Convex monotone semigroups and their generators with respect to Г-convergence
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Blessing, Jonas; Denk, Robert; Kupper, Michael; Nendel, Max Working Paper Convex monotone semigroups and their generators with respect to Г-convergence Center for Mathematical Economics Working Papers, No. 662 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Blessing, Jonas; Denk, Robert; Kupper, Michael; Nendel, Max (2022) : Convex monotone semigroups and their generators with respect to Г-convergence, Center for Mathematical Economics Working Papers, No. 662, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29614866 This Version is available at: https://hdl.handle.net/10419/273038 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/
662 February 2022 Convex Monotone Semigroups and their Generators with Respect to Γ-Convergence Jonas Blessing, Robert Denk, Michael Kupper 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
CONVEX MONOTONE SEMIGROUPS AND THEIR GENERATORS WITH RESPECT TO Γ-CONVERGENCE JONAS BLESSING∗,1, ROBERT DENK∗,2, MICHAEL KUPPER∗,3, AND MAX NENDEL∗∗,4 Abstract. We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to Γ-convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under the semigroup. To overcome this issue, we consider different versions of invariant Lipschitz sets which turn out to be suitable domains for weaker notions of the generator. The so-called Γ-generator is defined as the time derivative with respect to Γ-convergence in the space of upper semicontinuous functions. Under suitable assumptions, we show that the Γ-generator uniquely characterizes the semigroup and is determined by its evaluation at smooth functions. Furthermore, we provide Chernoff approximation results for convex monotone semigroups and show that approximation schemes based on the same infinitesimal behaviour lead to the same semigroup. Our results are applied to semigroups related to stochastic optimal control problems in finite and infinite-dimensional settings as well as Wasserstein perturbations of transition semigroups. Key words: Convex monotone semigroup, Γ-convergence, Lipschitz set, comparison principle, Chernoff approximation, optimal control, Wasserstein perturbation. MSC 2020: Primary 47H20; 47J25; Secondary 35K55; 35B20; 49L20. 1. Introduction The link between operator semigroups and abstract Cauchy problems through the infinitesimal behaviour of the semigroup is a classic question in the theory of partial differential equations. A fundamental result is that strongly continuous semigroups of bounded linear operators on Banach spaces are uniquely characterized by their infinitesimal generator. In a nonlinear setting, Alvarez et al. [1] provide an axiomatic foundation for viscosity solutions to fully nonlinear second-order partial differential equations based on monotone semigroups which are defined on the space of bounded uniformly continuous functions and satisfy suitable regularity and locality assumptions. This approach was picked up later by Biton [6] for semigroups on more general spaces of continuous functions with a certain behaviour at infinity. While these works mainly focus on the existence and axiomatization of second-order differential operators through semigroups, the uniqueness of the associated semigroups in terms of their generator is not yet fully clarified, cf. the discussion in [6, Section 5]. The key ideas of viscosity solutions are local comparisons with smooth functions and regularizations by introducing additional ∗Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany ∗∗Center for Mathematical Economics, Bielefeld University, 33615 Bielefeld, Germany E-mail addresses:1[email protected], 2[email protected], 3[email protected], 4[email protected]. Date: February 17, 2022. Financial support through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1283/2 2021 – 317210226 is gratefully acknowledged. 1
2 CONVEX MONOTONE SEMIGROUPS viscosity terms. We refer to Fleming and Soner [22, Chapter II.3] for an intuitive discussion on viscosity solutions in an operator-theoretic setting based on abstract dynamic programming principles. In the present paper we follow a different approach which is closer to the theory of linear semigroups and based on invariant sets which are suitable domains for a weaker definition of the generator. In order to provide uniqueness results for semigroups based on their infinitesimal behaviour, it is crucial that the domain of the generator is invariant under the semigroup. If we drop the linearity, the invariance may fail, see [17, Example 5.2]. Additionally, the domain might be even empty, see Crandall and Liggett [14, Section 4]. For convex semigroups, the invariance can, in general, only be guaranteed for spaces with order continuous norm, see [16]. In contrast to Lp-spaces and Orlicz hearts, which might be too large to handle the nonlinearity, spaces of continuous functions lack this property. Hence, we are looking for an invariant set on which we can define the generator in a weaker form such that the semigroup still represents the unique solution to the associated abstract Cauchy problem. Similar to Rademacher’s theorem, it turns out that Lipschitz continuous paths of the semigroup are differentiable in a weaker sense, i.e., the generator can be defined with respect to Γ-convergence. Let S:= (S(t))t≥0be a strongly continuous semigroup of convex monotone operators on a suitable space of continuous functions. On the so-called upper Lipschitz, which is invariant under the semigroup, we can define the upper Γ-generator A+ Γf:= Γ- lim sup h↓0 S(h)f−f h. Likewise we define the Γ-generator AΓfif the limes superior in the previous equation can be replaced by a Γ-limit. Since the function A+ Γfis merely upper semicontinuous, an extension of Sto the set of all upper semicontinuous functions is necessary in order to define the term S(t)A+ Γf. For that purpose, we follow the ideas of Beer [4]. Under the assumption that Sis continuous from above, there exists a unique extension which is upper semicontinuous with respect to Γ-convergence. The latter property is crucial for our work. Moreover, on the set of upper semicontinuous functions, the concept of Γ-convergence satisfies desirable stability properties, see, e.g., Dal Maso [15] and Rockafellar and Wets [38]. Our first main result is a comparison principle which implies, in particular, that convex monotone semigroups are uniquely determined by their upper Γ-generator on their upper Lipschitz set, see Theorem 2.10 and Corollary 2.11. Second, under additional assumptions, we show that AΓf= Γ- limn→∞ AΓfnif (fn)n∈Nis a suitable approximating sequence for f, see Theorem 3.4. Typically, the approximating sequence can be constructed via convolutions with mollifiers or sup-inf-convolutions. In particular, we obtain an explicit description of AΓfif, for smooth functions, the Γ-generator is given as a convex functional of certain partial derivatives. It was shown in Alvarez et al. [1] and Biton [6] that this is the case for typical fully nonlinear PDEs. Third, we study approximation schemes of the form S(t)f= lim l→∞ I(2−nlt)2nltf which are known as Chernoff approximations [11,12] or Trotter formulae [42,43]. In this case, key properties of Scan be obtained from the corresponding properties of I, see Theorem 4.3 and Theorem 4.6.
CONVEX MONOTONE SEMIGROUPS 3 The semigroups we consider are upper semicontinuous with respect to Γ-convergence and sequentially continuous in buc1, see Lemma 2.6 and Lemma 2.7. The idea of weakening topological properties of the semigroup is already present in the literature. Goldys and Kocan [24], van Casteren [44], Kunze [30] and Kraaij [28] study linear semigroups in strict topologies, see also Kraaij [26] and Yosida [46] for semigroups in locally convex spaces. Furthermore, equi-continuity in the strict topology is suitable for stability results. Kraaij [29] provides convergence results for nonlinear semigroups based on the connection between viscosity solutions to Hamilton–Jacobi equations and pseudoresolvents and, in [27], Γ-convergence of functionals on path-spaces is established. A classical approach to nonlinear semigroups concentrates on the study of maximal monotone or m-accretive operators, see, e.g., Barbu [2], Bénilan and Crandall [5], Brézis [9], Kato [25] and the references therein. However, there exist simple examples of operators which are accretive but not m-accretive, see, e.g., [17, Example 5.2]. This obstacle was one of the motivations for the study of viscosity solutions to fully nonlinear equations, cf. Lions [32], Crandall et.al. [13] and Evans [19, Section 4]. In Section 5, the abstract results are applied to several classes of examples. In Subsection 5.1, we show that dynamic stochastic optimal control problems for drift and volatility controlled diffusions can be approximated by iterating a corresponding static control problem, where we only take simple deterministic controls. Moreover, the study of the so-called symmetric Lipschitz set yields a regularity result for the corresponding PDE even in the degenerate case. In Subsection 5.2, we show that, in the sublinear case, the previous approximation result can be lifted to an infinite-dimensional setting. In Subsection 5.3, we show that non-parametric Wasserstein perturbations of transition semigroups asymptotically coincide with perturbations which have a finite-dimensional parameter space. As a byproduct, we recover the Talagrand T2inequality for the normal distribution. 2. Comparison of convex monotone semigroups on Lipschitz sets 2.1. Setup and Γ-convergence. Let (X, d)be a complete separable metric space and denote by B(x, r) := {y∈X:d(x, y)≤r}the closed ball with radius r≥0around x∈X. We endow the set of all functions with the pointwise order, i.e., f≤gif and only if f(x)≤g(x)for all f, g :X→[−∞,∞)and x∈X. All order-related notions (sup, inf, max, min, lim sup, etc.) for such functions are understood with respect to this order. We define f∨g:= max{f, g},f∧g:= min{f, g},f+:= f∨0and f−:= −(f∧0) for all f, g:X→[−∞,∞), where f−(x) := ∞if f(x) = −∞. We slightly relax the supremum norm in order to include unbounded functions with controlled growth behaviour at infinity. For this, we fix a bounded continuous function κ:X→(0,∞)and consider the κ-weighted supremum norm kfkκ:= sup x∈X|f(x)|κ(x)∈[0,∞]for all f:X→[−∞,∞). Let Cκbe the space of all continuous functions f:X→Rwith kfkκ<∞and Uκbe the set of all upper semicontinuous functions f:X→[−∞,∞)with kf+kκ<∞. If κ≡1, then k·kκis the usual supremum norm k·k∞,Cκcoincides with the space Cb of all bounded continuous functions and Uκis the set Ubof all upper semicontinuous functions which are bounded above by a real constant. Since the mapping Cκ→Cb, f 7→ fκ 1A sequence converges buc if and only if it is bounded and converges uniformly on compacts. On Polish spaces, a sequence converges buc if and only if it converges in the strict topology, see [39].
4 CONVEX MONOTONE SEMIGROUPS is an order-preserving linear isometric isomorphism, the space Cκis a Banach lattice. Note that Uκ= (Cκ)δ:= ninf n∈Nfn: (fn)n∈N⊂Cκo.2 Let (fn)n∈N⊂Uκbe a sequence and f∈Uκ. We write fn↓fif (fn)n∈Ndecreases pointwise to f. If fnand fare real-valued, we say that fn→funiformly on compacts if supx∈K|f(x)−fn(x)| → 0for every compact set K⊂X. Furthermore, a set F⊂Uκ is called bounded if supf∈Fkfkκ<∞and bounded above if supf∈Fkf+kκ<∞. Definition 2.1. For every sequence (fn)n∈N⊂Uκ, which is bounded above, we define Γ- lim sup n→∞ fn(x) := sup nlim sup n→∞ fn(xn): (xn)n∈N⊂Xwith xn→xo∈[−∞,∞) for all x∈X. Moreover, we say that f= Γ- limn→∞ fnwith f∈Uκif, for every x∈X, •f(x)≥lim supn→∞ fn(xn)for every sequence (xn)n∈N⊂Xwith xn→x, •f(x) = limn→∞ fn(xn)for some sequence (xn)n∈N⊂Xwith xn→x. For every t≥0and (fs)s≥0⊂Uκ, which bounded above, we define Γ- lim sup s→t fs:= sup nΓ- lim sup n→∞ fsn:sn→to∈Uκ. Furthermore, we say that f= Γ- lims→tfswith f∈Uκif f= Γ- limn→∞ fsnfor all sequences (sn)n∈N⊂[0,∞)with sn→t. For further details and a summary of basic results on Γ-convergence, we refer to Appendix A. The following geometric characterization of the Γ- lim sup is based on the work of Beer [4]. We will use it throughout this work to link Γ-convergence with monotone convergence and Γ-upper semicontinuity with continuity from above. Remark 2.2. For every f∈Uκ, there exists a family (fε)ε>0⊂Uκwith fε↓fas ε↓0and −1 ε≤fεκ≤ kf+kκ+εfor all ε > 0. Furthermore, let (fn)n∈N⊂Uκbe bounded above. Then, it holds Γ- lim supn→∞ fn≤f if and only if, for every ε > 0and K⊂Xcompact, there exists n0∈Nsuch that fn(x)≤fε(x)for all x∈Kand n≥n0. The family (fε)ε>0is explicitly constructed in Appendix A. 2.2. Convex monotone semigroups on Lipschitz sets. Let S:= (S(t))t≥0be a family of operators S(t): Cκ→Uκ. The norm generator is defined by A:D(A)→Cκ, f 7→ lim h↓0 S(h)f−f h, where the domain D(A)consists of all f∈Cκsuch that S(t)f∈Cκfor all t≥0, the previous limit exists with respect to the norm k · kκand S(s+t)f=S(s)S(t)f for all s, t ≥0. As previously discussed, the assumption S(t): D(A)→D(A)for all t≥0is, in general, too restrictive. We point out that this problem can not be avoided by restricting the semigroup and the generator to a subspace of Cκ. Hence, instead of considering the largest set on which the right derivative of the trajectories t7→ S(t)f exists with respect to the norm, we consider the largest set on which the trajectories are merely (upper) Lipschitz continuous. For these trajectories the right derivative can still be defined as a limes superior. This brings us to the following crucial definition. 2It holds f(x) = infn∈Nsupy∈X 1 κ(x)max{(fκ)(y),−n} − n2d(x, y)for all f∈Uκand x∈X.
CONVEX MONOTONE SEMIGROUPS 5 Definition 2.3. The upper Lipschitz set LS +consists of all f∈Cκsuch that (i) S(t)f∈Cκfor all t≥0, (ii) S(s+t)f=S(s)S(t)ffor all s, t ≥0, (iii) there exist h0>0and c≥0with S(h)f−fκ≤ch or all h∈[0, h0]. Furthermore, we define the upper Γ-generator by A+ Γf:= Γ- lim sup h↓0 S(h)f−f h∈Uκfor all f∈ LS +. By definition, it holds D(A)⊂ LS +and Af =A+ Γffor all f∈D(A). Furthermore, we will see in Section 3that, under additional assumptions, the limes superior in the previous definition can be replaced by a limit. The invariance of the upper Lipschitz set, i.e., S(t): LS +→ LS +for all t≥0, does not require further assumptions and is shown in Remark 2.9 below. While Af ∈Cκholds by definition, A+ Γfis not necessarily continuous and may take the value −∞. Hence, in order to to give the term S(t)A+ Γf a meaning, an extension of Sfrom Cκto Uκis necessary. Moreover, the argumentation in this article relies heavily on the fact that the extension is Γ-upper semicontinuous in time and continuous from above. This can be achieved under the following assumption which holds throughout the rest of this section. Let BCκ(0, r) := {f∈Cκ:kfkκ≤r} and BUκ(0, r) := {f∈Uκ:kfkκ≤r}be the closed balls with radius r≥0around zero in Cκand Uκ, respectively. Assumption 2.4. The family S:= (S(t))t≥0of operators S(t): Cκ→Uκsatisfies the following conditions: (S1) The operator S(t)is convex and monotone,3and S(t)0 = 0 for all t≥0. (S2) For every t≥0, the operator S(t)is continuous from above, i.e., S(t)fn↓S(t)f for all sequences (fn)n∈N⊂Cκand f∈Cκwith fn↓f. (S3) Γ- lim sups→tS(s)f≤S(t)fand S(0)f=ffor all t≥0and f∈Cκ. (S4) sups∈[0,t]supf∈BCκ(0,r)kS(s)fkκ<∞for all r, t ≥0. Since Cκis a Banach lattice and S(t)is convex, it is sufficient to require condition (S2) for some f∈Cκ, e.g., f= 0. Although, in many examples, Sis a priori only defined on a closed subspace C ⊂ Cκ, we assume that Sis defined on the whole space Cκ. The reason behind this is the fact that the property Uκ= (Cκ)δtogether with condition (S2) ensures the existence of a pointwise extension to Uκwhich is again continuous from above and satisfies the conditions (S1)-(S4) for all f∈Uκ. If Uκ=Cδ, it is sufficient to state Assumption 2.4 with Cinstead of Cκ. We conclude this subsection with the crucial observation that Sis simultaneously upper semicontinuous in the variables tand f, see Lemma 2.6 below. Moreover, we state some further comments about Definition 2.3 and Assumption 2.4. Let Bκbe the space of all Borel measurable functions f:X→[−∞,∞) with kf+kκ<∞. Remark 2.5. We discuss the extension of Sfrom Cκto Uκ. For a proof, we refer to Corollary C.3. (i) Fix t≥0. Since S(t): Cκ→Uκis continuous from above and Uκ= (Cκ)δ, there exists a unique extension S(t): Uκ→Uκwhich is continuous from above. The family of extended operators satisfies the conditions (S1)-(S4) with Uκinstead of Cκ. In addition, for every x∈X, the functional Uκ→[−∞,∞), f 7→ (S(t)f)(x) 3The mapping Cκ→[−∞,∞), f 7→ (S(t)f)(x)is convex and monotone for all x∈X.
6 CONVEX MONOTONE SEMIGROUPS can even be extended to Bκin such a way that, for every ε > 0and c≥0, there exists a compact set K⊂Xwith S(t)c κ1Kc(x)≤ε. Here, Kc:= X\K. (ii) Fix t≥0and K⊂Xcompact. Let (fn)n∈N⊂Uκbe a sequence with fn↓0. By part (i), the mapping [0, t]×K→R,(s, x)7→ S(s)fn(x) is upper semicontinuous for all n∈Nand decreases pointwise to zero as n→ ∞. It thus follows from Dini’s theorem that sup (s,x)∈[0,t]×KS(s)fn(x)↓0as n→ ∞. Moreover, for every ε > 0and c≥0, there exists a compact set K1⊂Xwith sup (s,x)∈[0,t]×KS(s)c κ1Kc 1(x)≤ε. Lemma 2.6. Let (fn)n∈N⊂Uκbe bounded above and (tn)n∈N⊂[0,∞)a convergent sequence. Define f:= Γ- lim supn→∞ fnand t:= limn→∞ tn. Then, Γ- lim sup n→∞ S(tn)fn≤S(t)f. Proof. Let ε∈(0,1] and K⊂Xcompact. By Remark 2.2, there exists n0∈Nwith fn(x)≤fε(x)for all x∈Kand n≥n0. We use the fact that fε≥ −1 /εκ together with the monotonicity of S(tn)to come up with S(tn)fn≤S(tn)fε+cε κ1Kcfor all n≥n0, where cε:= supn∈Nkf+ nkκ+1 /ε<∞. For all λ∈(0,1), the convexity of S(tn)implies S(tn)fε+cε κ1Kc≤λS(tn)1 λfε+ (1 −λ)S(tn)cε κ(1−λ)1Kc. Hence, it follows from Lemma A.1(iv) and Remark 2.5(i) that Γ- lim sup n→∞ S(tn)fn≤λS(t)1 λfε+ (1 −λ) sup n∈N S(tn)cε κ(1−λ)1Kc. Since K⊂Xis arbitrary, we can use Remark 2.5(ii) to conclude that Γ- lim sup n→∞ S(tn)fn≤λS(t)1 λfε. Furthermore, fεκ≤ kf+kκ+ε≤cεand the monotonicity of S(t)yield Γ- lim sup n→∞ S(tn)fn≤λS(t)1 λfε≤λS(t)fε+1 λ−1cε κ. Since S(t)is continuous from above, the right-hand side converges to S(t)fεas λ↑1. Thus, it follows from fε↓fthat Γ- lim sup n→∞ S(tn)fn≤S(t)fε↓S(t)fas ε↓0. Lemma 2.7. Let (fn)n∈N⊂Cκbe a bounded sequence and f∈Cκwith fn→f uniformly on compacts. Then, S(t)fn→S(t)funiformly on compacts for all t≥0. Proof. Let K⊂Xbe compact and ε > 0. Choose δ, λ ∈(0,1) with (1 −λ)kS(t)fkκ<ε 3and ck(1 −λ)f+δkκ<ε 3,(2.1)
CONVEX MONOTONE SEMIGROUPS 7 where the constant c≥0will be fixed later. By Remark 2.5(ii), there exists a compact set K1⊂Xwith K⊂K1such that sup x∈K(1 −λ)S(t)c (1−λ)κ1Kc 1(x)<ε 3. Since fn→funiformly on compacts, there exists n0∈Nwith fn≤f+δ+c1 κ1Kc 1for all n≥n0, where c1:= supn∈N2kfnkκ. Monotonicity and convexity of S(t)imply S(t)fn≤λS(t)f+δ λ+ (1 −λ)S(t)c1 (1−λ)κ1Kc 1for all n≥n0. By Lemma B.2(ii), there exists a constant c2≥0such that λS(t)f+δ λ−S(t)f κ≤c2k(1 −λ)f+δkκ+ (1 −λ)kS(t)fkκ. Now, choose δ, λ ∈(0,1) such that condition (2.1) is satisfied with c:= c1∨c2. Then, S(t)fn(x)≤S(t)f(x) + εfor all n≥n0and x∈K. Since (fn)n∈Nis bounded, we can change the role of fand fnin the previous considerations in order to obtain the reverse estimate. The following lemma shows that the upper Lipschitz set is invariant. Lemma 2.8. It holds S(t): LS +→ LS +for all t≥0. Proof. Let f∈ LS +. Choose h0∈(0,1] and c≥0such that S(h)f−fκ≤ch for all h∈[0, h0]. We use S(h)S(t)f=S(h+t) = S(t+h) = S(t)S(h)f, Lemma B.1, the monotonicity of S(t)and Lemma B.2 to estimate S(h)S(t)f−S(t)f hκ≤S(t)f+(S(h)f−f)+ h−S(t)fκ ≤c0 (S(h)f−f)+ h κ≤c0ch for all h∈(0, h0], where c0≥0is a constant independent of h∈(0, h0]. In the next remark, we give an outlook on stronger versions of the Lipschitz set. Furthermore, we discuss some results about the Lipschitz set from the theory of linear semigroups. Most of these results rely on the reflexivity of the underlying Banach space, a property which Cκdoes not have. Remark 2.9. (i) Similar to LS +, one can define the Lipschitz set LS, consisting of all f∈Cκwith •S(t)f∈Cκfor all t≥0, •S(s+t)f=S(s)S(t)ffor all s, t ≥0, •there exists h0>0and c≥0with kS(h)f−fkκ≤ch for all h∈[0, h0]. Furthermore, we define the symmetric Lipschitz set LS sym := {f∈ LS:−f∈ LS}. While S(t): LS→ LSfor all t≥0follows by a similar argumentation as in the proof of Lemma 2.8, the invariance of LS sym can, in general, not be guaranteed. However, in several examples, the symmetric Lipschitz set is invariant and can, in contrast to LS +and LS, be determined explicitly. This leads to regularity results for the associated semigroup, see [7,8] and Subsection 5.1.
14 CONVEX MONOTONE SEMIGROUPS Let (rl)l∈N⊂(0,∞)be a sequence satisfying condition (3.2). Then, for every l∈N, AΓfl(xl)≤sup z∈B(yl,rl)S(hml)f−f hml(z) + ε κ(yl)+ε ≤sup z∈B(x,δl)S(hml)f−f hml(z) + sup l∈N ε κ(yl)+ε, where δl:= d(x, yl) + rl→0. Since κ > 0is continuous, we have infl∈Nκ(yl)>0. It follows from Lemma A.1(vii) that g2(x) = Γ- lim l→∞ AΓfl(x) = lim l→∞ AΓfl(xl)≤Γ- lim l→∞ S(hml)f−f hml(x) = g1(x). Third, we show that f∈D(AΓ)with AΓf= Γ- limn→∞ AΓfn. From the first part, we know that every sequence (hm)m∈N⊂(0,∞)with hm→0has a subsequence which satisfies equation (3.3). A priori the choice of the subsequence and the limit g1depend on the choice of the sequence (hm)m∈N. However, we have g1=g2and the function g2 is independent of (hm)m∈N. Hence, Lemma A.1(ii) implies g1= Γ- lim h↓0 S(h)f−f h, i.e., f∈D(AΓ)with AΓf=g1. Since the limit in equation (3.4) is also independent of the choice of of subsequence, we obtain AΓf= limn→∞ AΓfn. 3.2. Convolution. In this subsection, we study two particular convolution schemes to generate approximating sequences (fn)n∈Nwhich satisfy condition (3.2). The first one, convolution with probability measures, works particularly well if X=Rdand the measure has a smooth density with respect to the Lebesgue measure. In this case, (fn)n∈N is a sequence of smooth functions and the generator can be a differential operator of arbitrary order. However, the condition fn∈D(AΓ)is no longer verifiable if Xis infinite-dimensional. The second one, sup-inf-convolution, is restricted to first-order equations but can be applied in separable Hilbert spaces. 3.2.1. Convolution with mollifiers. Let Xbe a separable Banach space. Suppose that there exists δ0>0with c:= sup x∈X sup y∈B(x,δ0) κ(x) κ(y)<∞.(3.5) To simplify the notation, we assume, w.l.o.g., that δ0:= 1. In the sequel, we fix a sequence (µn)n∈Nof probability measures on the Borel σ-algebra B(X)of Xwhich concentrate in the sense that µnB(0,1 /n)c= 0 for all n∈N.(3.6) For every n∈N,f∈Uκand x∈X, we define the convolution by (f∗µn)(x) := ˆX f(x−y)µn(dy)∈[−∞,∞). Condition (3.5) ensures that the previous integral is well-defined. Lemma 3.5. For every n∈N, the mapping Uκ→Uκ, f 7→ f∗µnis well defined and upper semicontinuous, i.e., for every sequence (fm)m∈N⊂Uκ, which is bounded above, Γ- lim sup m→∞ (fm∗µn)≤Γ- lim sup m→∞ fm∗µn. Furthermore, it holds Γ- lim supn→∞ (f∗µn)≤ffor all f∈Uκ.
CONVEX MONOTONE SEMIGROUPS 15 Proof. First, we show that Uκ→Uκ, f 7→ f∗µnis well defined. Let n∈N,f∈Uκ and x∈X. It follows from condition (3.5) and condition (3.6) that (f∗µn)(x)κ(x) = ˆB(0,1) f(x−y)κ(x−y)κ(x) κ(x−y)µn(dy)≤ckf+kκ. Moreover, let (xm)m∈N⊂Xbe a sequence with xm→x. Fatou’s lemma implies lim sup m→∞ (f∗µn)(xm)≤ˆX lim sup m→∞ f(xm−y)µn(dy)≤ˆX f(x−y)µn(dy) Second, we show that, for fixed n∈N, the convolution is upper semicontinuous. Let (fm)m∈N⊂Uκbe bounded above, x∈Xand (xm)m∈N⊂Xwith xm→x. Since supm∈Nkf+ mkκ<∞and ν(A) := ´A 1 κdµndefines a finite Borel measure, we can apply Fatou’s lemma to conclude that lim sup m→∞ (f∗µn)(xm) = lim sup m→∞ ˆX fm(xm−y)µn(dy) ≤ˆX lim sup m→∞ fm(xm−y)µn(dy)≤ˆXΓ- lim sup m→∞ fm(x−y)µn(dy). Third, we show that Γ- lim supn→∞ (f∗µn)≤ffor all f∈Uκ. Let x∈Xand (xn)n∈N⊂Xwith xn→x. Since fis upper semicontinuous, for every ε > 0, there exists n0∈Nsuch that f(xn−y)≤f(x) + εfor all n≥n0and y∈B(0,1 /n). Hence, (f∗µn)(xn) = ˆB(0,1 /n) f(xn−y)µn(dy)≤f(x) + εfor all n≥n0. Letting ε↓0yields lim supn→∞(f∗µn)(xn)≤f(x). For every x∈X, we define the shift operator τx: Uκ→Uκby (τxf)(y) := f(x+y)for all y∈X. Condition (3.7) in the following lemma is clearly satisfied if Sis translation invariant, i.e., τxS(t)f=S(t)(τxf). Moreover, in many examples, the condition holds at least for Lipschitz continuous functions. Lemma 3.6. Let f∈ LS +such that fn:= f∗µn∈D(AΓ)for all n∈N. Assume that, for every ε > 0, there exists δ > 0such that kτxS(t)f−S(t)(τxf)kκ≤εt for all t∈[0,1] and x∈B(0, δ).(3.7) Then, it holds f∈D(AΓ)and AΓf= Γ- limn→∞ AΓfn. Proof. We verify the assumptions of Theorem 3.4. First, we show that the sequence (fn)n∈Nis bounded and converges to funiformly on compacts. It follows from condition (3.5) that kf+ nkκ≤ckf+kκfor all n∈N. Moreover, for every compact set K⊂X, continuity of fimplies sup x∈K|fn(x)−f(x)| ≤ sup x∈KˆB(0,1 /n)|f(x−y)−f(x)|µn(dy)→0as n→ ∞. Second, we verify condition (3.2). To do so, let ε > 0. By condition (3.7), there exists n0∈Nsuch that S(t)(τ−yf)≤τ−yS(t)f+εt κfor all t∈[0,1] and y∈B0,1 n0.
16 CONVEX MONOTONE SEMIGROUPS For every h∈[0,1],n≥n0and x∈X, we use Jensen’s inequality and the monotonicity of S(h)to estimate (S(h)fn)(x) = S(h) ˆB(0,1 /n) (τ−yf)( ·)µn(dy)!!(x) ≤ˆB(0,1 /n)S(h)(τ−yf)(x)µn(dy)≤ˆB(0,1 /n)τ−yS(h)f(x) + εh κ(x)µn(dy) =S(h)fn+εh κ(x). It follows from the linearity of the convolution and condition (3.6) that S(h)fn−fn h≤S(h)f−f h∗µn+ε κ≤sup y∈B(·,1 /n)S(h)f−f h(y) + ε κ.(3.8) Third, we show that the sequence (AΓfn)n∈Nis bounded above. By inequality (3.8) with ε:= 1, Jensens’s inequality and condition (3.6), there exists n0∈Nwith k(S(h)fn−fn)+kκ≤ k(S(h)f−f)+∗µnkκ+h≤ck(S(h)f−f)+kκ+h. for all h∈[0,1] and n≥n0. Since f∈ LS +, we can choose h0>0and c0≥0such that sup n∈Nk(S(h)fn−fn)+kκ≤ck(S(h)f−f)+kκ+h≤(cc0+ 1)hfor all h∈(0, h0]. This shows that (A+ Γfn)n∈Nis bounded above. Now, Theorem 3.4 yields the claim. To discuss the assumption f∗µn∈D(AΓ), we consider only the finite-dimensional case and convolution with mollifiers. Denote by Lipbthe space of all bounded Lipschitz continuous functions f:Rd→Rand by C∞ bthe space of all bounded infinitely differentiable functions f:Rd→Rsuch that all derivatives are bounded. Moreover, let C∞ c be the set of all infinitely differentiable functions f:Rd→Rwith compact support. Remark 3.7. Let X:= Rd. Moreover, let η∈C∞ cwith η≥0,supp(η)⊂B(0,1) and ´Rdη(x) dx= 1. Define ηn(x) := ndη(nx)for all n∈Nand x∈Rd. For the measure µn(A) := ´Aηn(y) dy, the convolution f∗µnis given by (f∗ηn)(x) = ˆRd f(x−y)ηn(y) dy. We have f∗ηn∈C∞ bfor all f∈Lipband n∈N. Assume that S(t): Lipb→Lipbfor all t≥0,C∞ b⊂D(AΓ)and that condition (3.7) is satisfied for all f∈Lipb. Lemma 3.6 yields f∈D(AΓ)and AΓf= Γ- limn→∞ AΓ(f∗ηn)for all f∈ LS +∩Lipb. In particular, the Γ-generator is uniquely determined by the evaluation at smooth functions. Hence, the uniqueness result, Corollary 2.11, can be improved accordingly, i.e., equality of the generator on C∞ bensures equality of the semigroups on Lipb. Furthermore, Lemma 2.7 implies equality on Cκ. 3.2.2. Regularization with sup-inf-convolution. Let Xbe a separable Hilbert space with norm |·|. We fix κ≡1and denote by BUC the space of all bounded uniformly continuous functions f:X→R. For every n∈Nand f∈BUC, we define the sup-inf-convolution θnf(x) := sup y∈X inf z∈Xf(z) + n 2|y−z|2−n|y−x|2.(3.9) It is shown in [31] that θnf∈Lip1 bfor all n∈N, and limn→∞ kθnf−fk∞= 0 for all f∈BUC. Here, Lip1 bdenotes the space of all differentiable functions f∈Lipbsuch that the first derivative is again bounded and Lipschitz continuous.
CONVEX MONOTONE SEMIGROUPS 17 Lemma 3.8. For every r > 0,f, g ∈BBUC(0, r),n∈Nand x∈X, (θnf−θng)(x)≤sup y∈B(x,rn) (f−g)(y),where rn:= (√2 + 2)qr n. Proof. In equation (3.9), the supremum can be taken over the ball B(x, p2r/n)and the infimum over the ball B(y, p4r/n). Hence, we can estimate (θnf−θng)(x)≤sup y∈B(x,√2r/n) sup z∈B(y,√4r/n)F(y, z)−G(y, z) ≤sup y∈B(x,√2r/n) sup z∈B(y,√4r/n)f(z)−g(z)= sup z∈B(x,rn)f(z)−g(z), where F(y, z) := f(z)+ n 2|y−z|2−n|y−x|2and G(y, z) := g(z)+ n 2|y−z|2−n|y−x|2. As an application of Theorem 3.4, we obtain the following result. Condition (i) is typically satisfied for first order equations, where we have Lip1 b⊂D(AΓ). Lemma 3.9. Assume that S(t): BUC →BUC for all t≥0. Let f∈ LS +∩BUC with (i) θnf∈D(AΓ)for all n∈N, (ii) S(t)(θnf)≤θnS(t)ffor all n∈Nand t≥0. Then, it holds f∈D(AΓ)and AΓf= Γ- limn→∞ AΓ(θnf). Proof. We verify the assumptions of Theorem 3.4. Define fn:= θnffor all n∈N. By definition and [31], it holds supn∈Nkfnk∞≤ kfk∞and kθnf−fk∞→0. Moreover, we use condition (ii), condition (S4) and Lemma 3.8 to choose a sequence (rn)n∈N⊂(0,∞) with rn→0such that S(h)fn−fn≤θnS(h)f−fn≤sup y∈B(·,rn)S(h)f−f(y)for all h∈[0,1]. Hence, condition (3.2) is satisfied. Furthermore, since f∈ LS +, there exist h0∈(0,1] and c≥0with S(h)fn−fn h≤sup y∈B(·,rn) S(h)f−f h≤ch for all h∈(0, h0]. This shows that (A+ Γfn)n∈Nis bounded above. Theorem 3.4 yields the claim. 3.3. Connection to distributional derivative. Let X:= Rd. Moreover, let η∈C∞ c with η≥0,supp(η)⊂B(0,1) and ´Rdη(x) dx= 1. Denote by f∗ηnthe convolution, where ηn(x) := ndη(nx). Let I⊂Nd 0be an index set and H:RI→Rbe a convex function. Our goal is to identify AΓfwith g:= H((Dαf)α∈I)if the partial derivatives Dαfexist as regular distributions for all α∈Iand gis locally integrable. Since AΓfis, in contrast to g, upper semicontinuous, we want to replace gby its upper semicontinuous hull g. To do so, we have to choose a suitable representative of g. Let f:Rd→Rbe a locally integrable function. We define Xfas the set of all x∈Rd such that the limit ˜ f(x) := lim r↓0 B(x,r) f(y) dy exists.4We remark that the Lebesgue set of a function f, consisting of all x∈Rdwith ˜ f(x) = f(x), depends on the choice of the representative, while the set Xfdoes not. 4Denoting by λthe Lebesgue measure, the normalized integral is given by B(x,r) f(y) dy:= 1 λ(B(x, r)) ˆB(x,r) f(y) dy.
18 CONVEX MONOTONE SEMIGROUPS Furthermore, by the Lebesgue differentiation theorem, the complement of the Lebesgue set has measure zero, see [40, Corollary 3.1.6]. This implies λ(Xc f) = 0 and therefore Xfis dense in Rd. Theorem 3.10. Let I⊂Nd 0be an index set and H:RI→Rbe a convex function. Let f∈ LS +satisfy condition (3.7). For every n∈N, we define fn:= f∗ηnand assume that (i) Dαfand Dαfnexist as regular distributions for all α∈I, (ii) H((Dαf)α∈I)and H((Dαfn)α∈I)are locally integrable, (iii) fn∈D(AΓ)and AΓfn=H((Dαfn)n∈N). Furthermore, we define the functions g(x) := H(Dαf(x))α∈Ifor all x∈Rd, ˜g(x) := lim r↓0 B(x,r) g(y) dyfor all x∈Xg, g(x) := lim sup y∈Xg,y→xeg(y)for all x∈Rd. Then, it holds f∈D(AΓ)and (AΓf)(x) = g(x)for all x∈Rd. In particular, we have g(x)<∞for all x∈Rd. Proof. It follows from Lemma 3.6 that f∈D(AΓ)and AΓf= Γ- limn→∞ Afn. First, we show g≤AΓf. Define F:= (Dαf)α∈Iand Fn:= (Dαfn)α∈Ifor all n∈N. It holds Fn=F∗ηnfor all n∈N, where the convolution of the vector valued-function Fwith ηn is understood componentwise. Hence, we can use condition (i) and [40, Theorem 3.2.1] to obtain Fn→Falmost everywhere. Continuity of Hand condition (iii) imply g=H(F) = lim n→∞ H(Fn) = lim n→∞ AΓfn≤Γ- lim sup n→∞ AΓfn=AΓf almost everywhere. This yields the estimate B(x,r) g(y) dy≤ B(x,r) (AΓf)(y) dyfor all x∈Rdand r > 0. For every x∈Xg, it follows from the upper semicontinuity of AΓfthat ˜g(x) = lim r↓0 B(x,r) g(y) dy≤lim sup r↓0 B(x,r) (AΓf)(y) dy≤(AΓf)(x). In particular, ˜gis bounded above and therefore g(x)<∞for all x∈Rd. We use again that AΓfis upper semicontinuous in order to conclude that g(x) = lim sup y∈Xg,y→x ˜g(y)≤lim sup y∈Xg,y→x (AΓf)(y)≤(AΓf)(x)for all x∈Rd. Second, we show AΓf≤g. Let x∈Rdand ε > 0. Because of AΓf= Γ- limn→∞ AΓfn, there exists a sequence (xn)n∈N⊂Rdwith xn→xand (AΓfn)(xn)→(AΓf)(x). In addition, since gis upper semicontinuous, there exists δ > 0such that g(y)< g(x) + ε for all y∈B(x, δ). Choose n0∈Nwith B(xn,1 /n)⊂B(x, δ)for all n≥n0. Since gis
CONVEX MONOTONE SEMIGROUPS 19 locally integrable, it holds H(F) = g= ˜g≤galmost everywhere. It follows from the previous considerations and Jensen’s inequality that AΓfn(xn) = HFn(xn)=H ˆB(0,1 /n) F(xn−y)ηn(y) dy! ≤ˆB(0,1 /n) HF(xn−y)ηn(y) dy ≤ˆB(0,1 /n) g(xn−y)ηn(y) dy≤g(x) + ε. We obtain (AΓf)(x) = limn→∞ AΓfn(xn)≤g(x)for all x∈Rd. In several examples, the set LS sym ∩Lipbis invariant under Sand has an explicit representation by means of Sobolev spaces. In addition, it holds C∞ b⊂D(AΓ)and AΓf=H((Dαf)α∈I)for all f∈C∞ b. Hence, for every f∈ LS sym ∩Lipb, the previous theorem yields that u(t) := S(t)fsolves the equation Γ- lim h↓0 u(t+h)−u(t) h=H(Dαu(t))α∈Ifor all t≥0. These ideas can be carried out by using quite elementary calculations, see Subsection 5.1. Nonetheless, we want to mention a general result that locality of the generator implies the existence of a function Hwith Af =H((Dαf)α∈I)for sufficiently smooth f. Remark 3.11. Fix κ:= 1 and assume that Srestricted to BUC is a strongly continuous semigroup. Furthermore, let H:RI→Rbe a convex function satisfying Af =H((Dα)α∈I)for all f∈BUC∞,(3.10) where I:= {α∈Nd 0:|α| ≤ 2}and BUC∞denotes the space of all infinitely differentiable functions f∈BUC such that all partial derivatives are again in BUC. Then, Ais local in sense that, for fixed x∈Rd, it holds Af(x) = Ag(x)if f, g ∈BUC∞coincide on an open neighbourhood of x. On the other hand, it was shown in [6] that locality and some (technical) regularity of the semigroup already imply the existence of a convex function Hsatisfying equation (3.10). To formulate the conditions from [6], we denote by C∞the space of all infinitely differentiable functions f:Rd→R. For every sequence r:= (rn)n∈N⊂[0,∞), let Qr be the set of all f∈C∞ csuch that kDαfk∞≤rnfor all n∈Nand |α| ≤ n. We call Sregular if the following holds: Let (rn)n∈N⊂[0,∞),f∈BUC ∩C∞and K⊂Rd compact. Then, for every T≥0and ε > 0, there exists δ0>0such that, for all t∈[0, T],δ∈(0, δ0],g∈Qrand x∈K S(t)(f+δg)(x)−S(t))(x)−δg(x)≤εt. Moreover, we call Slocal if the following holds: Let x∈Rdand f, g ∈BUC∩C∞which coincide on an open neighbourhood of x. Then, for all ε > 0, there exists h0>0with |S(h)f−S(h)g|(x)< εh for all h∈[0, h0]. In particular, this condition implies that Af(x) = Ag(x)if f, g ∈D(A). In the sequel, let Sbe regular and local. Then, by [6, Theorem 3.1], there exists a continuous function H:Rd×RI→Rsuch that Af(x) = Hx, (Dαf(x))α∈Ifor all f∈BUC∞and x∈Rd.
20 CONVEX MONOTONE SEMIGROUPS It follows from the proof of [6, Theorem 3.1] that convexity of Simplies convexity of the mapping H(x, ·): RI→Rfor all x∈Rd. Furthermore, if Sis translation invariant, i.e., S(t)(τxf) = τxS(t)f, then Hdoes not depend on x∈Rd, see [6, Proposition 4.1]. 4. Generating families In many examples, Scan be approximated by iterating another family I:= (I(t))t≥0 of operators which do not form a semigroup but have (in contrast to S) an explicit representation. This leads to approximation schemes of the form S(t)f= lim l→∞ I(2−nlt)2nltf. (4.1) Furthermore, they can be used in order to construct nonlinear semigroups, see [8]. In this case, we call Iagenerating family and San associated semigroup. In view of the previous results, we expect that two generating families with the same infinitesimal behaviour lead to the same associated semigroup. Let Ibe a family of operators I(t): Cκ→Cκ. For every t≥0and n∈N, we define the partition πt n:= {k2−n∧t:k∈N0}and the iterated operator I(πt n) := I(2−n)kI(t−k2−n),where k:= max{n∈N0:k2−n≤t}. The Lipschitz set LIconsists of all f∈Cκfor which there exist h0>0and c≥0such that kI(h)f−fkκ≤ch for all h∈[0, h0]. Define R+:= {x∈R:x≥0}and denote by T:= {k2−n:k, n ∈N0}the set of all positive dyadic numbers. Assumption 4.1. Suppose that Isatisfies the following conditions: (i) I(0) = idCκ. (ii) The operator I(t)is convex and monotone with I(t)0 = 0 for all t≥0. (iii) There exists a function α:R+×R+→R+, which is non-decreasing in the second argument, such that, for all r, s, t ≥0, I(t): BCκ(0, r)→BCκ(α(r, t)) and α(α(r, s), t)≤α(r, s +t). (iv) For every r≥0, there exists ωr≥0such that kI(t)f−I(t)gkκ≤etωrkf−gkκfor all t∈[0,1] and f, g ∈BCκ(0, r). W.l.o.g. ,we assume that the mapping r7→ ωris non-decreasing. (v) There exists a countable set D ⊂ LIsuch that the sequence (I(πt n)f)n∈Nis uniformly equicontinuous for all (f, t)∈ D ×T . Moreover, for every f∈Cκ, there exists a sequence (fn)n∈N⊂ D with kfnkκ≤ kfkκfor all n∈Nand fn→f uniformly on compacts. (vi) For every t≥0,x∈Xand (fn)n∈N⊂Cκwith fn↓0, sup s∈[0,t] sup k∈NI(πs k)fn(x)↓0as n→ ∞. For every (f, t)∈ D × T , by condition (iii) and (v) and Lemma D.1, the sequence (I(πt n)f)n∈Nhas a subsequence which convergences uniformly on compacts. Hence, up to a subsequence, we can define Sby equation (4.1). If Iis translation-invariant and κ≡1, the verification of condition (v) is particularly simple for Lipschitz continuous functions. While many properties of Sfollow from the conditions (i)-(v) or even weaker conditions, the semigroup property is rather delicate and requires an additional assumption. However, condition (vi), which implies continuity from above, also guarantees the semigroup property. Another possibility to ensure the semigroup property is to assume norm convergence or monotone convergence, see [8,23,34].
CONVEX MONOTONE SEMIGROUPS 21 Remark 4.2. We mention two sufficient conditions for Assumption 4.1(vi). (i) Let ˜κ:X→(0,∞)be another bounded continuous function such that, for every ε > 0, there exists a compact set K⊂Xwith supx∈Kc˜κ(x) κ(x)≤ε. Furthermore, we assume that there exists a function ˜α:R+×R+→R+, which is non-decreasing in the second argument, such that, for all r, s, t ≥0and f∈Cκwith kfk˜κ≤r, kI(t)fk˜κ≤˜α(r, t)and ˜α(˜α(r, s), t)≤˜α(r, s +t).(4.2) Let (fn)n∈N⊂Cκbe a sequence with fn↓0. Dini’s theorem implies uniform convergence on compact sets and, therefore, kfnk˜κ→0. It follows from Assumption 4.1(ii), equation (4.2), [8, Lemma 2.7] and Lemma B.2(ii) that sup s∈[0,t] sup k∈NkI(πs k)fnk˜κ≤˜α(3kf1k˜κ, t)kfnk˜κ→0as n→ ∞. (ii) Assume that there exists a family Jof operators J(t): Cκ→Cκ, which are continuous from above, such that, for all s, t ≥0and f∈Cκ, I(t)f≤J(t)fand J(s)J(t)f≤J(s+t)f. In addition, we suppose that the mapping [0,∞)→R, f 7→ (J(t)f)(x)is upper semicontinuous for all f∈Cκand x∈X. The monotonicity of Iimplies I(πt n)f≤ J(t)ffor all t≥0,n∈Nand f∈Cκ. Let (fn)n∈N⊂Cκwith fn↓0. Then, for every t≥0and x∈X, Dini’s theorem implies sup s∈[0,t] sup k∈NI(πs k)fn(x)≤sup s∈[0,t] (J(t)fn)(x)↓0as n→ ∞. The next statement is a consequence of the results in [8] and Appendix C. For the reader’s convenience, we provide a proof in Appendix D. Theorem 4.3. Suppose that Isatisfies Assumption 4.1. Then, there exist a family S of operators S(t): Cκ→Cκand a subsequence (nl)l∈N⊂Nsuch that, uniformly on compacts, S(t)f= lim l→∞ I(πt nl)ffor all (f, t)∈Cκ×T.(4.3) Furthermore, Ssatisfies Assumption 2.4 and is a strongly continuous convex monotone semigroup on LS. It holds LI⊂ LS. In addition, for every f, g ∈Cκ, lim h↓0 I(h)f−f h−g κ implies lim h↓0 S(h)f−f h−g κ = 0. If the sequence (I(πt n)f)n∈Nis non-decreasing, it holds S(t)f= supn∈NI(πt n)f. In particular, the limit in equation (4.3) exists without choosing a convergent subsequence. This is, for instance, the case for Nisio semigroups, see [34,35]. Furthermore, under mild assumptions, one can show that S(t)f= supπ∈PtI(π)f, where Ptconsists of all finite partitions of the interval [0.t]. For details we refer to [8, Lemma 2.15]. In order to apply the results from Subsection 3.2.1, we subsequently only consider the case X:= Rd. Furthermore, we assume that there exists δ0>0with sup x∈Rd sup y∈B(x,δ0) κ(x) κ(y)≤1.(4.4) To simplify the notation, we assume, w.l.o.g., that δ0:= 1. For every r≥0, let Lipb(r) be the space of all r-Lipschitz functions f:X→Rwith kfk∞≤r. Assumption 4.4. Suppose that Isatisfies the following conditions:
22 CONVEX MONOTONE SEMIGROUPS (i) There exist c≥0and δ∈(0,1] such that kI(t)(τxf)−τxI(t)fkκ≤crt|x| for all t∈[0,1],x∈BRd(0, δ),r≥0and f∈Lipb(r). (ii) For every f∈C∞ b, we can define the norm limit I0(0)f:= lim h↓0 I(h)f−f h∈Cκ. (iii) There exists a function β:R+×R+→R+, which is non-decreasing in the second argument, such that, for all r, s, t ≥0, I(t): Lipb(r)→Lipb(β(r, t)) and β(β(r, s), t)≤β(r, s +t). If Isatisfies Assumption 4.1 and Assumption 4.4(ii), Theorem 4.3 yields C∞ b⊂D(A) and Af =I0(0)ffor all f∈C∞ b, where Adenotes the norm generator of S. In the sequel, we fix η∈C∞ cwith η≥0,supp(η)⊂BRd(0,1) and ´Rdη(x) dx= 1. For every n∈Nand x∈Rd, we define ηn(x) := ndη(nx)and denote by f∗ηnthe convolution, see Remark 3.7. Theorem 4.5. Suppose that Isatisfies Assumption 4.1 and Assumption 4.4. Then, it holds LS +∩Lipb⊂D(AΓ)and AΓf= Γ- limn→∞ A(f∗ηn)for all f∈ LS +∩Lipb. In addition, we have S(t): Lipb→Lipbfor all t≥0. Proof. We verify condition (3.7) for all f∈Lipb. Fix n∈Nand x∈BRd(0, δ). By induction, we show that, for all k∈N,r≥0and f∈BCκ(0, r)∩Lipb(r), kI(2−n)k(τxf)−τxI(2−n)kfkκ≤cβ(r, k2−n)ek2−nωα(r,k2−n)k2−n|x|.(4.5) For k= 1, the claim holds by Assumption 4.4(i). For the induction step, we assume that inequality (4.5) holds for some fixed k∈N. Let r≥0and f∈BCκ(0, r)∩Lipb(r). Assumption 4.1(iii), Assumption 4.4(iii) and condition (4.4) imply I(2−n)f, I(2−n)(τxf), τxI(2−n)f∈BCκ(0, α(r, 2−n)) ∩Lipb(β(r, 2−n)). We use [8, Lemma 2.7] and inequality (4.5) to conclude kI(2−n)k+1(τxf)−τxI(2−n)k+1fkκ ≤ kI(2−n)kI(2−n)(τxf)−I(2−n)k(τxI(2−n)f)kκ +kI(2−n)k(τxI(2−n)f)−τxI(2−n)kI(2−n)fkκ ≤ek2−nωα(α(r,2−n),k2−n)kI(2−n)(τxf)−τxI(2−n)fkκ +cβ(β(r, 2−n), k2−n)ek2−nωα(α(r,2−n),k2−n)k2−n|x| ≤cek2−nωα(r,(k+1)2−n)β(r, 2−n)2−n|x|+cβ(r, (k+ 1)2−n)ek2−nωα(r,(k+1)2−n)k2−n|x| ≤cβ(r, (k+ 1)2−n)ek2−nωα(r,(k+1)2−n(k+ 1)2−n|x|. Equation (4.3), inequality (4.5) and equation (D.6) imply kS(t)(τxf)−τxS(t)fkκ≤cβ(r, t)etωα(r,t)t|x| for all t≥0,x∈BRd(0, δ),r≥0and f∈BCκ(0, r)∩Lipb(r). In particular, we obtain that condition (3.7) is satisfied for all f∈Lipb. Lemma 3.6 implies LS +∩Lipb⊂D(AΓ) and AΓf= Γ- limn→∞ A(f∗ηn)for all f∈ LS +∩Lipb. The invariance of Lipbis a result of Assumption 4.4(iii), equation (4.3) and equation (D.6).
CONVEX MONOTONE SEMIGROUPS 23 If we define LI +and LI sym analogously to LS +and LS sym, then, similar to [8, Lemma 2.8 and Lemma 2.9], one can show that LI +⊂ LS +. Moreover, in many examples, it holds I(t)f≤S(t)ffor all t≥0and f∈Cκand thus LI +=LS +. Unfortunately, an explicit characterization of LI +is, in general, not possible. For that purpose, the symmetric Lipschitz set is more suitable, see Subsection 5.1. In [8, Section 5], the authors provide conditions which guarantee that LI sym =LS sym and S(t): LS sym → LS sym for all t≥0. Theorem 4.6. Let Iand Jbe two generating families of operators on Cκwhich satisfy Assumption 4.1 and Assumption 4.4. Denote by Sand T, respectively, the associated semigroups which exist by Theorem 4.3. (i) Let C ⊂ LT∩LS +∩Lipbsuch that T(t): C → C for all t≥0. Assume that I0(0)f≤J0(0)ffor all f∈C∞ b. Then, it holds S(t)f≤T(t)ffor all t≥0and f∈Cκ. (ii) Assume that LJ +∩Lipb⊂ LI +,I0(0)f≤J0(0)ffor all f∈C∞ band J(s+t)f≤J(s)J(t)ffor all s, t ≥0and f∈Cκ. Moreover, let the mapping [0,∞)→R, t 7→ (J(t)f)(x)be lower semicontinuous for all f∈Cκand x∈X. Then, it holds S(t)f≤T(t)ffor all t≥0and f∈Cκ. Proof. First, it follows from Theorem 4.3 and Assumption 4.4(ii) that Af =I0(0)f≤J0(0)f=Bf for all f∈C∞ b, where Aand Bdenote the norm generators of Sand T, respectively. Hence, for every f∈ C, Theorem 4.5 and Lemma A.1(iv) imply AΓf= Γ- lim n→∞ A(f∗ηn)≤Γ- lim n→∞ B(f∗ηn) = BΓf. We use Corollary 2.11 to conclude S(t)f≤T(t)ffor all t≥0and f∈C∞ b. Let f∈Cκ be arbitrary. Condition (4.4) ensures that there exists a sequence (fn)n∈N⊂C∞ bwith kfnkκ≤ kfkκfor all n∈Nand fn→funiformly on compacts. Lemma 2.7 implies that S(t)f= limn→∞ S(t)fn≤limn→∞ T(t)fn=T(t)f. Second, it follows from equation (4.3) that J(t)f≤limn→∞ J(πt n)f=T(t)ffor all t∈ T and f∈Cκ. For arbitrary times t≥0, the inequality J(t)f≤T(t)ffollows from the required lower semicontinuity of Jand condition (S3). We obtain LT∩Lipb⊂ LT +∩Lipb=LJ +∩Lipb⊂ LI +⊂ LS +. By Remark 2.9, Theorem 4.3 and Theorem 4.5, it holds T(t): LT∩Lipb→ LT∩Lipbfor all t≥0. Hence, the claim follows from part (i). 5. Examples 5.1. Control problems and upper semigroup envelopes. In this subsection, we show that value functions of stochastic optimal control problems can be approximated by a sequence of static optimization problems over increasingly finer partitions. The latter corresponds to the construction of so-called upper semigroup envelopes, cf. Nisio [35] and Nendel and Röckner [34]. For a representative class of optimal control problems, we thus show that the value function coincides with the upper envelope of a suitable family of penalized linear semigroups. Let (Wt)t≥0be a d-dimensional standard Brownian motion on a complete filtered probability space (Ω,F,(Ft)t≥0,P)satisfying the usual conditions. Denote by Sd +the
30 CONVEX MONOTONE SEMIGROUPS Typical examples include Koopman semigroups and transition semigroups of Lévy and Ornstein–Uhlenbeck processes, see [23]. Denote by LRthe Lipschitz set of R. Let C∞ 0 be the space of all infinitely differentiable functions f∈C0such that all derivatives are in C0, where C0consists of all continuous functions f:Rd→Rwith lim|x|→∞ f(x)=0. Assumption 5.7. Suppose that Rforms a semigroup. Furthermore, let (µt)t≥0and (ψt)t≥0satisfy the following conditions: (i) limt↓0´Rd|y|pdµt(y)=0. (ii) There exist r > 0and c≥0such that µt(B(0, r)c)≤ct for all t∈[0,1]. (iii) ψt(0) = 0 for all t≥0. (iv) There exists c≥0such that, for all x, y ∈Rdand t∈[0,1], |ψt(x)−ψt(y)−(x−y)| ≤ ct|x−y|. (v) For every f∈C∞ 0∩LR, the limit R0(0)f:= lim h↓0 R(h)f−f h∈Cb exists uniformly on compacts. For every x, y ∈Rdand t∈[0,1], the conditions (iii) and (iv) imply |ψt(x)−x| ≤ ct|x|and |ψt(x)−ψt(y)| ≤ ect|x−y|.(5.1) Lemma 5.8. It holds C∞ c⊂ LR. Proof. Let f∈C∞ c. Choose c, r ≥0such that Assumption 5.7(ii) and (iv) are satisfied. In addition, let supp(f)⊂B(0, r). By Assumption 5.7(v), there exist t0∈(0,1 /2c]and c0≥0with sup x∈B(0,4r)|(R(t)f−f)(x)| ≤ c0tfor all t∈[0, t0]. Inequality (5.1) implies |ψt(x) + y| ≥ rfor all x∈B(0,4r)c,y∈B(0, r)and t∈[0, t0]. Hence, for every x∈B(x, 4r)cand t∈[0, t0], it follows from Assumption 5.7(ii) that |(R(t)f)(x)−f(x)|≤kfk∞µtB(0, r)c≤ckfk∞t. We obtain kR(t)f−fk∞≤max{c0, ckfk∞}tfor all t∈[0, t0]. In the sequel, we consider a perturbation of the linear transition semigroup R, where we take the supremum over all probabilities which are sufficiently close to the reference measure µt. Recall that, in Subsection 5.1, we considered a Brownian motion with uncertain drift and volatility, where the uncertainty was parametrized by a finitedimensional parameter space. Here, the non-parametric uncertainty is instead given by an infinite-dimensional ball of probability measures. We define the p-Wasserstein distance by Wp(µ, ν) := inf π∈Cpl(µ,ν)ˆRd×Rd|y−z|pdπ(y, z)1 p for all µ, ν ∈ Pp, where Cpl(µ, ν)consists of all probability measures on B(Rd×Rd)with first marginal µand second marginal ν. Let ϕ: [0,∞)→[0,∞]be a convex lower semicontinuous function with ϕ(0) = 0 and ϕ(v)>0for some v > 0. Furthermore, we assume that that the mapping [0,∞)→[0,∞], v 7→ ϕv1/pis convex. The previous assumptions ensure that ϕ∗(w) := sup v≥0vw −ϕ(v)<∞for all w≥0.
CONVEX MONOTONE SEMIGROUPS 31 For every t≥0,f∈Cband x∈Rd, we define (I(t)f)(x) := sup ν∈PpˆRd f(ψt(x) + z) dν(z)−ϕtWp(µt, ν), where ϕt: [0,∞)→[0,∞]denotes the rescaled function ϕt(v) := tϕv t, t > 0, v ≥0, 0, t =v= 0, +∞, t = 0, v 6= 0. The following result is a consequence of [23, Theorem 3.13] and Lemma 3.6. Theorem 5.9. There exists a semigroup Son Cb, which satisfies Assumption 2.4, such that, uniformly on compacts, S(t)f= lim n→∞ I(πn)f= inf n∈NI(πn)ffor all f∈Cband t≥0.(5.2) Denoting by AΓthe Γ-generator of S, it holds C∞ 0∩LR⊂D(AΓ)and AΓf=R0(0)f+ϕ∗(|∇f|) = lim h↓0 S(h)f−f h uniformly on compacts for all f∈C∞ 0∩LR. Condition (3.7)is satisfied for all f∈Lipb. Furthermore, it holds S(t): Lipb→Lipband S(t): C0→C0for all t≥0. Proof. By [23, Theorem 3.13 and Remark 3.14], there exists a semigroup Son Cb which satisfies equation (5.2), Assumption 2.4 and the statement about the Γ-generator. Choose c≥0such that Assumption 5.7(iv) is satisfied. It follows from the proof of Theorem 4.5 that condition (3.7) holds for all f∈Lipbif we verify the following conditions: •I(t): BCb(0, r)→BCb(0, r)for all r, t ≥0, • kI(t)f−I(t)gk∞≤ kf−gk∞for all t≥0and f, g ∈Cb, •I(t): Lipb(r)→Lipb(ectr)for all r, t ≥0, • kτx(I(t)f)−I(t)(τxf)k∞≤crt|x|for all r, t ≥0,f∈Lipb(r)and x∈Rd. The first two statements follow immediately from the definition of I. For every r, t ≥0, f∈Lipb(r)and x, y ∈Rd, inequality (5.1) implies |(I(t)f)(x)−(I(t)f)(y)| ≤ sup ν∈PpˆRd|f(ψt(x) + z)−f(ψt(y) + z)|dν(z) ≤ectr|x−y|.(5.3) Furthermore, we use Assumption 5.7(iv) to estimate |(τx(I(t)f)−I(t)(τxf)|(y)≤sup ν∈PpˆRd|f(ψt(x+y) + z)−f(ψt(y) + x+z)|dν(z) ≤r|ψt(x+y)−ψt(y)−x| ≤ crt|x|. It remains to show S(t): C0→C0for all t≥0. Since Sis semigroup, it suffices to show S(t)f∈C0for all t∈[0,1 /2c]and f∈C0. Since S(t)is monotone and convex with S(t)0 = 0, it holds |S(t)f| ≤ S(t)|f| ≤ I(t)|f|. By [23, Lemma 3.4], there exists a constant c0≥0with I(t)|f|(x)≤sup {v∈Pp:Wp(µt,ν)≤c0}ˆRd|f(ψt(x) + y)|dν(y).
32 CONVEX MONOTONE SEMIGROUPS Let ε > 0. Since f∈C0and the Wasserstein ball {v∈ Pp:Wp(µt, ν)≤c0}is tight, we can choose r≥0with sup x∈B(0,r)c|f(x)| ≤ ε 2and sup Wp(µt,ν)≤c0 ν(B(0, r)c)kfk∞≤ε 2. For every x∈B(0,4r)cand y∈B(0, r), inequality (5.1) implies |ψt(x) + y| ≥ r. Thus, I(t)|f|(x)≤sup Wp(µt,ν)≤c0 ˆB(0,r)|f(ψt(x) + y)|dν(y) + ˆB(0,r)c|f(ψt(x) + y)|dν(y)! ≤sup Wp(µt,ν)≤c0ε 2+ν(B(0, r)c)kfk∞≤ε. We obtain S(t): C0→C0for all t≥0. In addition to the Wasserstein perturbation, we consider a perturbation which is parametrized only by drifts b∈Rd. For every t≥0,f∈Cband x∈Rd, we define (J(t)f)(x) := sup b∈RdˆRd f(ψt(x) + y+b) dµt(y)−ϕt(|b|t). We remark that Wp(µt, νb) = |b|tfor all b∈Rdand t≥0, where νb(A) := ˆRd 1A(b+y) dµt(y)for all A∈ B(Rd). Furthermore, the previous definition of Jis consistent with the one in Subsection 5.1 if we fix the volatility matrix aand choose µt:= P◦(√aWt)−1and L(b) := ϕ(|b|). Note that uncertainty in the volatility is not included in the setting of this subsection. Define Lip0:= Lipb∩C0. Theorem 5.10. There exists a family Tof operators T(t): Cb→Cb, which satisfy Assumption 2.4, such that, uniformly on compacts, T(t)f= lim n→∞ J(πt n)ffor all (f, t)∈Cb×T.(5.4) Furthermore, Tis a strongly continuous convex monotone semigroup on C0. Denoting by BΓthe Γ-generator of T, it holds C∞ 0∩LR⊂D(BΓ)and BΓf=R0(0)f+ϕ∗(|∇f|) = lim h↓0 T(h)f−f h uniformly on compacts for all f∈C∞ 0∩LR. Condition (3.7)is satisfied for all f∈Lip0. Furthermore, we have T(t): Lip0→Lip0for all t≥0. Proof. First, we verify Assumption 4.1. Clearly, the conditions (i)-(iv) are satisfied. Choose c≥0such that Assumption 5.7(iv) is satisfied. Similar to inequality (5.3), one can show that J(t): Lipb(r)→Lipb(ectr)for all r, t ≥0. In particular, the sequence (J(πt n)f)n∈Nis uniformly Lipschitz continuous for all t≥0and f∈Lipb. Assumption 4.1(vi) follows from Remark 4.2(ii), since J(t)f≤S(t)ffor all t≥0and f∈Cb. Next, we show that J(t): Cb→Cbfor all t≥0. Let t≥0and f∈Cb. Choose a bounded sequence (fn)n∈N⊂Lipbwith fn→funiformly on compacts. Lemma 2.7 yields J(t)fn→J(t)funiformly on compacts. Indeed, the corresponding proof only relies on the fact that J(t)is convex, monotone and continuous from above. Since J(t)fn∈Lipb⊂Cbfor all n∈N, we obtain J(t)f∈Cb. Moreover, it holds |J(t)f| ≤ J(t)|f| ≤ S(t)|f|and thus |J(πt n)f| ≤ S(t)|f| ∈ C0for all t≥0,n∈Nand
CONVEX MONOTONE SEMIGROUPS 33 f∈C0. It remains to show that LJ⊂C0is dense. For every c, t ≥0and f∈Lipb(c), it follows from [23, Equation (3.7)] that 0≤J(t)f−R(t)f≤I(t)f−R(t)f≤tϕ∗(c).(5.5) Hence, Lemma 5.8 implies C∞ c⊂ LR⊂ LJ. By Theorem 4.3, there exists a family T of operators T(t): Cb→Cband a subsequence (nl)l∈N, which satisfy Assumption 2.4, such that, uniformly on compacts, T(t)f= lim l→∞ J(πt nl)ffor all (f, t)∈Cb×T. Since (J(πt n)f)n∈Nis non-decreasing, the convergence holds without choosing a subsequence. In addition, Tis a strongly continuous convex monotone semigroup on C0and it holds T(t): Lip0→Lip0for all t≥0. Second, we show that C∞ 0∩LR⊂D(BΓ)and BΓf=R0(0)f+ϕ∗(|Df|) = lim h↓0 T(h)f−f h uniformly on compacts for all f∈C∞ 0∩LR. For every h > 0and b, x, y ∈Rd, we use Taylor’s formula to obtain |f(ψh(x) + y+bh)−f(ψh(x) + y)−h∇f(ψh(x) + y), bhi| ≤ kD2fk∞|b|2h2. As seen in the proof of [23, Lemma 3.6], it holds limh↓0|R(h)g−g|(x) = 0 for all g∈Lipb. Hence, we can estimate lim inf h↓0J(h)f−R(h)f h(x)≥lim inf h↓0ˆRdh∇f(ψh(x) + y, bidµh(y)−ϕ(|b|) = lim inf h↓0R(h)h∇f, bi(x)−ϕ(|b|) =h∇f(x), bi−ϕ(|b|). Taking the supremum over all b∈Rdyields lim inf h↓0J(h)f−R(h)f h(x)≥ϕ∗(|∇f|). We use the previous inequality, J(h)f≤T(h)f≤S(h)f, Assumption 5.7(v) and Theorem 5.9 to conclude lim h↓0 T(h)f−f h=R0(0)f+ϕ∗(|∇f|) uniformly on compacts. Third, we remark that the verification of condition (3.7) for all f∈Lip0is similar to the proof of Theorem 5.9. Moreover, it follows from J(t): Lip0(r)→Lip0(ectr)and the construction of Tthat T(t): Lip0(r)→Lip0(ectr)for all r, t ≥0. Theorem 5.11. It holds S(t)f=T(t)ffor all t≥0and f∈Cb. Proof. By construction, it holds 0≤T(t)f−R(t)f≤S(t)f−R(t)f≤I(t)f−R(t)f for all t≥0and f∈Lip0. Hence, inequality (5.5) implies LR∩Lip0=LS∩Lip0=LT∩Lip0. Let D:= LR∩Lip0. It follows from Remark 2.9(i) and Theorem 5.10 that T(t): D → D for all t≥0. Next, we show that AΓf=BΓffor all f∈ D. Let η∈C∞ cwith η≥0, supp(η)⊂B(0,1), and ´Rdη(x) dx= 1. Define ηn(x) := ndη(nx)and fn:= f∗ηnfor all n∈Nand x∈Rd. Then, it holds fn∈C∞ 0for all n∈N. Choose c, r ≥0such that
34 CONVEX MONOTONE SEMIGROUPS Assumption 5.7(iv) is satisfied and f∈Lip0(r). For every t≥0,n∈Nand x∈Rd, we use Fubini’s theorem to estimate |R(t)fn−(R(t)f)∗ηn|(x) ≤ˆB(0,1) ˆRd|f(ψt(x) + y−z)−f(ψt(x−z) + y)|dµt(y)ηn(z) dz≤crt. We obtain kR(t)fn−fnk∞≤ kR(t)f−fk∞+crt and, therefore, fn∈ LR. It follows from Lemma 3.6, Theorem 5.9 and Theorem 5.10 that AΓf= Γ- lim n→∞ AΓfn= Γ- lim n→∞ BΓfn=BΓf. Hence, Corollary 2.11 implies S(t)f≤T(t)ffor all t≥0and f∈C∞ 0∩LRwhile the inequality S(t)f≥T(t)fholds by construction. Now, let f∈Cbbe arbitrary and choose a bounded sequence (fn)n∈N⊂C∞ cwith fn→funiformly on compacts. We use Lemma 2.7 and Lemma 5.8 to conclude S(t)f= lim n→∞ S(t)fn= lim n→∞ T(t)fn=T(t)f. In the particular case that Scoincides with the entropic semigroup, as a byproduct of Theorem 5.11, we recover that µtsatisfies the Talagrand T2inequality, see [45, Chapter 22] and [41]. Denote by N(0, t1)the d-dimensional normal distribution with mean zero and covariance matrix t1, where 1∈Rd×dis the identity matrix. Corollary 5.12. It holds W2(ν, µt)≤p2tH(ν|µt)for all t≥0and ν∈P2, where H(ν|µt)denotes the relative entropy of νw.r.t. µt:= N(0, t). Proof. Choose ψt:= idRd,µt:= N(0, t1)and ϕ(v) := v2/2for all t, v ≥0. Moreover, let (Wt)t≥0be a d-dimensional standard Brownian motion on a complete filtered probability space (Ω,F,(Ft)t≥0,P)satisfying the usual conditions. We show that (T(t)f)(x) = ( ˜ S(t)f)(x) := 1 2log E[exp(2f(x+Wt))] for all t≥0,f∈Cband x∈Rd. By elementary calculation, see [7, Lemma 4.4] and the proof of [7, Theorem 4.5], one can show that J(t)f≤˜ S(t)fand thus T(t)f≤˜ S(t)ffor all t≥0and f∈Cb. To show the reverse inequality, we want to apply Corollary 2.11. By straightforward computations, one can show that ˜ Ssatisfies Assumption 2.4 and BUC2⊂D(˜ A)with ˜ Af =1 2∆f+|∇f|2for all f∈BUC2. Moreover, it follows from Itô’s formula and Theorem 5.3 that D:= LT sym ∩Lipb⊂ L˜ S +. Since ˜ Sand Tare translation invariant, Lemma 3.6 yields ˜ AΓf=BΓffor all f∈ D. Hence, Corollary 2.11 implies ˜ S(t)f=T(t)ffor all t≥0and f∈BUC2. For arbitrary f∈Cb, the equality follows by approximation. In addition, it follows from Theorem 5.11 that ˜ S(t)f=S(t)f≤I(t)ffor all t≥0and f∈Cb. Fenchel–Moreaus’s theorem yields W2(ν, µt)2 2t= sup f∈CbˆRd fdν−(I(t)f)(0) ≤sup f∈CbˆRd fdν−(˜ S(t)f)(0)=H(ν|µt).
CONVEX MONOTONE SEMIGROUPS 35 Appendix A. Γ-convergence Following the works of Beer [4], Dal Maso [15] and Rockafellar and Wets [38], we gather some basics about Γ-convergence. We remark that, in [15] and [38], all results are formulated for extended real-valued lower semicontinuous functions. However, a function f:X→[−∞,∞)is upper semicontinuous if and only if −f:X→(−∞,∞] is lower semicontinuous, and all results immediately transfer to our setting. Lemma A.1. Let (fn)n∈N⊂Uκand (gn)n∈N⊂Uκbe bounded above, and f, g ∈Uκ. (i) It holds Γ- lim supn→∞ fn∈Uκ. Furthermore, (fn)n∈N⊂Uκhas a Γ-convergent subsequence, i.e., Γ- limk→∞ fnk∈Uκexists for a subsequence (nk)k∈N. (ii) We have f= Γ- limn→∞ fnif and only if every subsequence (nk)k∈Nhas another subsequence (nkl)l∈Nwith f= Γ- liml→∞ fnkl. (iii) If fn↓f, then f= Γ- limn→∞ fn. (iv) It holds Γ- lim supn→∞(fn+gn)≤Γ- lim supn→∞ fn+ Γ- lim supn→∞ gn. Moreover, we have Γ- lim supn→∞ fn≤Γ- lim supn→∞ gnif fn≤gnfor all n∈N. (v) Assume that f∈Cκ,fn→funiformly on compacts and g= Γ- limn→∞ gn. Then, it holds f+g= Γ- limn→∞(fn+gn). (vi) If f= Γ- lim supn→∞ fnand g∈Cκ, then f∨g= Γ- lim supn→∞(fn∨g). (vii) It holds (Γ- lim supn→∞ fn)(x)≥lim supn→∞ supy∈B(x,δn)fn(y)for all x∈Xand (δn)n∈N⊂(0,∞)with δn→0. Proof. Part (i) follows from [15, Remark 4.11], [15, Theorem 4.16], and [15, Theorem 8.4]. For part (ii) and (iii), we refer to [15, Proposition 8.3] and [15, Proposition 5.4], respectively. Part (iv) and (vii) are direct consequences of the definition of the Γ- lim sup. In order to show part (v), let x∈Xand (xn)n∈N⊂Xwith xn→xand g(x) = limn→∞ gn(xn). For every n∈N, (f+g)(x)−(fn+gn)(xn)≤ |f(x)−f(xn)|+ sup y∈K|f(y)−fn(y)|+|g(x)−gn(xn)|, where K:= {xn:n∈N}∪{x}is compact. Since fn→funiformly on compact and fis continuous, the right-hand side converges to zero. We obtain f+g≤Γ- limn→∞(fn+gn) and the reverse inequality follows from part (iv). It remains to show part (vi). The inequality f∨g≤Γ- lim supn→∞(fn∨g)follows from part (iv). Let (xn)n∈N⊂Xand x∈Xwith xn→x. Continuity of gimplies lim sup n→∞ (fn∨g)(x) = lim k→∞(fnk∨g)(xk) = lim sup k→∞ fnk(xnk)∨g(x)≤(f∨g)(x), where (xnk)k∈Nis a suitable subsequence approximating the limes superior. Let f∈Uκand ε > 0. Following [4], the upper ε-parallel function to fis defined by fε:X→R, x 7→ 1 κ(x)sup y∈B(x,ε) max f(y)κ(y),−1 ε+ε. If closed bounded sets in Xare compact (e.g., if X=Rdor Xis compact), one can show that fεis upper semicontinuous, cf. the proof of [4, Lemma 1.3]. For a general metric space (X, d), however, the upper semicontinuity of fεcannot be guaranteed. For this reason, we consider the upper semicontinuous envelope of fε, which is defined by fε:X→R, x 7→ lim sup y→xfε(y) = inf δ>0sup y∈B(x,δ) fε(y).
36 CONVEX MONOTONE SEMIGROUPS Note that, for all x∈X, fε(x) = sup nlim sup n→∞ fε(xn): (xn)n∈N⊂Xwith xn→xo= Γ- lim sup n→∞ fε, where in the last expression fεstands for the constant sequence (fε)n∈N. We have the following geometric description of the Γ- lim sup. Lemma A.2. (i) For every f∈Uκand ε > 0, fε≤fε≤inf ε0>ε fε0and k(fε)+kκ≤ kf+kκ+ε. (A.1) Furthermore, it holds fε∈Uκfor all ε > 0and fε↓fas ε↓0. (ii) Let (fn)n∈N⊂Uκbe bounded above and f∈Uκ. Then, Γ- lim supn→∞ fn≤fif and only if, for every ε > 0and compact set K⊂X, there exists n0∈Nwith fn(x)≤fε(x)for all x∈Kand n≥n0.(A.2) Proof. First, we show inequality A.1. Fix f∈Uκ,ε > 0and x∈X. For every ε0> ε, fε(x)≤fε(x)≤sup y∈B(x,ε0−ε) fε(y) = sup y∈B(x,ε0−ε) 1 κ(y)sup z∈B(y,ε) max f(z)κ(z),−1 ε+ε ≤cε0(x) κ(x)sup y∈B(x,ε0−ε) sup z∈B(y,ε) max f(z)κ(z),−1 ε+ε ≤cε0(x) κ(x)sup z∈B(x,ε0) max f(z)κ(z),−1 ε0+ε0=cε0(x)fε0(x), where cε0(x) := supy∈B(x,ε0−ε) κ(x) κ(y). Continuity of κimplies cε0(x)↓1as ε0↓ε. Hence, taking the infimum over ε0> ε in the previous estimate yields the first part of inequality (A.1). Furthermore, we can estimate fεκ+(x)≤inf ε0>ε fε0κ+(x)≤inf ε0>ε sup y∈X (fκ)+(y) + ε0=kf+kκ+ε. In particular, we obtain fε∈Uκ, because fεis upper semicontinuous by definition. Second, we show that fδ↓fas δ↓0. Due to inequality (A.1) it is sufficient to prove fδ↓fas δ↓0. Since fκ is upper semicontinuous, for every x∈Xand ε > 0, there exists δ > 0such that (fκ)(y)≤(fκ)(x) + εfor all y∈B(x, δ). We obtain f(x)≤fδ(x) = 1 κ(x)sup y∈B(x,δ) max (fκ)(y),−1 δ+δ≤max f(x),−1 δ+δ+ε. This implies f(x)≤infδ>0fδ(x)≤f(x) + ε↓f(x)as ε↓0. Third, let (fn)n∈N⊂Uκbe bounded above and f∈Uκwith Γ- lim supn→∞ fn≤f. We follow the proof of [4, Lemma 1.5] to verify inequality (A.2). Let K⊂Xbe compact and ε > 0. Since Γ- lim supn→∞ fn≤fand κ > 0is continuous, we obtain lim sup n→∞ (fnκ)(xn)≤(fκ)(x)for all x∈Kand (xn)n∈N⊂Xwith xn→x. Hence, for every x∈K, there exist nx∈Nand rx∈(0, ε)such that (fnκ)(y)≤max (fκ)(x),−1 ε+εfor all n≥nxand y∈B(x, rx).
CONVEX MONOTONE SEMIGROUPS 37 By compactness of K, we can choose x1, . . . , xk∈Kwith K⊂Sk i=1 B(xi, rxi). Define n0:= nx1∨. . . ∨nxk. Let x∈Kand i∈ {1, . . . , k}with d(x, xi)< rxi< ε. We obtain fn(x)≤1 κ(x)max (fκ)(xi),−1 ε+ε≤fε(x)for all n≥n0. Fourth, let (fn)n∈N⊂Uκbe bounded above and f∈Uκsuch that inequality (A.2) holds. Let x∈Xand (xn)n∈N⊂Xwith xn→x. Since K:= {xn:n∈N}∪{x}is compact, for every ε > 0, there exists n0∈Nsuch that fn(xn)≤fε(xn)for all n≥n0. We obtain lim supn→∞ fn(xn)≤fε(x)for all ε > 0, because fεis upper semicontinuous. Hence, part (i) implies Γ- lim supn→∞ fn≤infε>0fε=f. We conclude this section by noting how the definition of fεsimplifies if (X, d)satisfies an additional geometric property. Lemma A.3. Assume that (X, d)has midpoints, i.e., for every x, z ∈Xand λ∈[0,1], there exists yλ∈Xwith d(x, yλ) = λd(x, z)and d(yλ, z) = (1 −λ)d(x, z). Then, it holds fε= infε0>ε fε0for all f∈Uκand ε > 0. Proof. Let x∈Xand ε0> ε. Since (X, d)has midpoints, for every z∈B(x, ε0)there exists y∈Xwith d(x, y)≤ε0−εand d(y, z)≤ε. Hence, we can estimate fε0(x) = 1 κ(x)sup z∈B(x,ε0)max f(z)κ(z),−1 ε0+ε0 =1 κ(x)sup y∈B(x,ε0−ε) sup z∈B(y,ε)max f(z)κ(z),−1 ε0+ε0 ≤cε0(x) sup y∈B(x,ε0−ε) 1 κ(y)sup z∈B(y,ε)max f(z)κ(z),−1 ε0+ε0, where cε0(x) := supy∈B(x,ε0−ε) κ(y) κ(x). Continuity of κimplies cε0↓1as ε0↓ε. We obtain inf ε0>ε fε0(x)≤inf ε0>ε sup y∈B(x,ε0−ε) fε(y) = fε(x). The reverse estimate follows from inequality (A.1). Appendix B. Basic convexity estimates Lemma B.1. Let φ: Cκ→Rbe a convex functional. Then, φ(f)−φ(g)≤λφf−g λ+g−φ(g)for all f, g ∈Cκand λ∈(0,1]. The previous statement remains valid if we replace Cκby Bκ. Proof. We use the convexity to estimate φ(f)−φ(g) = φλf−g λ+g+ (1 −λ)g−φ(g) ≤λφ f−g λ+g+ (1 −λ)φ(g)−φ(g) =λφf−g λ+g−φ(g).
38 CONVEX MONOTONE SEMIGROUPS Let Fκbe the space of all functions f:X→[−∞,∞)with kf+kκ<∞. Lemma B.2. Let Φ: Cκ→Fκbe a convex monotone operator with Φ(0) = 0. (i) For every r≥0, there exists c≥0such that kΦ(f)kκ≤ckfkκfor all f∈BCκ(0, r). One can choose c:= 1 rkΦ(r κ)+kκ. In particular, the function Φ(f)is real-valued, i.e., Φ(f): X→Rfor all f∈Cκ. (ii) For every r≥0, there exists c≥0such that kΦ(f)−Φ(g)kκ≤ckf−gkκfor all f, g ∈BCκ(0, r). One can choose c:= 1 rsupf0∈BCκ(0,3r)kΦ(f0)kκ<∞. The previous statements remain valid if we replace Cκby Bκ. Proof. First, let r > 0,f∈BCκ(0, r)and λ:= kfkκ/r. We use the fact that Φis convex and monotone with Φ(0) = 0 to estimate Φ(f) = Φλ1 λf+ (1 −λ)0≤λΦ1 λf≤λΦ(r κ=kfkκ rΦ(r κ. Moreover, it follows from the convexity of Φand Φ(0) = 0 that 0 = Φ(0) = Φ1 2f+1 2(−f)≤1 2Φ(f) + 1 2Φ(−f). We conclude (Φ(±f))(x)>−∞ for all x∈Xand −Φ(−f)≤Φ(f). Combining the previous estimates yields −kfkκ rΦ(r κ≤ −Φ(−f)≤Φ(f)≤kfkκ rΦ(r κ. Hence, it holds kΦ(f)kκ≤ckfkκwith c:= 1 rkΦ(r κ)+kκ<∞. For r= 0, the claim follows from Φ(0) = 0. Second, let r≥0and f, g ∈BCκ(0, r). We define Φf: Cκ→Fκ, f07→ Φ(f+f0)−Φ(f)for all f0∈Cκ. Note, that kΦ(f)kκ<∞by the first part and therefore Φ(f0)∈Fκfor all f0∈Cκ. Furthermore, it follows from the first part that kΦ(f)−Φ(g)kκ= Φf(f−g) κ≤1 2r Φf(2r κ)+ κkf−gkκ≤ckf−gkκ, where c:= 1 rsupf0∈BCκ(0,3r)kΦ(f0)kκ<∞. Appendix C. Extension of convex monotone functions Denote by ca+ κthe set of all Borel measures µ:B(X)→[0,∞]with ´X 1 κdµ < ∞. Let φ: Cκ→Rbe a convex monotone functional with φ(0) = 0. We define the convex conjugate of φby φ∗: ca+ κ→[0,∞], µ 7→ sup f∈Cκµf −φ(f),where µf := ˆX fdµ. Let Bκbe the space of all Borel measurable functions f:X→[−∞,∞)such that kf+kκ<∞. Denote by BBκ(0, r) := {f∈Bκ:kfkκ≤t}the closed ball with radius r≥0around zero. Using the ideas from [3], we obtain the following extension and dual representation result. Theorem C.1. Let φ: Cκ→Rbe a convex monotone functional with φ(0) = 0, which is continuous from above. Then, the following statements hold:
CONVEX MONOTONE SEMIGROUPS 39 (i) For every r≥0, there exists a σ(ca+ κ,Cκ)-compact convex set Mr⊂ca+ κwith φ(f) = max µ∈Mrµf −φ∗(µ)for all f∈BCκ(0, r). One can choose Mr:= {µ∈ca+ κ:φ∗(µ)≤φ(2r/κ)−2φ(−r/κ)}. (ii) Define φ1: Uκ→[−∞,∞), f 7→ inf{φ(g): g∈Cκ, g ≥f}. The functional φ1is convex, monotone and the unique extension of φ, which is continuous from above. In addition, φ1is admits the dual representation φ1(f) = max µ∈Mrµf −φ∗(µ)for all r≥0and f∈BUκ(0, r). (iii) Define φ2: Bκ→[−∞,∞), f 7→ limc→∞ supµ∈ca+ κµmax f, −c κ−φ∗(µ). The functional φ1is convex, monotone and an extension of φ. In addition, φ1is admits the dual representation φ2(f) = sup µ∈M0 rµf −φ∗(µ)for all r≥0and f∈BBκ(0, r), where M0 r:= Tε>0{µ∈ca+ κ:φ∗(µ)≤φ(2r/κ)−2φ(−r/κ) + ε}is σ(ca+ κ,Cκ)- compact and convex. In particular, for all ε > 0and r≥0, there exists a compact set K⊂Xwith φ2r κ1Kc< ε. Proof. First, we apply [3, Theorem 2.2] to obtain φ(f) = max µ∈ca+ κµf −φ∗(µ)for all f∈Cκ. Let r≥0and f∈BCκ(0, r). Choose µ∈ca+ κwith φ(f) = µf −φ∗(µ). It follows from the definition of φ∗and the monotonicity of φthat µ2r κ−φ2r κ≤φ∗(µ) = µf −φ(f)≤µr κ−φ−r κ. We obtain µr κ≤φ(2r κ)−φ(−r κ)and therefore φ∗(µ)≤φ2r κ)−2φ−r κ. Hence, φ(f) = max µ∈Mrµf −φ∗(µ)for all f∈BCκ(0, r), where Mr:= µ∈ca+ κ:φ∗(µ)≤φ2r/κ)−2φ−r/κ. Moreover, the set Mris convex and σ(ca+ κ,Cκ)-compact, see [3, Theorem 2.2]. Second, by monotonicity of φ, the functional φ1is monotone and an extension of φ. We show that φ1(f) = limn→∞ φ(fn)for all (fn)n∈N⊂Cκand f∈Uκwith fn↓f. By definition of the infimum, there exists a sequence (gk)k∈N⊂Cκsuch that φ(gk)→φ1(f) as k→ ∞. Let gk n:= fn∨gkfor all k, n ∈N. Since gk n↓gkas n→ ∞, and φis monotone and continuous from above, we obtain lim n→∞ φ(fn)≤lim n→∞ φ(gk n) = φ(gk)for all k∈N. Monotonicity of φ1implies φ1(f)≤limn→∞ φ(fn)≤limk→∞ φ(gk) = φ1(f). In particular, it follows that φ1is convex. Indeed, let f, g ∈Uκand λ∈[0,1]. Since Uκ= (Cκ)δ and Cκis directed downwards, there exist sequences (fn)n∈Nand (gn)n∈Nin Cκwith fn↓fand gn↓g. We obtain φ1(λf + (1 −λ)g) = lim n→∞ φ(λfn+ (1 −λ)gn)≤lim n→∞ λφ(fn) + (1 −λ)gn =λφ1(f) + (1 −λ)φ1(g).
46 CONVEX MONOTONE SEMIGROUPS For every n≥n0, we use Lemma B.1, inequality (D.3), inequality (D.8), the monotonicity of I(πs n)and r≤α(r, t)to estimate I(πs n)I(πt n)f−I(πs n)S(t)f≤λI(πs n)I(πt n)f−S(t)f λ+S(t)f−λI(πs n)S(t)f ≤λI(πs n)δ λκ +2α(r, t) λκ 1Kc+α(r, t) κ+λα(r, s +t) κ ≤λI(πs n)2α(r, t) κ+2c κ1Kc+ε κ. Furthermore, convexity of I(πs n), inequality (D.3) and inequality (D.7) imply λI(πs n)2α(r, t) κ+2c κ1Kc+ε κ≤λ 2I(πs n)4α(r, t) κ+λ 2I(πs n)4c κ1Kc+ε κ ≤λα(4α(r, t), s) 2κ+ε 2κ+ε κ≤2ε κ. Interchanging the roles of fand gin the previous estimate yields |I(πs n)I(πt n)f−I(πs n)S(t)f|<2ε κ(x)for all n≥n0. We conclude S(s+t)f=S(s)S(t)ffor all s, t ∈ T and f∈Cκ. Now, let s, t ≥0be arbitrary. Choose (sn)n∈N⊂[0, s]∩T and (tn)n∈N⊂[0, t]∩T with sn→sand tn→t. It follows from condition (S4), inequality (D.5) and equation (D.6) that S(s+t)f−S(s)S(t)f=S(s+t)f−S(sn+tn)f+S(sn)S(tn)f−S(sn)S(t)f +S(sn)S(t)f−S(s)S(t)f→0 uniformly on compacts. Sixth, for every f, g ∈Cκ, it follows from [8, Theorem 4.3 and Lemma 4.4] that lim h↓0 I(h)f−f h−g κ implies lim h↓0 S(h)f−f h−g κ = 0. In [8] it is assumed that the semigroup Sis strongly continuous. However, we obtain immediately from the proof of [8, Theorem 4.3] that the inequality S(t)f−f t−g κ≤lim sup s→t S(s)f−f s−g κ for all t > 0 is sufficient. The latter is ensured by equation (D.6). Furthermore, by Remark 2.9(i), it holds S(t): LS→ LSfor all t≥0. Hence, the previous results yield that Sis a strongly continuous convex monotone semigroup on LS. The inclusion LI⊂ LSfollows from [8, Lemma 2.13]. Appendix E. Proofs of Section 5.1 Proof of Lemma 5.2.First, for every c, t ≥0,r > 0and (a, b)∈ A, we show that cP(|Xa,b t| ≥ r)−Eˆt 0 L(as, bs) ds≤c r+L∗c rt.
CONVEX MONOTONE SEMIGROUPS 47 Chebyshev’s inequality, Jensen’s inequality and Itô’s isometry imply P(|Xa,b t| ≥ r)≤E[|Xa,b t|] r≤1 r E"ˆt 0 √asdWs 2#1 2 +Eˆt 0|bs|ds =1 r Eˆt 0|as|ds1 2 +Eˆt 0|bs|ds!≤1 r1 + Eˆt 0|as|+|bs|ds. Using the definition of L∗, we conclude cP(|Xa,b t| ≥ r)−Eˆt 0 L(as, bs)ds≤c r+L∗c rt. Second, we show that Sis continuous from above. Let t≥0,x∈Rdand (fn)n∈N⊂ BUC be a sequence with fn↓0. For every r > 0and (a, b)∈ A, we use the first part to estimate Efn(x+Xa,b t)−Eˆt 0 L(as, bs) ds =Ehfn(x+Xa,b t)1{|Xa,b t|<r}i+Ehfn(x+Xa,b t)1{|Xa,b t|≥r}i−Eˆt 0 L(as, bs) ds ≤sup y∈B(x,r) fn(y) + kf1k∞·P(|Xa,b t| ≥ r)−Eˆt 0 L(as, bs) ds ≤sup y∈B(x,r) fn(y) + kf1k∞ r+L∗kf1k∞ rt. It follows from L∗(ε)→0as ε↓0and Dini’s theorem that (S(t)fn)(x)↓0. Since Cb⊂Ub= (BUC)δ, the previous statement remains valid if we replace BUC by Cb. Third, for every c, t ≥0,δ∈(0,1],x∈Rd,(a, b)∈ A and f∈Lipb(c), we show that Eˆt 0 f(x+Xa,b s) ds≤f(x) + cδt+ccL δ2Eˆt 0ˆs 0 1 + L(au, bu) duds. We use Chebyshev’s inequality, Itô’s isometry and the definition of cLto estimate Eˆt 0 f(x+Xa,b s) ds ≤Eˆt 0f(x+Xa,0 s) + cˆs 0|bu|duds =Eˆt 0f(x+Xa,0 s)1{|Xa,0 s|<δ}+f(x+Xa,0 s)1{|Xa,0 s|≥δ}+cˆs 0|bu|duds ≤f(x) + cδt+cEˆt 0P(|Xa,0 s| ≥ δ) + ˆt 0|bu|duds ≤f(x) + cδt+c δ2Eˆt 0ˆs 0|au|+|bu|duds ≤f(x) + cδt+ccL δ2Eˆt 0ˆs 0 1 + L(au, bu) duds.
48 CONVEX MONOTONE SEMIGROUPS Fourth, let t≥0,x∈Rd,f∈BUC2and (a, b)∈ A with (S(t)f)(x)≤t+Ef(x+Xa,b t)−Eˆt 0 L(as, bs) ds. Assumption 5.1(i) implies E[f(x+Xa∗,b∗ t)] ≤(S(t)f)(x). Define cf:= 21 + kBa∗,b∗fk∞+L∗kD2fk∞∨2k∇fk∞. We use Itô’s formula and the definition of L∗to estimate 2Eˆt 0 L(as, bs) ds≤2t+ 2Ef(x+Xa,b t)−2Ef(x+Xa∗,b∗ t) ≤2t+ 2Eˆt 0 Bas,bsf(x+Xa,b s) ds−2Eˆt 0 Ba∗,b∗f(x+Xa∗,b∗ s) ds ≤21 + kBa∗,b∗fk∞t+Eˆt 0|as|·kD2fk∞+ 2|bs|·k∇fk∞ds ≤21 + kBa∗,b∗fk∞+L∗kD2fk∞∨2k∇fk∞t+Eˆt 0 L(as, bs) ds =cft+Eˆt 0 L(as, bs) ds. Rearranging the previous inequality yields Eˆt 0 L(as, bs) ds≤cft. Proof of Theorem 5.3.First, we show that Jsatisfies Assumption 4.1 and Assumption 4.4. Clearly, J(t)is convex and monotone for all t≥0. Moreover, Assumption 5.1(i) implies J(0) = idCband J(t)0 = 0 for all t≥0. For every t≥0and f, g ∈Cb, kJ(t)f−J(t)gk∞≤sup (a,b)∈Sd +×RdkTa,b(t)f−Ta,b(t)gk∞≤ kf−gk∞, where (Ta,b(t))t≥0denotes the linear semigroup given by Ta,b(t)f(x) := E[f(x+√aWt+bt)] for all t≥0, f ∈Cband x∈Rd. Since J(t)is translation invariant, we obtain J(t): Lipb(r)→Lipb(r)for all r, t ≥0. In particular, it follows by induction that the sequence (I(πt n)f)n∈Nis uniformly Lipschitz continuous for all t≥0and f∈Lipb. Let t≥0,x∈Rdand (fn)n∈N⊂BUC be a sequence with fn↓0. Lemma 5.2(i) and Dini’s theorem imply 0≤sup s∈[0,t]∩T sup k∈NJ(πt k)fn)(x)≤sup s∈[0,t] (S(s)fn)(x)↓0as n→ ∞. Indeed, by Theorem 5.4, the mapping t7→ (S(t)fn)(x)is continuous. Furthermore, it holds J(t)f≤S(t)ffor all t≥0and f∈BUC and Sis a semigroup on BUC. Since Cb⊂Ub= (BUC)δ, the previous statement remains valid if we replace BUC by Cb. Next, for every f∈BUC2, we show that lim h↓0 J(h)f−f h−sup (a,b)∈Sd +×Rd1 2∆af+∇bf−L(a, b) ∞ = 0.
CONVEX MONOTONE SEMIGROUPS 49 By Assumption 5.1(ii) and Itô’s formula, there exists r≥0such that J(h)f−f h−sup (a,b)∈Sd +×Rd1 2∆af+∇bf−L(a, b) ∞ ≤sup {|a|,|b|≤r} Ta,b(h)f−f h−1 2∆af−∇bf ∞ ≤sup {|a|,|b|≤r}ˆh 0k∇bf(·+Xa,b s)−∇bfk∞+1 2k∆af(·+Xa,b s)−∆afk∞ds. For every ε > 0, there exists δ > 0such that, for all (a, b)∈Sd +×Rdwith |a|,|b| ≤ r and s≥0, k∇bf(·+Xa,b s)−∇bfk∞+1 2k∆af(·+Xa,b s)−∆afk∞1{|Xa,b s|<δ}< ε. Furthermore, Chebyshev’s inequality implies sup {|a|,|b|≤r} P|Xa,b s| ≥ δ≤sup {|a|,|b|≤r} 2 δ2E[|√aWs|2] + |b|2s2→0as s↓0. It remains to show that J(t): Cb→Cbfor all t≥0. Let t≥0and f∈Cb. Choose a bounded sequence (fn)n∈N⊂Lipbwith fn→funiformly on compacts. Lemma 2.7 implies J(t)fn→J(t)funiformly on compacts. Indeed, the corresponding proof only relies on the fact that J(t)is convex, monotone and continuous from above. Since J(t)fn∈Lipb⊂Cbfor all n∈N, we obtain J(t)f∈Cb. By Theorem 4.3, there exists a family Tof operators T(t): Cb→Cbwhich satisfy the conditions from Theorem 5.3 except for the statement about the symmetric Lipschitz set. Second, we show that LT sym ∩BUC = LJ sym ∩BUC and T(t): LT sym ∩Lipb→ LT sym ∩Lipbfor all t≥0. By straightforward computations, one can verify the conditions (i)-(iv) from [8, Assumption 5.2] with J+(t)f:= J(t)fand J−(t)f:= −J(t)(−f)for all t≥0and f∈BUC. Moreover, it holds T(t)f= supn∈NJ(πt n)ffor all (f, t)∈BUC ×T and J(t)f≤T(t)f for all (f, t)∈BUC×R+. We remark that in [8] a slightly stronger assumption has been made, but it follows immediately from the corresponding proof that [8, Theorem 5.3] is still applicable. We obtain LT sym ∩BUC = LJ sym ∩BUC and T(t): LT sym ∩BUC → LT sym ∩BUC for all t≥0. In addition, the invariance of Lipbholds by Theorem 4.5. It remains to show the explicit representation of LT sym ∩Lipb=LJ sym ∩Lipb. Third, let f∈ LJ sym ∩Lipb. Choose c≥0and t0>0with kJ(t)f−fk∞≤ct and kJ(t)(−f) + fk∞≤ct for all t∈[0, t0]. Fix (a, b)∈Sd +×Rdwith L(a, b)<∞and t∈[0, t0]. It holds −c+L(a, b)t≤ −J(t)(−f) + f+L(a, b)t≤ −Ta,b(t)(−f) + f =Ta,b(t)f−f≤J(t)f−f+L(a, b)t≤c+L(a, b)t and thus kTa,b(t)f−fk∞≤(c+L(a, b))t. Let η∈C∞ cwith η≥0,supp(η)⊂B(0,1) and ´Rdη(x) dx= 1. Define ηn(x) := ndη(nx)and fn:= f∗ηnfor all n∈Nand x∈Rd.
50 CONVEX MONOTONE SEMIGROUPS We use Fubini’s theorem to estimate Ta,b(t)fn−fn(x) = EˆRd f(x+√aWt+bt −y)ηn(y)dy−fn(x) =ˆRd Ef(x+√aWt+bt −y)ηn(y)dy −fn(x) =(Ta,b(t)f−f)∗ηn(x)≤ Ta,b(t)f−f ∞≤c+L(a, b)t. It follows from fn∈BUC2⊂D(Ba,b)that kBa,bfnk∞≤c+L(a, b)for all n∈N.(E.1) In addition, for every n∈N, ∆afn=1 2∆afn+∇bfn+1 2∆afn+∇−bfn=Ba,bfn+Ba,−bfn.(E.2) Inequality (E.1) and equation (E.2) imply sup n∈Nk∆afnk∞≤ kBa,bfnk∞+kBa,−bfnk∞≤2c+L(a, b). By Banach-Alaoglu’s theorem, there exists g∈L∞such that ∆afnk→gin the weak*- topology for a suitable subsequence. Moreover, it follows from f∈Lipb=W1,∞that √aT∇fn= (√aT∇f)∗ηn→√aT∇fand thus f∈D(∆a)with ∆af=g. Since the supremum norm is lower semicontinuous w.r.t. weak*-topology, inequality (E.1) yields kBa,bfk∞≤c+L(a, b). We obtain f∈\ a∈SL D(∆a)∩W1,∞and sup (a,b)∈SL×RdkBa,bfk∞−L(a, b)<∞. Fourth, let f∈Ta∈SLD(∆a)∩W1,∞and assume that c:= sup (a,b)∈SL×RdkBa,bfk∞−L(a, b)<∞. Let t≥0,(a, b)∈SL×Rdand define fn:= f∗ηnfor all n∈N. It follows from Itô’s formula and Ba,bfn= (Ba,bf)∗ηnthat Ta,b(t)f−f−L(a, b)t= lim n→∞ Ta,b(t)fn−fn−L(a, b)t ≤sup n∈NkBa,bfnk∞−L(a, b)t ≤kBa,bk∞−L(a, b)t≤ct. This implies J(t)f−f≤ct. For the lower bound, we use L(a∗, b∗)=0to estimate J(t)f−f≥Ta∗,b∗(t)f−f≥ −kBa∗,b∗fk∞t. We obtain f∈ LJ. Applying the previous estimate on −fyields f∈ LJ sym. References [1] L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel. Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal., 123(3):199–257, 1993. [2] V. Barbu. Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York, 2010. [3] D. Bartl, P. Cheridito, and M. Kupper. Robust expected utility maximization with medial limits. J. Math. Anal. Appl., 471(1-2):752–775, 2019. [4] G. Beer. Upper semicontinuous functions and the Stone approximation theorem. J. Approx. Theory, 34(1):1–11, 1982.
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