operators; •In Section 4.3, we consider integral operators. 4.1. Deterministic Example: Drift terms and convex first-order Hamiltonians. In this section, we consider the deterministic part of the operator (4.1). Proposition 4.4. Suppose that Bis given by Bf(x) = ⟨b(x),∇f(x)⟩+H(∇f(x)) with the drift term x7→ b(x)locally, one-sided Lipschitz with constant Lb,K and ||b(x)|| ≤ cb 2(1 + ||x||)for some constant cb>0, and p7→ H(p)continuous and convex. Then, Bis compatible with both collections of Definition 4.1, cf. Assumption 3.5 (b), and convex semi-monotone. Furthermore, V= log(1 + x2 2)is a Lyapunov function: sup x∈E BV(x)<∞. Proof. Convex semi-monotonicity: Clearly, Bis locally first-order with Bf(x) = ⟨b(x),∇f(x)⟩+H(∇f(x)) = B(x, ∇f(x)). Additionally, for any compact set K⊆E α > 0, and x, x′∈K, we have B(x, α(x−x′)) −B(y, α(x−x′)) = b(x), α(x−x′)+H(α(x−x′)) −b(x′), α(x−x′)−H(α(x−x′)) =b(x)−b(x′), α(x−x′) +H(α(x−x′)) −H(α(x−x′)) ≤αLb,Kd2(x, x′), establishing semi-monotonicity. As convexity of p7→ B(x, p)is immediate, we conclude that Bis convex semi-monotone. Lyapunov control: Using that V(x) = log 1 + x2 2,∇V(x) = 2x 2+|x|2is bounded as a function of x,bhas linear growth, and that His continuous, we find that sup x∈E BV(x) = sup x∈Eb(x),2x 2 + |x|2+H2x 2 + |x|2<∞. Compatibility: We show the compatibility of B, cf. Assumption 3.5 (b), by evaluation of the perturbation and containment function in the operator. Using ξz(x) = 1 2d2(x, z)and ζz,p(x) = ⟨p, x −z⟩, we find for z0, z1, z ∈Eand p∈B1(0) B(Ξz0,p,z1◦sz)(x) = ⟨b(x),(x−z−z0) + p+ (x−z−z1)⟩ +H((x−z−z0) + p+ (x−z−z1)) , which is continuous in (x, z0, p, z1, z)as band Hare continuous. For V(x) = log 1 + 1 2x2 and z∈E, we find B(V◦sz)(x) = b(x),2(x−z) 2 + |x−z|2+H2(x−z) 2 + |x−z|2, which is continuous in (x, z)as band Hare continuous. Thus, Bis compatible. □
A COMPARISON PRINCIPLE BASED ON COUPLINGS 15 4.2. Stochastic Example: Diffusion operators. In this section, we focus on diffusion operators of the form Af(x) = 1 2Tr Σ(x)ΣT(x)D2f(x), where Σ(x)is a positive semi-definite matrix for each fixed x∈E. Our main goal is to construct a controlled growth coupling for the operator A. To illustrate the idea behind our approach, consider the simpler case of the Laplacian operator A0f(x) = 1 2Tr(D2f(x)), which is the infinitesimal generator of Brownian motion. The well-known synchronous coupling of two Brownian motions started from xand x′, respectively, is given by (X(t), X′(t)) = (x+B(t), x′+B(t)) with B(t)a standard Brownian motion, having generator b A0g(x, x′) = 1 2(∂x+∂x′)2g(x, x′), which satisfies b A0d2= 0. Aiming to generalize this, we rewrite b A0g(x, x′) = Tr CCTD2g(x, x′)with C=1 √2 1 1 1 1 . In general we obtain the following result. Proposition 4.5. Suppose that Ais given by Af(x) = 1 2Tr Σ(x)ΣT(x)D2f(x) with Σ(x)positive semi-definite for all x∈E,x7→ Σ(x)locally Lipschitz with constant LΣ,K and ||b(x)|| ≤ cΣ 2(1 + ||x||)for some constant cΣ>0. Consider b Af(x, y) := Tr b Σ2(x, x′)D2f(x, x′), where b Σ2(x, y) := Σ(x)ΣT(x) Σ(x′)ΣT(x) Σ(x)ΣT(x′) Σ(x′)ΣT(x′). Then, Ais compatible, cf. Assumption 3.5 (a), linear on its domain, and admitting the controlled growth coupling b A. Furthermore, V= log(1 + x2 2)is a Lyapunov function: sup x AV(x)<∞. For the proof we make use of the following auxiliary lemma. Lemma 4.6. For each x∈E, let B(x)be a positive semi-definite matrix and consider Af(x) = 1 2Tr B(x)D2f(x). For any x, x′∈E, let b B(x, x′)be a positive semi-definite matrix having block-structure b B(x, x′) = B(x)B(x, x′) B(x, x′)TB(x′). Define b Af(x, x′) := 1 2Tr b B(x, x′)D2f(x, x′). Then, b Ais a coupling of A.
16 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Proof. b A(f1⊕f2)(x, y) = 1 2Tr b B(x, y)D2(f1⊕f2)(x, y) =1 2Tr B(x)D2f(x)+1 2Tr B(y)D2f(y) =Af1+Af2 and it satisfies the maximum principle. □ Proof of Proposition 4.5.Controlled growth coupling: By Lemma 4.6,b Ais a coupling for A. We thus verify that b Ahas controlled growth. Consider α > 1,K⊆Ea compact set, and x, x′, y, y′∈K. Then, b Aα 2d2 x−y,x′−y′(x, x′) = 1 2Tr b Σ2(x, x′)D2α 2d2 x−y,x′−y′(x, x′) =1 2Tr b Σ2(x, x′)α 1 − 1 − 1 1 (x, x′) =α 2Tr((ΣT(x)−ΣT(y))(Σ(x)−Σ(y))) ≤αL2 Σ,Kd2(x, x′), establishing controlled growth. Lyapunov control: Using V(x) = log(1+ x2 2)and the fact that Σhas linear growth, we find that sup x∈E AV(x) = sup x∈E 1 2Tr Σ(x)ΣT(x)D2V(x)<∞.(4.2) Compatibility: Using ξz(x) = 1 2d2(x, z),ζz,p(x) = ⟨p, x −z⟩and V(x) = log(1 + x2 2), we find for z0, z1, z ∈Eand p∈B1(0) A(Ξ ◦sz)(x) = 2 Tr(Σ(x)ΣT(x)), A(V◦sz)(x) = 1 2Tr Σ(x)ΣT(x)D2(V◦sz)(x), which, by an analogous calculation as in equation (4.2), is continuous in (x, z0, p, z1, z) and (x, z). Consequently, Ais compatible. □ 4.3. Stochastic Example: Integral operators. In this section, we cover examples of spatially inhomogeneous Lévy processes that have generators of the type Af(x) = Zf(x+z)−f(x)−χB1(0)(z)⟨z,∇f(x)⟩µx(dz),(4.3) where χB1(0)(z) = l(|z|)for some smooth non-decreasing function lsatisfying l= 1 on a neighborhood of 0and l(r) = 0 for r≥1. We next specify the space from which we can take our jump measures µx. For this, we need to control the mass close to 0as for large values of z. The following function controls both: W(z) := χB1(0)(z)|z|2+ (1 −χB1(0)(z)) log 1 + |z|2. We take the family of jump measures {µx}x∈Efrom the set of equivalence classes MW(Rq) := M(Rq)/∼with M(Rq):=µ∈M(Rq)ZW(z)µ(dz)<∞, where M(Rq)is the set of all Borel measures on Rqand where µ∼νif and only if µ|Rq\{0}=ν|Rq\{0}.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 17 We topologize the set MW(Rq)by the weak topology σWinduced by the pairings µ7→ Zg(z)µ(dz)∀g∈CW,(4.4) where CW:=(g∈C(Rq)g(0) = 0,and sup z=0 |g(z)| W(z)<∞.). Below, we construct controlled growth couplings for operators of the type (4.3). To clarify the concepts, we consider the example of an uncompensated process, i.e., having an operator of the type Af(x) = Zf(x+z)−f(x)µx(dz). Couplings for this type of operator are of the form b Af(x, x′) = Zf(x+z1, y +z2)−f(x, x′)πx,x′(dz1,z2),(4.5) where πx,x′couples µxand µx′. In the following example, we illustrate the need of being able to couple jumps synchronously. Example 4.7 (Random Walk).Consider the simple random walk on Rmaking jumps of size 1, i.e µx=µ=δ−1+δ1leading to the operator Af(x) = [f(x−1) + f(x+ 1) −2f(x)] . Well known couplings include walks with simultaneous jumps but independent directions, fully independent jumps, and synchronous jumps. The corresponding generators are given as in (4.5) with jump measures π1:= µ⊗µ, π2:= δ(−1,0) +δ(1,0) +δ(0,−1) +δ(0,1), π3:= δ(−1,−1) +δ(1,1), respectively. This leads to the operators b A1f(x, x′) = f(x−1, x′−1) + f(x−1, x′+ 1) +f(x+ 1, x′−1) + f(x+ 1, x′+ 1) −4f(x, x′), b A2f(x, x′) = f(x−1, x′) + f(x+ 1, x′)−2f(x, x′) +f(x, x′−1) + f(x, x′+ 1) −2f(x, x′), b A3f(x, x′) = f(x−1, x′−1) + f(x+ 1, x′+ 1) −2f(x, x′). Only for the final example, we see that b A3d2≤0, pointing at the necessity of the alignment of jumps. Note that the third coupling above has different total mass, and we thus work outside the realm of the typical notion of couplings of probability measures. A second feature of coupling jump measures, not present in the example above, is that we can make one process jump, whereas the other does not. We formalize this in the following definition. Definition 4.8. Let µ, ν ∈ MW(Rq). We say that π∈M(Rq×Rq)is an extended coupling of µand ν, if π((A\{0})×Rq) = µ(A\{0})∀A∈ B(Rq), π(Rq×(B\{0})) = ν(B\{0})∀B∈ B(Rq).
18 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Remark 4.9. A variant of this coupling was introduced in [21]. There mass can be moved to the boundary of a domain. In our context, this boundary is the point 0. Definition 4.10. Let x7→ µxbe a map from Einto M(Rq). Let (x, x′)7→ πx,x′be a map from E2into M(Rq×Rq). (a) We say that (x, x′)7→ πx,x′is an extended coupling of x7→ µx, if for all x, x′∈E, we have that πx,x′is an extended coupling of µxand µ′ x. (b) We say that (x, x′)7→ πx,x′is locally Lipschitz, if, for any compact set K⊆E, there exits a constant Lπ,K such that, for x, x′∈K, we have Zd2(z1,z2)πx,x′(dz1,dz2)≤Lπ,Kd2(x, x′). Remark 4.11. Note that conditions (12),(34), and (35) in [5] for µand jcorrespond to our choice of MW(Rq)and locally Lipschitz extended coupling πx,x′. Remark 4.12. Let η:R→Rbe any locally Lipschitz map with local Lipschitz constants Lη,K. Set µx:= δη(x) 1 η(x)=0(x)and πx,x′=δ(η(x),η(x′)). Then, (x, x′)7→ πx,x′ is a locally Lipschitz coupling of x7→ µxwith Lπ,K =Lη,K. The main proposition of this subsection below aims to show that integral operators of the form (4.3) can be treated analogous to the other examples above. We work with the second collection of penalization functions, cf. Definition 4.1, to avoid integrability issues. Proposition 4.13. Consider Af(x) = Zf(x+z)−f(x)−χB1(0) ⟨z,∇f(x)⟩µx(dz). Suppose there exists a σW-continuous map x7→ µxin MW(Rq), cf. (4.4), and that there exists a locally Lipschitz extended coupling (x, x′)7→ πx,x′of x7→ µxwith Lipschitz constant Lπ,K and, for bχ(z1,z2) := χB1(0)(z1)χB1(0)(z2), set b Ag(x, x′) := Zhg(x+z1, x′+z2)−g(x, x′) −bχ(z1,z2)(z1,z2)T,∇g(x, x′)iπx,x′(dz1,dz2). Assume furthermore that sup x∈EZlog 1 + 1 2|z|2+⟨x, z⟩ 1 + 1 2|x|2!µx(dz)<∞. Then, Ais compatible, cf. Definition 3.5 (a), and linear on its domain admitting the controlled growth coupling b A. Furthermore, V= log(1 + x2 2)is a Lyapunov function: sup x∈E AV(x)<∞. Remark 4.14. Corresponding to Remark 3.3, we refer to [6, Corollary 2.3] for a uniqueness result for a Lévy process martingale problem. The proof of Proposition 4.13 is based on the following two auxiliary lemmas. In the first, we obtain bounds on the integrand of our operator acting on the Lyapunov function V. In the second, we compute the integrand of our Lévy type operator acting on the shifted squared metric. We prove these two lemmas following the proof of Proposition 4.13.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 19 Lemma 4.15. Fix x, z ∈E. (a) For z∈Rq, we have −log 1 + 1 2|x−z|2≤V◦sz(x+z)−V◦sz(x) ≤log 1 + 1 2|z|2+⟨x−z, z⟩ 1 + 1 2(x−z)2! ≤log 1 + |z|2. (b) For z∈B1(0), we have |V◦sz(x+z)−V◦sz(x)−⟨z,∇(V◦sz)(x)⟩| ≤ 1 2|z|2. Lemma 4.16. We have 1 2d2 x−y,x′−y′(x+z1, x′+z2)−1 2d2 x−y,x′−y′(x, x′) −bχ(z1,z2)z1 z2,∇1 2d2 x−y,x′−y′(x, x′) ≤1−1 2bχ(z1,z2)d2(z1,z2) + (1 −bχ(z1,z2))1 2d2(y, y′). Proof of Proposition 4.13.Controlled growth coupling: As (x, x′)7→ πx,x′is a locally Lipschitz extended coupling of x7→ µx, cf. Definition 4.10, we have that b Ais a coupling. Thus, we need to verify the controlled growth property of b A. Let x, x′, y, y′∈Kfor K⊆Ea compact set. Using Lemma 4.16, we then have b Aα 2d2 x−y,x′−y′(x, x′)≤α 2Z1−1 2bχ(z1,z2)d2(z1,z2)πx,x′(dz1,dz2) +α 2Z(1 −bχ(z1,z2))1 2d2(y, y′)πx,x′(dz1,dz2) ≤α 2Lπ,Kd2(x, x′) + α 4c′ πd2(y, y′), where the second inequality is due to the local Lipschitz property of the map (x, x′)7→ πx,x′and c′ π>0exists since, for every x, x′∈E,πx,x′∈M(Rq×Rq). As such, A admits the controlled growth coupling b A. Lyapunov control: Using Lemma 4.15, we find sup x∈E AV(x)≤sup x∈EZ(1 −χB1(0)) log(1 + |z|2) + χB1(0)|z|2µx(dz)<∞. Compatibility: We start by establishing the continuity of (x, z)7→ A(V◦sz)(x). Let (xn, zn)converge to (x, z). We aim to apply Lemma C.1 with X=Rq\{0},νn=µxn, and ϕn(z) := V◦szn(xn+z)−V◦szn(xn)−χB1(0) ⟨z,∇(V◦szn)(xn)⟩, ϕ∞(z) := V◦sz(x+z)−V◦sz(x)−χB1(0) ⟨z,∇(V◦sz)(x)⟩. As ϕnis continuous, it remains to show that supn∈Nsupz=0 |ϕn(z)| W(z)<∞. By Lemma 4.15, we can estimate |ϕn(z)| ≤ χB1(0) 1 2|z|2+ (1 −χB1(0)) max −log 1 + 1 2|xn−zn|2,log 1 + |z|2. Since (xn, zn)is convergent, hence bounded, we obtain the desired estimate. Continuity of (x, z)7→ A(V◦sz)(x)now follows by Lemma C.1.
20 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Using the particular form of Ξz0,p,z1, cf. Definition 4.1, one readily verifies that the map (x, z0, p, z1, z)7→ A(Ξz0,p,z1◦sz) (x)is continuous with an analogous argumentation. □ Proof of Lemma 4.15.Let y=x−z, then we can write V◦sz(x+z)−V◦sz(x) = log 1 + 1 2(y+z)2−log 1 + 1 2|y|2= log 1 + 1 2|z|2+⟨y, z⟩ 1 + 1 2|y|2!. Applying Young’s inequality to ⟨y, z⟩leads to the upper bound V◦sz(x+z)−V◦sz(x)≤log 1 + |z|2+1 2|y|2 1 + 1 2|y|2!= log 2 + |z|2−1 1 + 1 2|y|2!≤log 1 + |z|2. Using that the first term is positive, we obtain the lower bound V◦sz(x+z)−V◦sz(x)≥log 1− 1 2|y|2 1 + 1 2|y|2!=−log 1 + 1 2|y|2. This establishes (a). For the proof of (b), we apply Taylor’s Theorem to obtain |V◦sz(x+z)−V◦sz(x)−⟨∇(V◦sz)(x),z⟩| ≤ 1 2|z|2sup z∈B1(0) sup i,j ∇2 i,jV(y+z) ≤1 2|z|2, which follows by a direct inspection of ∇2 i,jV(x) = 2δi,j 1 + 1 2|x|2−2xixj (1 + 1 2|x|2)2. □ Proof of Lemma 4.16.Evaluating the shift maps, calculating the gradient of the squared Euclidean distance, and expanding the squares leads to 1 2d2 x−y,x′−y′(x+z1, x′+z2)−1 2d2 x−y,x′−y′(x, x′) −bχ(z1,z2)z1 z2,∇1 2d2 x−y,x′−y′(x, x′) =1 2d2(y+z1, y′+z2)−1 2d2(y, y′)−bχ(z1,z2)y−y′,z1−z2 =1 2d2(z1,z2) + y−y′,z1−z2−bχ(z1,z2)y−y′,z1−z2 ≤1−1 2bχ(z1,z2)d2(z1,z2) + (1 −bχ(z1,z2))1 2d2(y, y′), where in the second equality we use properties of the Euclidean distance dand the final line is due to Young’s inequality. □ 5. Construction of test functions In classical proofs of comparison principles, the approach to estimate sup u−vfor a subsolution uand supersolution vis variable doubling or quadruplication, cf. [4, Theorem 3.1] or [16, introduction of Section 3]: For α > 1 sup x∈E u(x)−v(x)≤sup x,x′∈E u(x)−v(x′)−α 2d2(x, x′).(5.1)
A COMPARISON PRINCIPLE BASED ON COUPLINGS 21 Letting α→ ∞, forces optimizing points, if they exist, of the right-hand side together. In addition, by varying either of the two components, one obtains basic test functions in terms of α 2d2for the use in the definition of the suband supersolution properties of uand v. To ensure that optimizers in (5.1) exist, we will consider instead, for small ε > 0, the following problem that includes the containment function Vand upper bounds sup u−v up to a term of order ε: sup x∈E 1 1−εu(x)−1 1 + εv(x) ≤sup x,x′∈E 1 1−εu(x)−1 1 + εv(x′)−α 2d2(x, x′)−ε 1−εV(x)−ε 1 + εV(x′).(5.2) The particular form of the factors 1−εand 1 + εis motivated by convexity based arguments, which will show up in the proofs of Proposition 6.3 and Theorem 3.1 below. The procedure in (5.2) would be sufficient for a standard, first-order Hamilton–Jacobi equation. The test functions produced by this procedure, however, will not be sufficient to treat second-order or integral operators. This problem was considered in [16] and [5]. We will follow their approach by considering a quadruplication of variables, which we also phrase in terms of supand inf-convolutions. We then perform a Jensen-type perturbation. As we aim to unify proofs for both integral and differential operators, we revisit the full proof and state our result in terms of test functions. In Propositions 5.1 and 5.3 below, which can be considered to be an extended twovariable variant of the Crandall–Ishii construction [16, Theorem 3.2], we start out by considering the optimization (5.2) in terms of the supand inf-convolution of uand v, respectively, effectively leading to a quadruplication problem, see (5.3) below. We then perform the Jensen perturbation, see (5.4). The rest of the proposition deals with various properties of the optimizers in relation to uand v. In Proposition 5.3, we carry out an additional layer of smoothing operations to obtain C∞-test functions. Consequently, we can move away from the notion of solutions in terms of suband superjets, which is of paramount importance to effectively treat diffusive and jump-type processes in a common framework. For readability, we express suprema and infima using ⌈·⌉ and ⌊·⌋, respectively, as defined in Section 2.1. Proposition 5.1 (Construction of optimizers).Let ube bounded and upper semicontinuous, vbe bounded and lower semi-continuous, Vbe a containment function as in Definition 2.12, and {ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C1(E)be collections of functions as in Definition 2.13. Fix ε∈(0,1) and φ∈(0,1]. Then, there exist compact sets Kε,0⊆Kε⊆Eand, for any α > 1, three pairs of variables (yα,0, y′ α,0),(yα, y′ α),(xα, x′ α)in E2and pα, p′ α∈B1/α(0) such that the following four sets of properties hold. Properties of yα,0, y′ α,0: The variables yα,0, y′ α,0optimize ⌈Λα⌉, where Λα(y, y′) := 1 1−εPα[u](y)−1 1 + εPα[v](y′)−α 2d2(y, y′) −ε 1−ε(1 −φ)V(y)−ε 1 + ε(1 −φ)V(y′)(5.3) and satisfy the following property (a) yα,0, y′ α,0∈Kε,0.
22 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Properties of yα, y′ αand pα, p′ α: The pair yα, y′ αoptimizes Λα−ε 1−εφΞ0 1−ε 1 + εφΞ0 2(5.4) and uniquely optimizes Λα−ε 1−εφΞ1−ε 1 + εφΞ2(5.5) where Λαis as in (5.3)and Ξ0 1(y):= Ξ0 yα,0,pα(y),Ξ0 2(y′):= Ξ0 y′ α,0,p′ α(y′), Ξ1(y):= Ξyα,0,pα,yα(y),Ξ2(y′):= Ξy′ α,0,p′ α,y′ α(y′) as in Definition 2.13. Moreover, the optimizers yα, y′ αof (5.4)and (5.5)satisfy (b) We have d(yα, yα,0)≤1 α, d(y′ α, y′ α,0)≤1 α. (c) Pα[u]and Pα[v]are twice differentiable in yαand y′ α, respectively. Properties of xα, x′ α: The variables xα, x′ αoptimize Pα[u](yα) = u(xα)−α 2d2(xα, yα), Pα[v](y′ α) = v(x′ α) + α 2d2(x′ α, y′ α), and satisfy (d) xαand x′ αare the unique optimizers in the definition of Pα[u](yα)and Pα[v](y′ α), respectively. (e) We have that u(xα)−Pα[u]◦sxα−yα(xα) = ⌈u−Pα[u]◦sxα−yα⌉, v(x′ α)−Pα[v]◦sx′ α−y′ α(x′ α) = v−Pα[v]◦sx′ α−y′ α. Behaviour as α→ ∞: (f) We have limα→∞ αd2(yα,0, y′ α,0)=0. (g) We have lim α→∞ αd(xα, yα) + dyα, y′ α+dy′ α, x′ α2= 0. (h) xα, yα, y′ α, x′ α∈Kε. In addition, the following estimate on u−vholds: For any compact set K⊆E, there is a compact set b K=b K(K, ε, u, v)given by b K:= z∈EV(z)≤||u||+||v|| ε+⌈V⌉K, such that (i) For any compact set K⊆E, ⌈u−v⌉K≤1 1−εu(xα)−1 1 + εv(xα) + ε(cε,φ +o(1)) , where cε,φ := 2 1−ε2(1 −φ)⌈V⌉K−1 1−εu−1 1 + εvK , and o(1) is in terms of α→ ∞ for fixed εand φ.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 23 (j) Any limit point of the sequence (xα, yα, yα,0, y′ α,0, y′ α, x′ α)as α→ ∞ is of the form (z, z, z, z, z, z)with z∈b K. Figure 1visualizes the relation between the different optimizing points. xαyαy′ αx′ α yα,0y′ α,0 5.2.(b) 5.1.(g) 5.2.(b) 5.1.(b) 5.1.(f) 5.1.(b) Figure 1. Relation between the optimizing points with a note which parts of the propositions give us distance control. The proof of Proposition 5.1 uses various properties of supand inf-convolutions, which we gather in the next lemma. Its proof is relegated to Appendix D.2. Lemma 5.2. Let u:E→Rbe bounded and upper semi-continuous and v:E→Rbe bounded and lower semi-continuous. For α > 1, set Pα[u](y):= sup x∈Enu(x)−α 2d2(x, y)o=lu−α 2d2(·, y)m,(5.6) Pα[v](y):= inf x∈Env(x) + α 2d2(x, y)o=ju+α 2d2(·, y)k.(5.7) Then, (a) we have ||Pα[u]|| ≤ ||u|| and ||Pα[v]|| ≤ ||v||. (b) for any x, y ∈Esuch that Pα[u](y) = u(x)−α 2d2(x, y), we have α 2d2(x, y)≤u(x)−u(y). Similarly, for any x, y ∈Ewith Pα[v](y) = v(x) + α 2d2(x, y), we have α 2d2(x, y)≤v(y)−v(x). (c) Pα[u]and −Pα[v]are decreasing in α. (d) Pα[u]and −Pα[v]are semi-convex with semi-convexity constant α. As a consequence, both are locally Lipschitz continuous. (e) if Pα[u]is differentiable at y0, then there exists a unique optimizer x0in (5.6) such that Pα[u](y0) = u(x0)−α 2d2(x0, y0) and DPα[u](y0) = α(x0−y0). Similarly, if Pα[v]is differentiable at y0, then there is a unique optimizer x0in (5.7)such that Pα[v](y0) = v(x0) + α 2d2(x0, y0) and DPα[v](y0) = −α(x0−y0). Proof of Proposition 5.1.Proof of (a):As uand vare bounded, by Lemma 5.2 (a), the same holds for ||Pα[u]|| and ||Pα[v]||. Using that Vhas compact sublevelsets, cf. Definition 2.12, the existence of optimizers (yα,0, y′ α,0)for ⌈Λα⌉follows. The definition of Λαand the convolutions Pα[u]and Pα[v]imply that ε 1−ε(1 −φ)V(yα,0) + ε 1 + ε(1 −φ)V(y′ α,0)≤1 1−ε⌈u⌉− 1 1 + ε⌊v⌋−⌈Λα⌉.(5.8)
30 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Case y∈Ac 1and y′∈A2:We have b f1(y)−b f2(y′)−α 2d2(y, y′)≤b f1(y)−b f2(y′) <f1(yα)−f2(y′ α)−⌊f2⌋−α 2d2(yα, y′ α)−f2(y′) =f1(yα)−f2(y′ α)−α 2d2(yα, y′ α)−f2(y′)−⌊f2⌋ ≤f1(yα)−f2(y′ α)−α 2d2(yα, y′ α). Case y∈A1and y′∈Ac 2:Follows analogously to the case y∈Ac 1and y′∈A2. Case y∈Ac 1and y′∈Ac 2:We have b f1(y)−b f2(y′)−α 2d2(y, y′) ≤b f1(y)−b f2(y′) <f1(yα)−f2(y′ α)−⌊f2⌋−α 2d2(yα, y′ α) −f2(y′ α)+(⌈f1⌉−f1(yα)) + α 2d2(yα, y′ α) ≤f1(yα)−f2(y′ α)−2α 2d2(yα, y′ α)−f2(y′ α)−⌊f2⌋−(⌈f1⌉−f1(yα)) ≤f1(yα)−f2(y′ α)−α 2d2(yα, y′ α). We conclude that the pair (yα, y′ α)is also the unique optimizer of lb f1−b f2−α 2d2m. Applying the shift maps sxα−yαand sx′ α−y′ α, respectively, we find that (xα, x′ α)uniquely optimize lf1◦sxα−yα−f2◦sx′ α−y′ α−α 2d2 xα−yα,x′ α−y′ αm. Additionally, as M1≥m1and M2≤m2, we have b f1,b f2∈C∞ c(E), establishing (a). We next prove (b). As r≤Ω− M1(r), 1 1−εPα[u](y)−ε 1−ε(1 −φ)V(y)−ε 1−εφΞ1(y)=Π1(y)≤Ω− M1◦Π1(y)≤b f1(y), which, after rearrangement of terms, implies (b). We proceed with the proof of (c). By (b) and Proposition 5.1 (e), f†(x)−f†(xα) = b f†◦sxα−yα(x)−b f†◦sxα−yα(xα) ≥(Pα[u]◦sxα−yα) (x)−(Pα[u]◦sxα−yα) (xα) ≥u(x)−α 2d2(x, sxα−yα(x)−u(xα)−α 2d2(xα, sxα−yα(xα) =u(x)−u(xα) with equality uniquely realized at xα, establishing (c). We conclude with the proof of (d). First of all, note that the equality of first and second order derivatives for f†and b f†as well as for f‡and b f‡follows by the chain rule. The expressions for Db f†(yα)and Db f‡(y′ α)follow from (b) and Proposition 5.1 (c) and (d).□
A COMPARISON PRINCIPLE BASED ON COUPLINGS 31 6. Proof of the strict comparison principle In this section, we prove Theorem 3.1. The proof is based on a variant of the variable quadruplication procedure on the basis of sup x∈E 1 1−εu(x)−1 1 + εv(x) ≤sup x,,y,y′x′∈E 1 1−εu(x)−1 1 + εv(x′)−α 2(1 −ε)d2(x, y)−α 2d2(y, y′) −α 2(1 + ε)d2(y′, x′)−ε 1 + εV(x)−ε 1 + εV(x′), which we have formalized in terms of test functions f†, f‡in Propositions 5.1 and 5.3. In a first step, we relate suband supersolutions for the Hamilton–Jacobi equation for Hto those for H+and H−: This will be carried out in Lemma 6.1. A second step is to show that f†∈ D(H+)and f‡∈ D(H−): This will be carried out in Lemma 6.2. After establishing these technical points, we proceed to frame the comparison principle in terms of an estimate on H+f† 1−ε−H−f‡ 1 + ε.(6.1) This reduction will be carried out in Proposition 6.3, the statement of which is more involved than typically in the literature, but leads to the improved strict comparison principle. Its formulation and proof hinges on the use of Vas a Lyapunov function. The statements of Lemmas 6.1,6.2, and Proposition 6.3 can be found in Section 6.1, their proofs in Section 6.2. We finish in Section 6.3 by estimating (6.1) in two steps leading to our final result. We first establish in Lemma 6.4 that the pre-factors (1 −ε)−1and (1 + ε)−1work well with the combinations of functions that define f†, f‡in Proposition 5.3. We conclude this section with the proof of Theorem 3.1, where we use this split, the coupling assumption on A, the semi-monotonicity of B, modulus of continuity control on Iand, again, that Vis a Lyapunov function to arrive at our final result. 6.1. Comparison in terms of estimating the difference of Hamiltonians. We start with connecting the notion of suband supersolutions for Hto those for H+and H−, respectively. Lemma 6.1. Let Hand Hsatisfy Assumption 3.4. Then, for any h∈Cb(E)and λ > 0, we have the following: (a) Any viscosity subsolution of f−λHf =his also a viscosity subsolution of f−λH+f=h. (b) Any viscosity supersolution of f−λHf =his also a viscosity supersolution of f−λH−f=h. The proof follows in Section 6.2 below. In the next lemma we show that the test functions that we constructed in the previous section are in the domain of H+and H−. Lemma 6.2. Let Hbe an operator satisfying Assumptions 3.4 and 3.5. Let b f†, f†and b f‡, f‡be as in Proposition 5.3. Then, b f†, f†∈ D(H+)and b f‡, f‡∈ D(H−). The proof of the lemma is outlined in Section 6.2 below. We next state our key proposition, which relates the strict comparison principle to an estimate on the difference of Hamiltonians.
32 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Proposition 6.3. Let H⊆C(E)×C(E)satisfy Assumptions 3.4 and 3.5. Let h1, h2∈ Cb(E), and λ > 0. Consider the equations f−λH+f≤h1,(6.2) f−λH−f≥h2.(6.3) Let uand vby viscosity suband supersolutions to (6.2)and (6.3), respectively. For each ε∈(0,1),φ∈(0,1] and α > 1, consider the construction of optimizers xα, x′ αand test functions f†, f‡as in Propositions 5.1 and 5.3. Suppose there exists a map ε7→ C0 ε, and for any ε∈(0,1) a non-negative map φ7→ Cε,φ satisfying lim supε↓0C0 ε<∞and limφ↓0Cε,φ = 0 such that lim inf α→∞ H+f†(xα) 1−ε−H−f‡(x′ α) 1 + ε≤εC0 ε+Cε,φ.(6.4) Then, for any compact set K⊆Eand ε∈(0,1), sup x∈K u(x)−v(x)≤εCε+ sup x∈ b K h1(x)−h2(x), where b Kε:= b Kε(K, u, v)and Cε:= Cε(K, u, v, h1, h2)are given by b Kε:= z∈EV(z)≤||u||+||v|| ε+⌈V⌉K, Cε:= λC0 ε+2 1−ε2⌈V⌉K+1 1−ε||h1||+1 1−ε||h2||−1 1−εu−1 1 + εvK . In particular, the strict comparison principle holds for (6.2)and (6.3). 6.2. Proof of Lemmas 6.1,6.2, and Proposition 6.3. Proof of Lemma 6.1.We only prove the first statement, the second one follows analogously. Let ube a subsolution to f−λHf =hand let (f, g)∈H+. Our claim thus follows if there exists x0satisfying u(x0)−f(x0) = ⌈u−f⌉,(6.5) u(x0)−λg(x0)≤h(x0).(6.6) As uis upper semi-continous and bounded, and fhas compact sublevel sets, the existence of x0satisfying (6.5) is immediate. We thus proceed with (6.6) using the sequential upward denseness of D(H)in D(H+), cf. Assumption 3.4 (c). Set a:= f(x0) + ⌈u⌉−u(x0), A := {x|f(x)≤a}. We can thus find (fa, ga)∈Hwith fasatisfying (fa(x) = f(x)if x∈A, a<fa(x)≤f(x)if x /∈A. We first establish that u(x0)−fa(x0) = ⌈u−fa⌉.(6.7) Using (6.5) and that f=faon A, (6.7) follows by verifying that u(x)−fa(x)< u(x0)−f(x0), x ∈Ac, which follows from the definition of a: u(x)−f(x)< u(x)−a =u(x)−(f(x0) + ⌈u⌉−u(x0)) =u(x0)−f(x0)−(⌈u⌉−u(x)) ≤u(x0)−f(x0).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 33 Thus, by (6.7), we can use the subsolution inequality for (fa, ga)in the point x0. We obtain: u(x0)−λga(x0)≤h(x0).(6.8) Recalling that fa(x0) = f(x0)and fa≤f, we have fa(x0)−f(x0) = ⌈fa−f⌉. Using the positive maximum principle for H, cf. Assumption 3.4 (a), thus yields ga(x0)≤g(x0).(6.9) Combining (6.8) and (6.9), leads to u(x0)−λg(x0)≤u(x0)−λga(x0)≤h(x0), establishing (6.6) and consquently that uis a subsolution to f−λH+f=h.□ Proof of Lemma 6.2.As f1, f2,b f1,b f2∈C∞ c(E), it follows by Assumption 3.4 (b) that f1, f2,b f1,b f2∈ D(H). By compatibility, cf. Assumption 3.5, we have V◦sz,Ξ◦sz∈ D(H). By Assumption 3.4 (e) and the fact that Vhas compact sublevel sets, cf. Definition 2.12, we thus have (1 −φ)V◦sz+φΞ◦sz∈ D(H+). Consequently, b f†, f†∈ D(H+)and b f‡, f‡∈ D(H−)by Assumption 3.4 (f).□ Proof of Proposition 6.3.Let ube a subsolution of f−λH+f=h1and va supersolution of f−λH−f=h2. Consider the constructions in Propositions 5.1 and 5.3 for the subsolution u, supersolution vand ε∈(0,1) and φ∈(0,1]. By Lemma 6.2, we have f†∈ D(H+)and f‡∈ D(H−)and, by Proposition 5.3 (c), we find that (xα, x′ α)are the unique optimizers in u(xα)−f†(xα) = ⌈u−f†⌉, v(x′ α)−f‡(x′ α) = ⌈v−f‡⌉, which, by the suband supersolution properties for H+and H−, respectively, and Lemma D.1, implies that u(xα)−λH+f†(xα)≤h1(xα), v(x′ α)−λH−f‡(x′ α)≥h2(x′ α).(6.10) By Proposition 5.1 (i), we find ⌈u−v⌉K≤1 1−εu(xα)−1 1 + εv(x′ α) + ε(cε,φ +o(1)) , where cε,φ := 2 1−ε2(1 −φ)⌈V⌉K−1 1−εu−1 1 + εvK ,(6.11) and o(1) is in terms of α→ ∞. Using (6.10), we estimate ⌈u−v⌉K≤1 1−εu(xα)−1 1 + εv(x′ α) + ε(cε,φ +o(1)) ≤1 1−εh1(xα)−1 1 + εh2(x′ α) + λH+f†(xα) 1−ε−H−f‡(x′ α) 1 + ε+ε(cε,φ +o(1)) ≤h1(xα)−h2(x′ α) + λH+f†(xα) 1−ε−H−f‡(x′ α) 1 + ε +ε 1−ε||h1||+ε 1 + ε||h2||+ε(cε,φ +o(1)) .
34 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL We next expand cε,φ from (6.11). Furthermore, taking lim infα→∞ on the right-hand side, using Proposition 5.1 (j) to treat the difference h1−h2, and (6.4) to treat the difference of Hamiltonians, we find ⌈u−v⌉K≤ ⌈h1−h2⌉b K+λ(εC0+Cε,φ) + ε 1−ε||h1||+ε 1 + ε||h2|| +ε2 1−ε2(1 −φ)⌈V⌉K−1 1−εu−1 1 + εvK. As φ∈(0,1] was arbitrary, we can take the limit for φ↓0, which leads to ⌈u−v⌉K≤ ⌈h1−h2⌉b K +ελC0 ε+2 1−ε2⌈V⌉K+1 1−ε||h1||+1 1 + ε||h2||−1 1−εu−1 1 + εvK, establishing the claim. □ 6.3. Proof of Theorem 3.1.We start with an auxiliary lemma that provides a detailed decomposition of the operators Aand Bevaluated in the test functions. Lemma 6.4. Let Aand Bboth satisfy Assumption 3.4 and Assumption 3.5 (a) and (b), respectively. Fix z0, z1∈Rqand p∈Rq. Let Ξ = Ξz0,p,z1as in Definition 2.13 and, for b f∈C∞ c(E),ε∈(0,1), and φ∈(0,1], set b f†:= (1 −ε)b f+ε(1 −φ)V+εφΞ, b f‡:= (1 + ε)b f−ε(1 −φ)V−εφΞ. For z∈E, set f†=b f†◦sz, and f‡=b f‡◦sz. Then, the following statements hold: (a) f†∈ D(A+)and f‡∈ D(A−). Suppose furthermore that Ais linear on its domain, then A+f† 1−ε=A(b f◦sz) + ε 1−ε(1 −φ)A+(V◦sz) + ε 1−εφA(Ξ ◦sz),(6.12) A−f‡ 1 + ε=A(b f◦sz)−ε 1 + ε(1 −φ)A+(V◦sz)−ε 1 + εφA(Ξ ◦sz), (b) f†,b f†∈ D(B+)and f‡,b f‡∈ D(B−). Suppose furthermore that Bis convex, then for any x, y such that z=x−y, we have B+f† 1−ε(x)≤1 1−εB+f†(x)−B+b f†(y)+Bb f(y)(6.13) +ε 1−ε(1 −φ)B+V(y) + ε 1−εφB+Ξ(y), B−f‡ 1 + ε(x)≥1 1 + εB−f‡(x)−B−b f‡(y)+Bb f(y) −ε 1 + ε(1 −φ)B−V(y)−ε 1 + εφB−Ξ(y). Proof. The domain statements f†∈ D(A+),f‡∈ D(A−),f†,b f†∈ D(B+)and f‡,b f‡∈ D(B−)follow by Lemma 6.2. The four statements in (6.12) and (6.13) follow from linearity of A+and convexity of B+.□ Proof of Theorem 3.1.To prove inequality (3.3), and consequently the strong comparison principle for the Hamilton–Jacobi equation in terms of H, it suffices by Lemma 6.1 and Proposition 6.3 to establish (6.4), which we repeat for readability: lim inf α→∞ H+f†(xα) 1−ε−H−f‡(x′ α) 1 + ε≤εC0 ε+Cε,φ.(6.14)
A COMPARISON PRINCIPLE BASED ON COUPLINGS 35 Let θ∗ 1,α ∈Θ1be such that H+f†(xα) = sup θ1∈Θ1 inf θ2∈Θ2{Aθ1,θ2f†(xα) + Bθ1,θ2f†(xα)−I(xα, θ1, θ2)} = inf θ2∈Θ2nAθ∗ 1,α,θ2f†(xα) + Bθ∗ 1,α,θ2f†(xα)−I(xα, θ∗ 1,α, θ2)o. Such optimizer exists by the compactness of Θ1and the lower semi-continuity of Iin θ1assumed in (d). By Isaacs’ condition (a), we can write H−f‡(x′ α) = inf θ2∈Θ2 sup θ1∈Θ1Aθ1,θ2f‡(x′ α) + Bθ1,θ2f‡(xα)−I(x′ α, θ1, θ2). Then, by compactness of Θ2and the upper semi-continuity of Iin θ2assumed in (d), we can find θ∗ 2,α ∈Θ2such that H−f‡(x′ α) = sup θ1∈Θ1nAθ1,θ∗ 2,α f‡(x′ α) + Bθ1,θ∗ 2,α f‡(x′ α)−I(x′ α, θ1, θ∗ 2,α)o. Consequently, we can estimate 1 1−εH+f†(xα)−1 1 + εH−f‡(x′ α)≤1 1−εAθ∗ 1,α,θ∗ 2,α f†(xα)−1 1 + εAθ∗ 1,α,θ∗ 2,α f‡(x′ α) |{z } (1) +1 1−εBθ∗ 1,α,θ∗ 2,α f†(xα)−1 1 + εBθ∗ 1,α,θ∗ 2,α f‡(x′ α) | {z } (2) +1 1 + εI(x′ α, θ∗ 1,α, θ∗ 2,α)−1 1−εI(xα, θ∗ 1,α, θ∗ 2,α) | {z } (3) . We treat (1),(2), and (3) separately. Note, that due the compactness of Θ1and Θ2, the sequences of optimizers θ∗ 1,α and θ∗ 2,α converge to some θ∗ 1and θ∗ 2, respectively. Estimate (1):Using the expansions of A+f†and A−f‡obtained in Lemma 6.4 we find Aθ∗ 1,α,θ∗ 2,α f†(xα) 1−ε− Aθ∗ 1,α,θ∗ 2,α f‡(x′ α) 1 + ε=Aθ∗ 1,α,θ∗ 2,α,+f†(xα) 1−ε−Aθ∗ 1,α,θ∗ 2,α,−f‡(x′ α) 1 + ε ≤Aθ∗ 1,α,θ∗ 2,α f1(xα)−Aθ∗ 1,α,θ∗ 2,α f2(x′ α) +ε 1−ε(1 −φ)Aθ∗ 1,α,θ∗ 2,α,+(V◦sxα−yα) (xα) +ε 1 + ε(1 −φ)Aθ∗ 1,α,θ∗ 2,α,+V◦sx′ α−y′ α(x′ α) +ε 1−εφAθ∗ 1,α,θ∗ 2,α (Ξ1◦sxα−yα) (xα) +ε 1 + εφAθ∗ 1,α,θ∗ 2,α Ξ2◦sx′ α−y′ α(x′ α).(6.15) We first consider the terms involving Vand Ξ. By Proposition 5.1 (j), we have that, along subsequences, the optimizers (xα, yα, yα,0, y′ α,0, y′ α, x′ α)converge to (z, z, z, z, z, z) with z∈b Kand pα, p′ α∈B1/α(0). Then, using the compatibility of Aθ1,θ2, cf. Assumption 3.5, we find
36 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL lim inf α→∞ ε 1−ε(1 −φ)Aθ∗ 1,α,θ∗ 2,α,+(V◦sxα−yα) (xα) +ε 1 + ε(1 −φ)Aθ∗ 1,α,θ∗ 2,α,+V◦sx′ α−y′ α(x′ α) +ε 1−εφAθ∗ 1,α,θ∗ 2,α (Ξ1◦sxα−yα) (xα) + ε 1 + εφAθ∗ 1,α,θ∗ 2,α Ξ2◦sx′ α−y′ α(x′ α) ≤2ε 1−ε2(1 −φ)Aθ∗ 1,θ∗ 2,+(V)(z) + φAθ∗ 1,θ∗ 2(Ξz,0,z)(z).(6.16) Next, we consider the second line in (6.15). Using that, for all θ1,θ2,Aθ1,θ2has a controlled growth coupling b Aθ1,θ2with a modulus uniform in θ1and θ2satisfying the maximum principle and Proposition 5.3 (a), we find Aθ∗ 1,α,θ∗ 2,α f1(xα)−Aθ∗ 1,α,θ∗ 2,α f2(x′ α) = b Aθ∗ 1,α,θ∗ 2,α (f1⊖f2) (xα, x′ α) ≤b Aθ∗ 1,α,θ∗ 2,α α 2d2 xα−yα,x′ α−y′ α(xα, x′ α) ≤ωb A, b Kαd(xα, yα) + d(yα, y′ α) + d(y′ α, x′ α)2 +d(xα, yα) + d(yα, y′ α) + d(y′ α, x′ α),(6.17) which converges to 0as α→ ∞ by Proposition 5.1 (g). Estimate (2):By using the expansions of B+f†and B−f‡obtained in Lemma 6.4, we find Bθ∗ 1,α,θ∗ 2,α f†(xα) 1−ε−Bθ∗ 1,α,θ∗ 2,α f‡(x′ α) 1 + ε=Bθ∗ 1,α,θ∗ 2,α,+f†(xα) 1−ε−Bθ∗ 1,α,θ∗ 2,α,−f‡(x′ α) 1 + ε ≤Bθ∗ 1,α,θ∗ 2,α b f1(yα)−Bθ∗ 1,α,θ∗ 2,α b f2(y′ α) +1 1−εBθ∗ 1,α,θ∗ 2,α,+f†(xα)−Bθ∗ 1,α,θ∗ 2,α,+b f†(yα) +1 1 + εBθ∗ 1,α,θ∗ 2,α,−b f‡(y′ α)−Bθ∗ 1,α,θ∗ 2,α,−f‡(x′ α) +ε 1−ε(1 −φ)Bθ∗ 1,α,θ∗ 2,α,+V(yα) + ε 1 + ε(1 −φ)Bθ∗ 1,α,θ∗ 2,α,+V(y′ α) +ε 1−εφBθ∗ 1,α,θ∗ 2,α Ξ1(yα) + ε 1 + εφBθ∗ 1,α,θ∗ 2,α Ξ2(y′ α). Again, by sending α→ ∞, using Proposition 5.1 (j), and the compatibility of Bθ1,θ2, cf. Assumption 3.5, we obtain that lim inf α→∞ ε 1−ε(1 −φ)Bθ∗ 1,α,θ∗ 2,α,+V(yα) + ε 1 + ε(1 −φ)Bθ∗ 1,α,θ∗ 2,α,+V(y′ α)(6.18) +ε 1−εφBθ∗ 1,α,θ∗ 2,α Ξ1(yα) + ε 1 + εφBθ∗ 1,α,θ∗ 2,α Ξ2(y′ α) ≤2ε 1−ε2(1 −φ)Bθ∗ 1,θ∗ 2,+(V)(z) + φBθ∗ 1,θ∗ 2(Ξz,0,z)(z).
A COMPARISON PRINCIPLE BASED ON COUPLINGS 37 Using that, for all θ1,θ2,Bθ1,θ2is semi-monotone with Bθ1,θ2and the expressions for the gradients obtained in Proposition 5.3, we find that 1 1−εBθ∗ 1,α,θ∗ 2,α,+f†(xα)−Bθ∗ 1,α,θ∗ 2,α,+b f†(yα) +Bθ∗ 1,α,θ∗ 2,α b f1(yα)−Bθ∗ 1,α,θ∗ 2,α b f2(y′ α) +1 1 + εBθ∗ 1,α,θ∗ 2,α,−b f‡(y′ α)−Bθ∗ 1,α,θ∗ 2,α,−f‡(x′ α) =1 1−εBθ∗ 1,α,θ∗ 2,α (xα, α(xα−yα)) −Bθ∗ 1,α,θ∗ 2,α (yα, α(xα−yα)) +Bθ∗ 1,α,θ∗ 2,α (yα, α(yα−y′ α)) −Bθ∗ 1,α,θ∗ 2,α (y′ α, α(yα−y′ α)) +1 1 + εBθ∗ 1,α,θ∗ 2,α (yα, α(y′ α−x′ α)) −Bθ∗ 1,α,θ∗ 2,α (x′ α, α(y′ α−y′ α)).(6.19) By the semi-monotonicity property of Bθ1,θ2, (6.19) is bounded by 1 1−εωB, b K(d(xα, yα) + αd2(xα, yα)) + ωB, b K(d(yα, y′ α) + αd2(yα, y′ α)) +1 1 + εωB, b K(d(y′ α, x′ α) + αd2(y′ α, x′ α)).(6.20) Thus, taking the lim infα→∞ gives 0by Proposition 5.1 (g). Estimate (3):We have 1 1 + εI(x′ α, θ∗ 1,α, θ∗ 2,α)−1 1−εI(xα, θ∗ 1,α, θ∗ 2,α) =I(x′ α, θ∗ 1,α, θ∗ 2,α)−I(xα, θ∗ 1,α, θ∗ 2,α)−ε 1−εI(xα, θ∗ 1,α, θ∗ 2,α)−ε 1 + εI(x′ α, θ∗ 1,α, θ∗ 2,α). By assumption, Iadmits a modulus of continuity ωI,K, uniform in θ1,θ2, implying 1 1 + εI(x′ α, θ∗ 1,α, θ∗ 2,α)−1 1−εI(xα, θ∗ 1,α, θ∗ 2,α) ≤ωI, b K(d(xα, x′ α)) −ε 1−ε(1 −φ)I(xα, θ∗ 1,α, θ∗ 2,α)−ε 1 + ε(1 −φ)I(x′ α, θ∗ 1,α, θ∗ 2,α). Sending α→ ∞, using the lower semi-continuity of I, and using Proposition 5.1 (j), we find lim inf α→∞ 1 1 + εI(x′ α, θ∗ 1,α, θ∗ 2,α)−1 1−εI(xα, θ∗ 1,α, θ∗ 2,α) ≤lim inf α→∞ ωI, b K(d(xα, x′ α)) + lim sup α→∞ −ε 1−ε(1 −φ)I(xα, θ∗ 1,α, θ∗ 2,α)−ε 1 + ε(1 −φ)I(x′ α, θ∗ 1,α, θ∗ 2,α) ≤ − 2ε 1−ε2(1 −φ)I(z, θ∗ 1, θ∗ 2). (6.21)
38 SERENA DELLA CORTE, FABIAN FUCHS, RICHARD C. KRAAIJ, AND MAX NENDEL Conclusion: Putting together (6.16), (6.17), (6.18), (6.20), and (6.21), we can conclude that lim inf α→∞ H+f†(xα) 1−ε−H−f‡(x′ α) 1 + ε≤2ε 1−ε2(1 −φ)Aθ∗ 1,θ∗ 2,+V(z) + φAθ∗ 1,θ∗ 2(Ξz,0,z)(z) +2ε 1−ε2(1 −φ)Bθ∗ 1,θ∗ 2,+V(z) + φBθ∗ 1,θ∗ 2(Ξz,0,z)(z) −2ε 1−ε2(1 −φ)I(z, θ∗ 1, θ∗ 2) ≤2ε 1−ε2(1 −φ)(Aθ∗ 1,θ∗ 2,++Bθ∗ 1,θ∗ 2,+)(V)−I(·, θ∗ 1, θ∗ 2) +2ε 1−ε2φ(Aθ∗ 1,θ∗ 2+Bθ∗ 1,θ∗ 2)(Ξ·,0,·)b Kε ≤ε2 1−ε2cV+2 1−ε2φ(Aθ∗ 1,θ∗ 2+Bθ∗ 1,θ∗ 2)(Ξ·,0,·)b Kε ≤εC0 ε+Cε,φ with cVgiven by (3.1), and C0 εand Cε,φ defined via the last two lines. The estimate on the difference of Hamiltonians (6.14) and thus (6.4) of Proposition 6.3 are satisfied. As a consequence our final estimate (3.3) and, consequently, the strong comparison principle follow. □ Appendix A. The Jensen perturbation The main result of this section is Proposition A.1 that allows us to perturb a semiconvex function with a unique extreme point such that we get a new extreme point close by, in which the function is twice differentiable. The result is a variant of the well-known perturbation result by Jensen, see e.g. [16, Lemma A.3]. Proposition A.1. Fix η > 0. Let ϕ:E×E→Rbe bounded above and semi-convex with convexity constant κ≥1. Suppose that (x0, y0)is an optimizer of ϕ(x0, y0) = ⌈ϕ⌉. Let R > 0,{ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C1(E)and semi-concavity constant κξbe as in Definition 2.13. Fix ε1, ε2>0such that 1−(ε1+ε2)κξ>0. Furthermore, define for p= (p1, p2)∈ Rq×Rqthe perturbed functions ϕp(x, y) := ϕ(x, y)−ε1(ξx0(x) + ζx0,p1(x)) −ε2(ξy0(y) + ζy0,p2(y)) .(A.1) Then there exist p1, p2∈Bη(0), and a pair (x1, y1)∈Bη(x0)×Bη(y0)globally maximizing ϕpat which ϕpis twice differentiable. Corollary A.2. For η > 0,pand (x1, y1)as in Proposition A.1, we have 0≤ −ε1(ξx0(x1) + ζx0,p1(x1)) −ε2(ξy0(y1) + ζy0,p2(y1)) ≤ε1η+εη2,(A.2) and ⌈ϕ⌉ ≤ ϕp,ε(x1, y1)≤ ⌈ϕ⌉+ε1η+εη2.(A.3) The proof of the perturbation proposition is based partly on results from set-valued analysis. To facilitate the proof, we first introduce the necessary auxiliary definitions and results.
A COMPARISON PRINCIPLE BASED ON COUPLINGS 39 Definition A.3. A set-valued function Γ : A⇒Bis called upper hemi-continuous at a∈A, if, for all open neighbourhoods V⊆Bof Γ(a)(meaning that Γ(a)⊆V), there exists a neighbourhood Uof asuch that, for all x∈U, we have Γ(x)⊆V. If A, B are metric, this can equivalently formulated in terms of sequences: A setvalued map Γ : A⇒B, which takes closed values, is upper hemi-continuous at a, if, for any sequence an→aand bn∈Γ(an)satisfying bn→b, we have b∈Γ(a). We say that Γis upper hemi-continuous, if it is upper hemi-continuous at all points. Lemma A.4. Let Kbe a compact metric space and let Ξbe a metric space. For any ξ∈Ξ, let ϕξ∈C(K)and suppose that the map ξ7→ ϕξis continuous from Ξto C(K), endowed with the supremum norm on K. Then the set-valued map Opt : Ξ ⇒Kdefined by Opt(ξ) := {x∈K|ϕξhas a maximum at x} is upper hemi-continuous. Proof. The result follows immediately from Berge’s Maximum Theorem [1, Theorem 17.31] with ξ7→ imageϕξ(K)being the relevant set-valued map. □ Remark A.5. In the proof below, we will make use of the notion of a lim sup of sets. For a sequence of sets (An)n∈Ndenote lim sup n→∞ An=\ n∈N[ m≥n Am to be interpreted as x∈lim supn→∞ Anif and only if there are infinitely many n∈N such that x∈An. The following proof is a variant of the proof of [16, Lemma A.3] and [11, Theorem 2.3.3]. Proof of Proposition A.1.For notational convenience, we will write w= (x, y)and w0= (x0, y0). Let R > 0and {ζz,p}z∈E,p∈Rq⊂C(E)and {ξz}z∈E⊂C(E)be two collections of functions as in Definition 2.13. Without loss of generality, we can assume that R≥η. We start out by making z0the unique optimizer by replacing ϕby b ϕ(w) = ϕ(w)−ε1ξx0(x)−ε2ξy0(y). Note that as 1−(ε1+ε2)κξ>0the map b ϕis semi-convex and bounded from above with a unique optimizer w0. Our next step is to locally, linearly perturb b ϕto obtain ϕpas in equation (A.1). This procedure produces a new optimizer close to w0in which the perturbed function ϕpis twice differentiable. To further facilitate the analysis of optimizers, we smoothen out ϕ. To that end, let Cδ:Cb(E)→C2 b(E)be a mollifier with supδ>0||Cδf|| <∞and Cδf→funiformly on compacts as δ↓0. Define ϕp,δ(w):= (Cδϕ)(w)−ε1(ξx0(x) + ζx0,p1(x)) −ε2(ξy0(y) + ζy0,p2(y)) , where we will read C0= 1 such that ϕp,0=ϕpand ϕ0,0=b ϕ. We next study the optimizers for the map (p, δ)7→ ϕp,δ on Ξ=(B1(0) ×B1(0))×[0,1] using Berge’s Maximum Theorem with K=BR(w0). Set Opt(p, δ):=nw∈BR(w0)ϕp,δ has a local maximum at w∈BR(w0)o. First note that the local nature of the problem can be removed due to the fact that the perturbations all vanish in w0, whereas they add up to something negative outside the
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