Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit S(t):=limn→∞(I(t2−n))2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S(t):=\lim _{n\to \infty }(I(t 2^{-n}))^{2^{n}}$\end{document} exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by limh↓0I(h)x−xh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim _{h\downarrow 0}\tfrac {I(h)x-x}{h}$\end{document}, whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.


Introduction
Let X be a complete metric space and (I(t)) t≥0 a family of Lipschitz continuous mappings I(t) : X → X. We are interested in the question whether one can construct an associated semigroup (S(t)) t≥0 , which satisfies S(t)x = lim n→∞ I(2 −n t) 2 n x. (1.1) Formulas of this type are called Chernoff approximation. In his monograph [8], Chernoff generalized his previous work [7] and the results by Trotter [33,34]: under additional stability conditions on the approximating sequence (I(2 −n t) 2 n x) n∈N and the assumption that (I(t)) t≥0 is strongly continuous, it was shown in [8,Theorem 2.5.3] that the family (S(t)) t>0 is a strongly continuous semigroup of Lipschitz continuous mappings.
In this article, we provide a detailed study how a modified version of the Chernoff approximation can be used for the construction of nonlinear semigroups. The key idea is to find properties of (I(t)) t≥0 , which are preserved during the iteration, and can therefore be transferred to the semigroup (S(t)) t≥0 . Typically, the family (I(t)) t≥0 has a representation, which allows for explicit verification of these properties. Compared to formula (1.1), we take the limit only along a convergent subsequence, i.e., S(t)x = lim l→∞ I(2 −n l t) 2 n l x. (1.2) Clearly, under suitable compactness and separability assumptions, we can choose a diagonal sequence for all (x, t) in a countable dense subset D × T of X × R + . Hence, Date: December 2, 2021. We thank Robert Denk, Stephan Eckstein, Karsten Herth, Markus Kunze, Max Nendel, Reinhard Racke and Liming Yin for helpful comments and discussions. Furthermore, we thank an anonymous referee for valuable comments and feedback on an earlier version of the paper.
we have to extend the mapping D × T → X, (x, t) → S(t)x to the closure X × R + . To do so, we need sufficient Lipschitz continuity of (I(t)) t≥0 in the variables (x, t) ∈ D×T , which is preserved during the construction. In several examples, the boundedness and Lipschitz assumptions on the family (I(t)) t≥0 are easily verified. To prove the relative compactness of the sequence (I(2 −n t) 2 n x) n∈N is more involved and depends on the choice of the space X. The described construction leads to a strongly continuous semigroup (S(t)) t≥0 of locally Lipschitz continuous operators S(t) : X → X.
Another focus of this work is the connection between the local behaviour of (I(t)) t≥0 and (S(t)) t≥0 . Let X be a Banach space. If (I(t)) t≥0 is a family of linear contractions, it has been shown in [8,Theorem 3.7] that the infinitesimal generator of (S(t)) t≥0 is an extension of the derivative I ′ (0). In this spirit, in Theorem 4.3, we provide an abstract technical condition, which ensures that For instance, this result always holds, if (I(t)) t≥0 consists of convex monotone operators. We also discuss whether (S(t)) t≥0 represents the unique solution to the abstract Cauchy problem ∂ t u(t) = Au(t) for all t ≥ 0, u(0) = x, where A denotes the generator of (S(t)) t≥0 and x ∈ D(A). Unlike to the theory of linear semigroups, this is not immediately clear and depends strongly on additional properties of X. In particular, one has to verify S(t) : D(A) → D(A) for all t ≥ 0, which is in general wrong, see [13]. In contrast, the invariance of the symmetric Lipschitz set [13] does not depend on X, but rather on convexity and monotonicity of (S(t)) t≥0 . We provide sufficient conditions on (S(t)) t≥0 for the invariance and investigate how these conditions can be derived from (I(t)) t≥0 . Furthermore, the symmetric Lipschitz set can be completely described by (I(t)) t≥0 , and therefore explicitly determined in examples, see Subsection 6.1. Finally, we want to emphasise that the results on the generator and the symmetric Lipschitz set do not require norm convergence of the approximating sequence as in equation (1.2). The present approach is inspired by the Nisio semigroup [26], where the sequence (I(2 −n t) 2 n x) n∈N is non-decreasing. In this case, the limit in equation (1.2) does not depend on the choice of the convergent subsequence, i.e., equation (1.1) and equation (1.2) are equivalent, see Subsection 2.2. The Nisio semigroup is obtained from the family I(t)x := sup λ∈Λ S λ (t)x, where (S λ ) λ∈Λ is a family of monotone linear semigroups. In case that each (S λ (t)) t≥0 is the semigroup of a Markov process, the respective Nisio semigroup corresponds to a stochastic process under parameter uncertainty, see [12,24]. Typical examples include Brownian motions with drift or volatility uncertainty, see [9,29,30], and Lévy processes with uncertainty in the Lévy triplet, see [18,19,22,25]. Compared to the Nisio semigroup, the construction based on the norm convergence as in equation (1.2) does not rely at all on monotonicity and we do not require (I(t)) t≥0 to be the supremum over a family of monotone linear semigroups. For instance, this is illustrated in Subsection 6.5 by means of reaction-diffusion equations, where the operators I(t) are neither convex nor monotone.
In the literature, the Chernoff approximation is mainly used as a tool for finding approximative representations of semigroups or solutions of evolution equations, for which the existence is already established. In many applications the goal is to find an approximation of the form (1.1) to give explicit representations of semigroups, see, e.g., [6,27,31]. For a survey of Chernoff approximations of operator semigroups we refer to [5], see also [17] for an overview on different kinds of approximations. A classical approach to nonlinear PDEs and the respective nonlinear semigroups is based on the theory of maximal monotone or m-accretive operators, see [1,2,4,16,20]. For HJB-type equations, as outlined in [13], it is rather delicate to verify the m-accreditivity. Furthermore, the theory of backward stochastic differential equations provides a powerful tool for the representation of second order quasi-linear equations, see, e.g., [11,15], as well as [21,32] for fully nonlinear equations.
The paper is organized as follows. In Section 2, we introduce the basic assumptions and state the main result on the existence of nonlinear semigroups. In Section 3, we investigate how to verify the imposed compactness assumption for certain spaces of continuous functions. In Section 4, we prove that the generator of (S(t)) t≥0 is an extension of the derivative I ′ (0). In Section 5, we study the symmetric Lipschitz set. Finally, in Section 6 the main results are illustrated with several examples.

Construction of nonlinear semigroups
Let (X, d) be a complete metric space. We denote by B(x, r) := {y ∈ X : d(x, y) ≤ r} the closed ball with radius r ≥ 0 around x ∈ X, and by R + := {x ∈ R : x ≥ 0} the positive real numbers including zero. Let (I(t)) t≥0 be a family of operators I(t) : X → X satisfying the following boundedness and Lipschitz conditions. Assumption 2.1. The family (I(t)) t≥0 satisfies the following properties: where α : R + × R + → R + is a function, which is non-decreasing in the second argument, and satisfies α(α(r, s), t) ≤ α(r, s + t) for all r, s, t ≥ 0.

Remark 2.2.
(i) It holds α(r, t) ≥ r for all r, t ≥ 0, because the non-decreasingness of α in the second argument and I(0) = id X imply α(r, t) ≥ α(r, 0) ≥ r. Furthermore, we assume w.l.o.g. that the mapping R + → R + , r → w r is non-decreasing. (ii) Let (I(t)) t≥0 be a strongly continuous semigroup of bounded linear operators on a Banach space (X, · ). Then, there exist M ≥ 1 and ω ∈ R such that I(t)x ≤ M e ωt x for all t ≥ 0 and x ∈ X, see [28]. If (I(t)) t≥0 is quasicontractive, i.e. M = 1, Assumption 2.1(iii) is satisfied. In the nonlinear case, the exponent ω r might depend on r. Since we are only interested in the short time behaviour of (I(t)) t≥0 , we do not require this property for t > 1. Definition 2.3. Let (J(t)) t≥0 be a family of operators J(t) : X → X. The Lipschitz set L J consists of all x ∈ X, for which there exist t 0 > 0 and c ≥ 0 with For convex semigroups, the Lipschitz set was introduced in [13]. The Lipschitz set allows us to establish strong continuity of the semigroup S, and will be used in Subsection 6.1 to prove a regularity result.
Let T := {k2 −n : k, n ∈ N 0 } be the set of all positive dyadic numbers, which is countable and dense in R + . We define the partitions π t n := {k2 −n ∧ t : k ∈ N 0 } and the iterated operators I(π t n ) := I(2 −n ) 2 n t for all t ∈ T and n ∈ N with 2 n t ∈ N. The verification of the following compactness assumption will be discussed in Section 3 for certain spaces of continuous functions.
Assumption 2.4. There exists a countable set D ⊂ L I , which is dense in X. Furthermore, the sequence (I(π t n )x) n∈N is relatively compact in X for all (x, t) ∈ D × T . The previous assumption implies the existence of a subsequence (n l ) l∈N ⊂ N such that we can define S(t)x := lim l→∞ I(π t n l )x for all (x, t) ∈ D × T . Then, since the Lipschitz continuity of (I(t)) t≥0 in the variables (x, t) ∈ D × T from Assumption 2.1 and Definition 2.3 is preserved during the iteration and in the limit, we can extend the mapping D × T → X, (x, t) → S(t)x to the closure D × T = X × R + . This leads to the first main result.
Theorem 2.5. Suppose that (I(t)) t≥0 satisfies Assumption 2.1 and Assumption 2.4. Then, there exists a family (S(t)) t≥0 of operators S(t) : X → X with the following properties: (i) There exists a subsequence (n l ) l∈N ⊂ N such that (ii) The family (S(t)) t≥0 forms a semigroup, i.e., (iii) For every r, t ≥ 0, x, S(t)y ≤ e tω α(r,t) d(x, y) for all x, y ∈ B(x 0 , r).
(iv) The semigroup (S(t)) t≥0 is strongly continuous, i.e., the mapping is continuous for all x ∈ X. (v) For every r ≥ 0 and x ∈ B(x 0 , r) ∩ L I , there exists c ≥ 0 such that We call (I(t)) t≥0 a generating family of the semigroup (S(t)) t≥0 , and (S(t)) t≥0 an associated semigroup to the family (I(t)) t≥0 . Corollary 2.6. It holds L I ⊂ L S and S(t) : L S → L S for all t ≥ 0.
Proof. The inclusion L I ⊂ L S follows from Theorem 2.5 (v) and S(0) = id X . Fix x ∈ L S and choose t 0 > 0 and c ≥ 0 such that We use Theorem 2.5(iii), inequality (2.2) and inequality (2.1) to conclude that d S(s)S(t)x, S(t)x = d S(t)S(s)x, S(t)x ≤ e tω α(α(r,s),t) d S(s)x, x ≤ ce tω α(r,s+t) s for all t ≥ 0 and s ∈ [0, t 0 ].

2.1.
Proof of Theorem 2.5. In the sequel, we establish a series of lemmas, which prove Theorem 2.5. We will always suppose that Assumption 2.1 and Assumption 2.4 hold.
Lemma 2.7. For every r ≥ 0, t ∈ T and n ∈ N with 2 n t ∈ N, for all x, y ∈ B(x 0 , r). Proof. Fix n ∈ N. First, we show by induction that For k = 1, the claim holds by Assumption 2.1(ii). For the induction step, suppose that inequality (2.3) holds for some fixed k ∈ N and all r ≥ 0. We combine this with Assumption 2.1(ii) and inequality (2.1) to conclude that for all r ≥ 0 and x ∈ B(x 0 , r). Second, we show by induction that, for all k ∈ N, r ≥ 0 and x, y ∈ B(x 0 , r), For k = 1, the claim follows from Assumption 2.1(iii) and α(r, 2 −n ) ≥ r. For the induction step, suppose that inequality (2.4) holds for some fixed k ∈ N, all r ≥ 0 and x, y ∈ B(x 0 , r). Combing this with Assumption 2.1(ii) and (iii), inequality (2.1), and the non-decreasingness of the mappings r → ω r and t → α(r, t) yields for all r ≥ 0 and x, y ∈ B(x 0 , r).
By definition, for every s, t ∈ T and n ∈ N with 2 n s, 2 n t ∈ N, I π s+t n = I π s n I π t n . (2.5) Then, for every T ≥ 0, s, t ∈ [0, T ] ∩ T and n ∈ N with 2 n s, 2 n t ∈ N and 2 −n ≤ t 0 , d I(π s n )x, I(π t n )x ≤ ce T ω α(r,T ) |s − t]. Proof. Fix n ∈ N with 2 −n ≤ t 0 . First, we show by induction that d I(2 −n ) k x, x ≤ ce k2 −n ω α(r,k2 −n ) k2 −n for all k ∈ N. (2.7) For k = 1, the claim holds by inequality (2.6). For the induction step, assume that inequality (2.7) holds for some fixed k ∈ N. We combine this with Lemma 2.7, inequality (2.6), inequality (2.1), and the non-decreasingness of the mappings r → ω r and t → α(r, t) to estimate Second, we show that, for all k, l ∈ N with k ≥ l, It follows from equation (2.5), Lemma 2.7, inequality (2.1), and the non-decreasingness of the mappings r → ω r and t → α(r, t) that By using Assumption 2.4 and choosing a diagonal sequence for the countable set D × T , there exists a subsequence (n l ) l∈N ⊂ N such that the limit exists for all (x, t) ∈ D × T . In particular, the mapping S(·)x : [0, T ] ∩ T → X has a unique continuous extension to [0, T ], which satisfies the previous inequality for all s, t ∈ [0, T ]. (ii) For every r, t ≥ 0, d S(t)x, S(t)y ≤ e tω α(r,t) d(x, y) for all x, y ∈ B(x 0 , r) ∩ D.
In particular, the mapping S(t) : B(x 0 , r) ∩ D → X has a unique continuous extension to B(x 0 , r), which satisfies the previous inequality for all x, y ∈ B(x 0 , r).
Proof. First, we fix r, T ≥ 0, x ∈ B(x 0 , r) ∩ D, and choose t 0 > 0 and c ≥ 0 such that Now, let t ≥ 0 be arbitrary and choose a sequence (t n ) n∈N ⊂ [0, t] ∩ T with t n → t. We use part (i), inequality (2.9) and the non-decreasingness of α in the second argument to estimate d S(t)x, S(t)y = lim n→∞ d S(t n )x, S(t n )y ≤ sup n∈N e tnω α(r,tn) d(x, y) ≤ e tω α(r,t) d(x, y).
The existence and uniqueness of the extension follows, because B(x 0 , r) ∩ D ⊂ B(x 0 , r) is dense and the mapping S(t) : B(x 0 , r) ∩ D → X is Lipschitz continuous.
Equation (2.8) implies lim l→∞ d S(t)x, I(π t n l )x = 0. Second, we fix r ≥ 0 and x ∈ B(x 0 , r). It follows from the first part and Lemma 2.7 that Now, let t ≥ 0 be arbitrary and choose a sequence (t n ) n∈N ⊂ [0, t] ∩ T with t n → t. We use Lemma 2.10, inequality (2.10) and the non-decreasingness of α in the second argument to conclude that Lemma 2.12. It holds S(0) = id X and S(s + t) = S(s)S(t) for all s, t ≥ 0.
Proof. It follows from Assumption 2.1(i) and the construction that S(0) = id X . Fix x ∈ X and define r := d(x 0 , x). First, let s, t ∈ T . Equation (2.5) and Lemma 2.11 imply lim l→∞ d S(s + t)x, I(π s n l )I(π t n l )x = lim l→∞ d S(s + t)x, I(π s+t n l )x = 0.
Furthermore, it follows from Lemma 2.7 and Lemma 2.11 that d S(s)S(t)x, I(π s n l )I(π t n l )x ≤ d S(s)S(t)x, I(π s n l )S(t)x + d I(π s n l )S(t)x, I(π s n l )I(π t n l )x ≤ d S(s)S(t)x, I(π s n l )S(t)x + e sω α(α(r,t),s) d S(t)x, I(π t n l → 0 as l → ∞. Second, let s, t ≥ 0 be arbitrary, define T := s + t + 1 and choose sequences (s n ) n∈N and (t n ) n∈N in [0, T ] ∩ T with s n → s and t n → t. We use the first part, Lemma 2.9(ii), Lemma 2.10 and Lemma 2.11 to estimate Proof. Fix r, T ≥ 0, x ∈ B(x 0 , r) ∩ L I and choose t 0 > 0 and c ≥ 0 such that It follows from Lemma 2.11 and Lemma 2.8 that, for all s, t ∈ [0, T ] ∩ T , Now, let s, t ∈ [0, T ] be arbitrary and choose sequences (s n ) n∈N and (t n ) n∈N in [0, T ]∩ T with s n → s and t n → t. We use Lemma 2.10 and inequality (2.11) to conclude that 2.2. Discussion and comparison with Nisio semigroup. In the previous subsection, we only considered dyadic partitions, but actually the same iterations can be made with arbitrary partitions. The proofs do not change, except for being notationally more complicated. For every t ≥ 0, we denote by P t the set of all partitions π = {t 0 , . . . , t n } with 0 = t 0 < t 1 < . . . < t n = t. Moreover, we define the iterated operators For later reference, we state the following version of Lemma 2.7 and Lemma 2.8.
A priori the construction of an associated semigroup (S(t)) t≥0 to a generating family (I(t)) t≥0 depends on the choice of the partitions and the convergent subsequence. However, it is possible that , the convergence holds without choosing a subsequence. For instance, if the sequence In particular, Nisio semigroups fall into this category, as we will see in the next lemma and subsequent remark. Furthermore, if the semigroup represents the unique solution to a PDE, the construction is independent of the choice of the convergent subsequence. For details, we refer to Subsection 4.2. For the following lemma and subsequent remark, let X be a Banach lattice. An operator Lemma 2.15. Let (I(t)) t≥0 a family of operators I(t) : X → X, which satisfy Assumption 2.1 and Assumption 2.4, and (S(t)) t≥0 an associated semigroup as in Theorem 2.5.
In addition, we make the following assumptions: (i) I(t) is monotone and continuous from below for all t ≥ 0.
x for all (x, t) ∈ X × T , i.e., the convergence holds without choosing a subsequence.
Proof. By induction, it follows from condition (ii) that the sequence (I(π t n )x) n∈N is nondecreasing for all (x, t) ∈ X × T . Moreover, by Theorem 2.5, there exists a subsequence (n l ) l∈N ⊂ N such that S(t)x = lim l→∞ I(π t n l )x for all (x, t) ∈ X × T . Since X is a Banach lattice, we obtain In addition, condition (iii) and strong continuity of (S(t)) t≥0 imply We use the monotonicity of I(s) and the semigroup property of (S(t)) t≥0 to conclude x for all s, t ≥ 0 and x ∈ X.
By induction, we obtain T (t)x ≤ S(t)x for all (x, t) ∈ X × R + with equality for t ∈ T . It remains to show that the mapping R + → X, t → T (t)x is continuous for all x ∈ X. Condition (ii) implies that the set {I(π) : π ∈ P t } is directed upwards and, by assumption, the operator I(t) is continuous from below for all t ≥ 0. Hence, the family (T (t)) t≥0 forms a semigroup, and we can use Remark 2.14(ii) to show that the mapping R + → X, t → T (t)x is locally Lipschitz continuous for all x ∈ L I . Since L I ⊂ X is dense, it follows from Remark 2.14(i) that the mapping t → T (t)x is continuous for all x ∈ X, see the proof of Lemma 2.10. Furthermore, it holds S(t)x = lim n→∞ I(π t n )x for all (x, t) ∈ X × T , because the limit along a subsequence in Theorem 2.5(i) does not depend on the choice of the convergent subsequence.
Remark 2.16. Let (S λ ) λ∈Λ be a family of linear semigroups on X, which satisfy the following conditions: (i) S λ (t) is monotone and continuous from below for all λ ∈ Λ and t ≥ 0.
x is well-defined for all t ≥ 0. Moreover, we assume that, for every subset Y ⊂ X such that the supremum sup Y ∈ X exists, it holds sup Y ≤ sup x∈Y x . For instance, the supremum norm has this property, but not the L p -norm. For every t ≥ 0 and x, y ∈ X, we use the assumption on the norm and condition (ii) to estimate Hence, Assumption 2.1 is satisfied. Furthermore, condition (i) implies that (I(t)) t≥0 satisfies the first two conditions of Lemma 2.15. In many examples, the forth condition of Lemma 2.15 follows from the assumptions, which are already necessary for the construction of the semigroup (S(t)) t≥0 , while the third condition requires a mild additional assumption. If (I(t)) t≥0 satisfies the assumptions of Lemma 2.15, the associated semigroup (S(t)) t≥0 from Theorem 2.5 equals the family (T (t)) t≥0 , defined by It is also possible to weight the linear semigroups in the definition of I(t) with a penalization term. This leads to semigroups, which are not sublinear but merely convex.

Relative compactness based on Arzéla-Ascoli's theorem
Let C be the space of all continuous functions f : R d → R m , including the subsets C ∞ , Lip and C 0 of all functions, which are infinitely differentiable, Lipschitz continuous and vanish at infinity, respectively. Furthermore, let L ∞ be the space of all bounded (not necessarily measurable) functions f : R d → R m , endowed with the supremum norm where |·| denotes the Euclidean norm. We define C b := C ∩L ∞ , Lip b := Lip ∩L ∞ and Lip 0 := Lip ∩C 0 . In addition, for every c ≥ 0, we denote by Lip(c) the set of all c-Lipschitz continuous functions. For every c ≥ 0, let 3.1. Semigroups on C 0 . We give explicit conditions, how to verify the assumptions of Section 2 for a family of translation-invariant contractions, which will be illustrated in Section 6. We start with an application of Arzéla-Ascoli's theorem.
Proof. By assumption, the sequence (I(π t n ) n∈N is equicontinuous. Moreover, it follows from Lemma 2.7 that (I(π t n ) n∈N is bounded. Note that Lemma 2.7 is a consequence of Assumption 2.1(ii) and independent of the other assumptions in Section 2. By using Arzéla-Ascoli's theorem and choosing a diagonal sequence, we obtain a function g ∈ C such that I(π t n l )f → g as l → ∞ uniformly on compact sets for a suitable subsequence. It remains to show lim This inequality is preserved in the limit, i.e., sup x∈K c |g(x)| ≤ ε /2. We obtain Let (I(t)) t≥0 be a family of operators I(t) : C 0 → L ∞ , which satisfy the following conditions: Then, it holds I(t) : C 0 → C 0 and I(t) : Lip 0 (c) → Lip 0 (c) for all c, t ≥ 0. Furthermore, the family (I(t)) t≥0 satisfies Assumption 2.1 and Assumption 2.4.
and therefore I(t)f ∈ Lip b (c), by property (i). Moreover, property (vi) yields We obtain I(t)f ∈ Lip 0 (c) and conclude I(t) : Second, Assumption 2.1 is satisfied, because of the properties (i)-(iii). Furthermore, by property (v), there exists a countable set D ⊂ Lip 0 ∩L I , which is dense in C 0 . It remains to verify the assumptions from Lemma 3.1 for all (f, t) ∈ D × T . Fix (f, t) ∈ D × T and choose c ≥ 0 with f ∈ Lip 0 (c). By induction, it follows from In particular, the sequence (I(π t n )f ) n∈N is equicontinuous. Furthermore, we use condition (vi) and I(2 −n )f ∈ Lip 0 (c) to estimate is not a suitable choice. Thus, we modify the supremum norm with a weight function κ, following the setting of Nendel and Röckner [24]. The verification of Assumption 2.4 becomes particularly simple, also in the translation-invariant case. Let κ : R d → (0, ∞) be a continuous function function, vanishing at infinity. Let C κ be the space of all continuous functions f : is an isomorphism, which is linear, isometric and preserves the pointwise order, the space C κ is a Banach lattice. Furthermore, we define UC κ as the closure of Lip b in C κ . In particular, the space UC κ is a Banach lattice. Lemma 3.3. Let (I(t)) t≥0 be a family of operators I(t) : UC κ → UC κ . Assume that there exists a function ρ : Then, the sequence (I(π t n )f ) n∈N is relatively compact in UC κ for all f ∈ Lip b and t ∈ T . Proof. First, we show that Lip b (c) ⊂ UC κ is compact for all c ≥ 0. Let c ≥ 0 and (f n ) n∈N ⊂ Lip b (c) be a sequence. By using Arzéla-Ascoli's theorem and choosing a diagonal sequence, we obtain a function f ∈ C such that f n l → f as l → ∞ uniformly on compact sets for a suitable subsequence.
Second, let f ∈ Lip b (c) for some c ≥ 0 and t ∈ T . By induction, it follows from the assumptions on I and ρ that I(π t n )f ∈ Lip b (ρ(c, t)) for all n ∈ N with 2 n t ∈ N. The first part yields that the sequence (I(π t n )f ) n∈N is relatively compact in UC κ .
Let C ∞ c be the space of all infinitely differentiable functions f : R d → R m with compact support.
Lemma 3.4. Assume that ϕ is infinitely differentiable. Then, the space C ∞ c ⊂ UC κ is dense and the mapping ϕ : UC κ → C 0 , f → f κ is an isomorphism.
Proof. It follows from lim |x|→∞ κ(x) = 0 and the continuity of κ that ϕ(Lip b ) ⊂ C 0 . We conclude ϕ(UC κ ) ⊂ C 0 , because ϕ : UC κ → C b is isometric and Lip b ⊂ UC κ is dense. It remains to show ϕ(UC κ ) = C 0 . Let f ∈ C 0 and choose a sequence (f n ) n∈N ⊂ C ∞ c with f − f n ∞ → 0. Since κ is smooth, it holds fn /κ ∈ C ∞ c for all n ∈ N. Furthermore, the sequence ( fn /κ) n∈N ⊂ UC κ is a Cauchy sequence and the limit g := lim n→∞ fn /κ ∈ UC κ exists, because ϕ is isometric. We obtain

Infinitesimal generator
Throughout this section, we assume that X a Banach space with norm · . We investigate the relation between the local behaviour of a generating family (I(t)) t≥0 and an associated semigroup (S(t)) t≥0 . The technical condition (4.2) in Theorem 4.3 will be discussed in Subsection 4.1.
Assumption 4.1. Let (I(t)) t≥0 be a family of operators, which satisfy Assumption 2.1 with x 0 := 0, and (S(t)) t≥0 a strongly continuous semigroup on X. In addition, we assume that for all t ∈ T \{0} and x, y ∈ X. (4.1) Remark 4.2. If (I(t)) t≥0 satisfies both Assumption 2.1 and Assumption 2.4, and (S(t)) t≥0 is an associated semigroup as in Theorem 2.5, inequality (4.1) is clearly satisfied. But requiring norm convergence I(π t n )x → S(t)x is an unnecessarily strong assumption for the next theorem. For instance, if X is a space of continuous functions endowed with the supremum or κ-norm, inequality (4.1) is satisfied if we have mere pointwise convergence I(π t n l )f → S(t)f . In particular, Theorem 4.3 is applicable for Nisio semigroups. Theorem 4.3. Suppose that Assumption 4.1 holds. Let x, y ∈ X such that, for every ε > 0, there exists t 0 > 0 with Then, Proof. Fix ε > 0 and choose r ≥ 0 with x, y ∈ B(0, r). By assumption, there exists t 0 ∈ (0, 1] such that and First, we show by induction that For k = 1, the claim holds by inequality (4.4). To prove the induction step, we assume for some fixed k ∈ N that Let n ∈ N with (k + 1)2 −n ≤ t 0 . It holds The first term is further decomposed as We use Lemma 2.7 and inequality (4.4) to estimate (4.10) Note that Lemma 2.7 relies only on Assumption 2.1, but not on Assumption 2.4. Combining inequality (4.5), equation (4.9) and inequality (4.10) yields Furthermore, it follows from inequality (4.7), equation (4.8) and inequality (4.11) that Second, we show that the right-hand side of equation (4.3) holds. Inequality (4.1) and inequality (4.6) imply Now, let t ∈ (0, t 0 ] be arbitrary and choose a sequence (t n ) n∈N ⊂ (0, t 0 ] ∩ T with t n → t. Since (S(t)) t≥0 is strongly continuous, we obtain 4.1. Condition (4.2). If I(t) is convex and monotone for all t ≥ 0, inequality (4.2) is always satisfied. Furthermore, we will see in Subsection 6.5 an example, where I(t) has none of these two properties.
Lemma 4.4. Let X be a Banach lattice and (I(t)) t≥0 be a family of convex monotone operators I(t) : X → X, which satisfy Assumption 2.1 with x 0 := 0. Furthermore, let L I ⊂ X be dense. Then, condition (4.2) holds for all x, y ∈ X.
Proof. We argue similar as in the proof of [24,Proposition 3.9]. Let x, y ∈ X and k, n ∈ N. Since the operator I(2 −n ) k is convex, the mapping is convex and maps zero to zero. This implies Hence, for λ := 2 −n , we obtain It remains to show Let x ∈ X and ε > 0. We define r := x + 1 and choose δ ∈ (0, 1] with (e ω α(r,1) + 1)δ ≤ ε 2 . (4.12) Since L I ⊂ X is dense, there exists y ∈ B(x, δ) ∩ L I . By Lemma 2.8, there exists c ≥ 0 with I(π t n )y − y ≤ ce tω α(r,t) t for all t ≥ 0. Let t 0 > 0 such that (4.13) We use Lemma 2.7, inequality (4.12), inequality (4.13) and the non-decreasingness of α in the second argument to conclude for all t ∈ [0, t 0 ] ∩ T and n ∈ N with 2 n t ∈ N. Note that Lemma 2.7 and Lemma 2.8 rely only on Assumption 2.1, but not Assumption 2.4.
for all x, y ∈ X and k, n ∈ N.

4.2.
Invariance of the domain and uniqueness. Let (S(t)) t≥0 be a strongly continuous semigroup. Furthermore, we assume that, for every r, T ≥ 0, there exists c ≥ 0 such that S(t)x − S(t)y ≤ c x − y for all t ∈ [0, T ] and x, y ∈ B(0, r). (4.14) The local behaviour of the semigroup (S(t)) t≥0 is determined through the infinitesimal generator where the domain D(A) consists of all x ∈ X for which previous limit exists.
Lemma 4.6. For every x ∈ D(A) and t ≥ 0, Proof. Fix x ∈ D(A) and t ≥ 0. For every h > 0, It follows from inequality (4.14) and the definition of the generator that Hence, the claimed equivalence holds by definition of the generator.
Remark 4.7. Let X be a Banach lattice and (S(t)) t≥0 a semigroup of convex monotone operators. Then, the quotient on the right hand side of (4.15) is non-decreasing in h > 0 and bounded from below. Hence, if the norm is order continuous, the limit on the right hand side of (4.15) exists, and the domain is invariant. For details, we refer to [14]. Typical examples are L p -spaces and Orlicz hearts, see [35], while spaces of continuous functions with the supremum or κ-norm do not have this property. Thus, the domain of a nonlinear semigroup is in general not invariant, see [13] for a counter example. One possibility to overcome this problem is the extension of the semigroup to a space with order continuous norm, see [3]. Another one is to weaken the definition of the generator by using monotone convergence, see [13]. One can also rely on viscosity solutions, see [10,12,24]. Finally, we want to mention an upcoming paper, where we use Γ-convergence to study the generator on the symmetric Lipschitz set, which will be introduced in Section 5.
The same arguments as in [14,Theorem 3.5] lead to the following uniqueness result. For the sake of completeness, we provide a proof.
Theorem 4.8. Let x ∈ X and y : R + → X be a continuous function with y(0) = x and y(t) ∈ D(A) for all t ≥ 0. Furthermore, we assume that where the existence of the limit is part of the assumption.Then, it holds y(t) = S(t)x for all t ≥ 0.
Proof. We fix t ≥ 0 and define g : [0, t] → X, s → S(t − s)y(s). First, we show For every s ∈ [0, t] and h ∈ (0, t − s], By assumption, it holds Hence, it follows from inequality (4.14) that Second, we show that g is continuous. We have already established that g is right continuous, i.e., we only have to show left continuity. For every s ∈ [0, t], continuity of y, strong continuity of S and inequality (4.14) imply y(s) = lim h↓0 S(h)y(s − h). Hence, it follows from inequality (4.14) that Third, following the proof of [28, Lemma 1.1 in Section 2], one can show that every continuous function with vanishing right derivative is constant. In particular, we obtain y(t) = g(t) = g(0) = S(t)x. Remark 4.9. If the domain is invariant and dense, the semigroup (S(t)) t≥0 is uniquely determined through its generator. In general, we do not know if the left derivative of the function y in the previous theorem exists and the abstract Cauchy problem y ′ (t) = Ay(t) for all t ≥ 0, y(0) = x, has classical solution. But we know that there exists at most one solution, and if the solution exists it depends locally Lipschitz continuous on the initial value x. If the norm is order continuous, the existence of a solution is also known, see [14].

Symmetric Lipschitz set
Throughout this section, let (S(t)) t≥0 and (S ± (t)) t≥0 be three semigroups on a Banach lattice X. If we choose S + (t) := S(t) and S − (t)x := −S(t)(−x) for all t ≥ 0 and x ∈ X, the set t≥0 . In examples, this set can be determined explicitly, see Subsection 6.1. Furthermore, by defining the generator w.r.t a weaker norm or pointwise almost everywhere, for elements of the symmetric Lipschitz set, the generator can be determined explicitly, see [3,13]. In order to solve the corresponding Cauchy problem for all positive times, it is crucial to show that the symmetric Lipschitz set is invariant under the semigroup. Compared to the invariance of the domain, the invariance of the symmetric Lipschitz set does not depend on the underlying space, i.e., it also holds for spaces of continuous function. Furthermore, for every r, T ≥ 0, there exists c ≥ 0 such that, for all t ∈ [0, T ] and x, y ∈ B(0, r), Then, it holds L S + ∩ L S − ⊂ L S and S(t) : Proof. First, we show L S + ∩ L S − ⊂ L S . Let x ∈ L S + ∩ L S − . By definition, there exist t 0 > 0 and c ≥ 0 such that

Furthermore, by assumption, it holds
We obtain the estimate Second, we show S(t) : L S + ∩ L S − → L S + ∩ L S − for all t ≥ 0. Let x ∈ L S + ∩ L S − and t ≥ 0. By definition and the first part, there exist t 0 > 0 and c ≥ 0 such that It follows from the assumptions of the theorem that, for all s, t ≥ 0, We use inequality (5.1) and inequality (5.2) to estimate for all s ∈ [0, t 0 ] and t ≥ 0. Here, c 1 is a constant such that inequality (5.1) holds with T := t 0 and r := max{ S(t)x , S ± (t)x }.
The following assumption on the families (I(t)) t≥0 and (I ± (t)) t≥0 ensures that, if existing, the associated semigroups (S(t)) t≥0 and (S ± (t)) t≥0 satisfy the assumptions of Theorem 5.1. Furthermore, they allow us to study the relation between the Lipschitz sets of the generating families and the associated semigroups.
Assumption 5.2. Let (I(t)) t≥0 and (I ± (t)) t≥0 be three families of monotone operators on X, which satisfy Assumption 2.1 with x 0 := 0. We assume, for all s, t ≥ 0 and x ∈ X, Proof. W.l.o.g. we assume that (I(t)) t≥0 and (I ± (t)) t≥0 satisfy Assumption 2.1 with the same function α and the same constant ω r for all r ≥ 0. First, we show that . It follows from Assumption 5.2 and Remark 2.14(ii) that where r := x . On the other hand, let x ∈ L S + ∩ L S − . By definition, there exist t 0 > 0 and c ≥ 0 such that and therefore Third, we show S(s)S − (t)x ≤ S − (t)S(s)x and S + (s)S(t)x ≤ S(t)S + (s)x for all s, t ≥ 0 and x ∈ X. We use Assumption 5.2(ii) and the monotonicity of (I(t)) t≥0 and (I − (t)) t≥0 to estimate for all s 1 , s 2 , t 1 , t 2 ≥ 0. It follows by induction that for all s, t ≥ 0, π s ∈ P s , π t ∈ P t and x ∈ X. Hence, Assumption 5.2 and the monotonicity of I(π s ) and I − (π t ) imply Similarly, we use Assumption 5.2(ii) and the monotonicity of (I(t)) t≥0 and (I + (t)) t≥0 to obtain I + (π s )I(π t )x ≤ I(π t )I + (π s )x for all s, t ≥ 0, π s ∈ P s , π t ∈ P t and x ∈ X. Moreover, Assumption 5.2(iii) implies that the set {I(π) : π ∈ P t } is directed upwards for all t ≥ 0, and Assumption 5.2(iv) ensures that I + (π) is continuous from below for all s ≥ 0 and π ∈ P s . We combine this with the monotonicity of I(π t ) to conclude for all s, t ≥ 0 and x ∈ X. Fifth, inequality (5.1) follows from Assumption 2.1 and Remark 2.14(i). Hence, we can apply Theorem 5.1 and obtain S(t) : x ∈ D}. Let (S(t)) t≥0 and (S ± (t)) t≥0 be associated semigroups as in Theorem 2.5. Then, by construction, it holds S + (t) = S(t) and S − (t)x = −S(t)(−x) for all (x, t) ∈ X × R + . The condition S(t)x = sup π∈Pt I(π)x has already been discussed in Subsection 2.2. If this equation holds, we conclude We will see in Subsection 6.1 that the verification of Assumption 5.2(i)-(iv) is straightforward, if I(t) is a supremum over linear semigroups. Basically, condition (ii) is the consequence of interchanging a supremum with an infimum at the cost of an inequality. Furthermore, it is clear that Theorem 5.3 remains true without Assumption 5.2(iii) and (iv), if I(t) = I + (t) for all t ≥ 0.

Examples
We illustrate our main results with several examples. First, we consider Nisio semigroups, which have already been discussed in Subsection 2.2. This first type of examples is illustrated by a convex version of the g-expectation as well as a sublinear version the geometric Brownian motion and the G-expectation. Second, we start with a linear semigroup (S 0 (t)) t≥0 and consider the generating family (I(t)) t≥0 given as the perturbation where Ψ : X → X is a Lipschitz continuous mapping. If we choose X := R d and S 0 (t) := id R d , for every x ∈ R d , we obtain the unique solution to the ODE The main example of this second type are reaction-diffusion equations, where the operators I(t) are neither convex nor monotone. Note that the results in Section 2 do not rely at all on these properties. The verification of condition (4.2) becomes more complicated, but is possible by proving a suitable recursion formula for the iterated operators I(2 −n ) k .
The theory and the examples in this paper are independent of results from the established PDE theory. For a common approach to nonlinear parabolic equations we refer to [23]. There, short time existence is proved by a fixed-point argument, and, as long as the solution does not blow up, long time existence follows. In the semi-linear case, blow up in finite time is excluded, if the the non-linearity is locally Lipschitz continuous and does not grow faster than linear. While the approach in [23] relies strongly on a priori estimates in suitable function spaces, we use stochastic representations for the generating family (I(t)) t≥0 and Itô calculus. 6.1. Convex g-expectation. In this subsection, we construct a semigroup, which corresponds to a Brownian motion with uncertain drift. The generator is a semi-linear second order differential operator, where the first-order non-linearity corresponds to the uncertainty in the drift. Let (W t ) t≥0 be a d-dimensional Brownian motion on a probability space (Ω, F, P). Furthermore, let L : For every t ≥ 0, f ∈ C 0 (R d ; R) and x ∈ R d , we define where E[X] denotes the expectation of random variable X : Ω → R. For every λ ∈ R d , we denote by (S λ (t)) t≥0 the linear semigroup given by Moreover, we can write I(t)f = sup λ∈R d I λ (t)f for all t ≥ 0 and f ∈ C 0 by defining The first-order non-linearity will be described by the function where ·, · denotes the Euclidean inner product on R d . Note that we are not restricted to non-linearities with linear growth. For example, power functions H(x) := |x| p with p ≥ 1 are included in our setting. Proof. Let c ≥ 0, f ∈ Lip 0 (c) and λ 0 ∈ R d with L(λ 0 ) < ∞, which exists by assumption. For every λ ∈ R d , t ≥ 0 and x ∈ R d , Since the assumption on L implies L(λ 0 ) + c|λ − λ 0 | − L(λ) → −∞ as |λ| → ∞, the claim follows.
Let C 2 0 be the space of all twice continuously differentiable functions f ∈ C 0 such that the first and second derivative vanish at infinity. For every λ ∈ R d , we denote by A λ the generator of (S λ (t)) t≥0 . It follows from Ito's formula that Theorem 6.2. The family (I(t)) t≥0 satisfies the assumptions of Lemma 3.2. Hence, Theorem 2.5 yields an associated semigroup (S(t)) t≥0 on C 0 . Moreover, it holds C 2 0 ⊂ D(A) with Proof. First, we verify the conditions (i)-(iv) of Lemma 3.2.
(ii) For every t ≥ 0 and f ∈ C 0 , the first assumption on L implies We obtain By taking the supremum over λ ∈ R d and changing the role of f and g, we obtain Second, we show that Let f ∈ C 2 0 . By Lemma 6.1, there exists r ≥ 0 such that Moreover, the constant r ≥ 0 can be chosen such that because ∇f ∞ < ∞ and L growths faster than linear. Hence, it follows from Itô's formula that where X λ s := W s + λs for all λ ∈ R d and s ≥ 0. Since f ∈ C 2 0 , for every ε > 0, there exists δ > 0 such that for all λ ∈ R d and s ≥ 0. Furthermore, Chebyshev's inequality implies sup λ∈B(0,r) Third, we verify the conditions (v) and (vi) of Lemma 3.2. It follows immediately from inequality (6.1) that C 2 0 ⊂ Lip 0 ∩L I . In particular, we can choose a countable set D ⊂ Lip 0 ∩L I , which is dense in C 0 . It remains to show condition (vi). Let c ≥ 0 and choose r ≥ 0 such that I(t)f = sup λ∈B(0,r) I λ (t)f for all t ≥ 0 and f ∈ Lip 0 (c), which exists due to Lemma 6.1. We define for all t ≥ 0, f ∈ C + 0 and x ∈ R d . The family (T c (t)) t≥0 is a monotone semigroup on C + 0 . It remains to show |I(t)f | ≤ T c (t)|f | for all f ∈ Lip 0 (c) and t ≥ 0.
Let f ∈ Lip 0 (c) and t ≥ 0. We use W t ∼ N (0, t1), where N (0, t1) denotes the normal distribution, and the formula for its moment generating function to estimate for all λ ∈ B(0, r). Taking the supremum yields Furthermore, by assumption, there exists λ 0 ∈ R d with L(λ 0 ) = 0. W.l.o.g. we can assume r ≥ |λ 0 | and obtain Forth, by Theorem 2.5, there exists an associated semigroup (S(t)) t≥0 on C 0 . In particular, Assumption 4.1 is satisfied. In addition, it follows from Lemma 4.4 that condition (4.2) holds for all f, g ∈ C 0 . Hence, Theorem 4.3 implies C 2 0 ⊂ D(A) with Let L ∞ be the space of all bounded, Borel measurable functions f : R d → R, where two of them are identified if they coincide Lebesgue almost everywhere. Moreover, we denote by W 1,∞ the corresponding first order Sobolev space. For f ∈ W 1,∞ we say that ∆f exists in L ∞ if there exists a function g ∈ L ∞ with In this case, since g is unique Lebesgue almost everywhere, we define ∆f := g. Here, C ∞ c denotes the set of all infinitely differentiable functions ϕ : R d → R with compact support. Theorem 6.3. In addition to previous assumptions, we assume that L satisfies • sup λ∈B(0,r)∩{L<∞} L(λ) < ∞ for all r > 0, • there exists ε > 0 with sup {|λ|=ε} L(λ) < ∞.
Then, it holds S(t) : L S sym → L S sym for all t ≥ 0, and L S sym = L I sym = f ∈ W 1,∞ ∩ C 0 : ∆f exists in L ∞ .
Since C 2 0 ⊂ C 0 is dense and (I(t)) t≥0 is a family of contractions, the mapping I(·)f is also continuous for arbitrary f ∈ C 0 , see the proof of Lemma 2.10. It remains to show that, for every t ≥ 0, the operator is well-defined. Lemma 3.2 implies {I(π) : π ∈ P t } ⊂ Lip 0 (c), and therefore T (t)f ∈ Lip 0 (c) for all c, t ≥ 0 and f ∈ Lip 0 (c). Since T (t) : C 0 → L ∞ is continuous and Lip 0 ⊂ C 0 is dense, we obtain T (t) : C 0 → C 0 for all t ≥ 0. We can apply Lemma 2.15 and conclude that S(t)x = T (t)x for all (x, t) ∈ X × R + . Hence, Assumption 5.2 is satisfied and Theorem 5.3 yields L S = L I and S(t) : L S sym → L S sym for all t ≥ 0.
Third, we show L I sym ⊂ {f ∈ W 1,∞ ∩ C 0 : ∆f exists in L ∞ }. Let f ∈ L I sym . By definition, there exist t 0 > 0 and c ≥ 0 such that and therefore S λ (t)f − f ∞ ≤ (c + L(λ))t. Let η ∈ C ∞ c with supp(η) ⊂ B(0, 1) and R d η(x) dx = 1. For every n ∈ N and x ∈ R d , we define η n (x) := n d η(nx) and Moreover, for every n ∈ N, we have the identities 3) It follows from inequality (6.2) and inequality (6.3) that sup n∈N ∆f n ∞ < ∞, since the assumptions on L ensure the existence of λ ∈ R d with L(±λ) < ∞. Furthermore, we use inequality (6.2), inequality (6.4) and the assumption on L to estimate By Banach-Alaoglu's theorem, there exist functions g, g i ∈ L ∞ such that ∆f n k → g and ∂ i f n → g i as k → ∞ in the weak*-topology for a suitable subsequence. This implies f ∈ {h ∈ W 1,∞ ∩ C 0 : ∆h exists in L ∞ } with ∆f = g and ∂ i f = g i for i = 1, . . . , d.
Forth, we show {f ∈ W 1,∞ ∩ C 0 : ∆f exists in L ∞ } ⊂ L I sym . Let f ∈ W 1,∞ ∩ C 0 such that ∆f exists in L ∞ . By Lemma 6.1, there exists r ≥ 0 such that Let t ≥ 0, λ ∈ B(0, r) and f n := f * η n for all n ∈ N. We use Itô's formula, ∇f n = (∇f ) * η n and ∆f n = (∆f ) * η n to estimate Taking the supremum over λ ∈ B(0, r) yields For the lower bound, we choose λ ∈ R d with L(λ) = 0 and obtain This shows f ∈ L I . Applying the same argument on −f yields f ∈ L I sym .
(i) For functions f ∈ L S sym the Laplacian and gradient are defined in the distributional sense and 1 2 ∆f + H(∇f ) ∈ L ∞ . By extending the semigroup (S(t)) t≥0 from C 0 to an exponential Orlicz heart, one can show that the Cauchy problem has a unique classical solution, which is represented by the extended semigroup.
Here, the initial value f ∈ L S sym is chosen such that the generator Af is defined w.r.t. to the Orlicz norm, which is weaker than the supremum norm. For details, we refer to [3]. A sublinear version of this example had previously been studied in [14]. (ii) The explicit description of the symmetric Lipschitz set in the previous theorem relies only on elementary estimates and Banach-Alaoglu's theorem. Nonetheless, by using the results in [23], one can improve the regularity. It holds i.e., the symmetric Lipschitz set equals the domain of the Laplacian in C 0 .
Proof. Let {f ∈ p≥1 W 2,p loc ∩ C 0 : ∆f ∈ L ∞ }. By [23,Theorem 3.1.7], it holds f ∈ W 1,∞ , and therefore f ∈ L S sym . Now, let f ∈ L I sym and f n := f * η n for all n ∈ N. Fix p > d. By [23,Theorem 3.1.6], there exist c ≥ 0 and ε > 0 such that, for all n ∈ N, This implies sup n∈N f n W 2,p (B(0,r)) < ∞ for all r ≥ 0 and n ∈ N. Taking the limit n → ∞ yields f ∈ W 2,p (B(0, r)) for all r ≥ 0, i.e., f ∈ W 2,p loc . Since W 2,q loc ⊂ W 2,p loc for all p ≤ q, we obtain f ∈ p≥1 W 2,p loc . The last part of the claim follows from [23, Theorem 3.1.7]. 6.2. Geometric Brownian motion. In this subsection, we construct a semigroup, which corresponds to a geometric Brownian motion with uncertain drift and volatility. Based on the Nisio approach, this example has been studied in [24], where the authors obtain a viscosity solution for the associated Cauchy problem. In contrast to the previous example, we have to weaken the supremum norm by adding a weight function.
Let (W t ) t≥0 be a 1-dimensional Brownian motion on a probability space (Ω, F, P) and p > 1. We choose the weight function Let Λ ⊂ R × R + be a bounded set. For every f ∈ UC κ (R d ; R), t ≥ 0 and x ∈ R, we define where X λ,x t := xX λ t and X λ t := exp µ − σ 2 2 t + σW t for λ := (µ, σ 2 ). Furthermore, for every λ ∈ Λ, let (S λ (t)) t≥0 be the linear semigroup given by t ) for all t ≥ 0, f ∈ UC κ and x ∈ R. We start with two auxiliary lemmas.
Since the right hand side of the previous inequality converges to zero as t ↓ 0, we obtain the first part of the claim. Furthermore, by Lemma 6.5, the set (X λ t ) q − 1 : t ∈ [0, 1], λ ∈ Λ is bounded in L 2 (P), and therefore uniformly integrable. Hence, the second part of the claim follows from the first one, similar to the Vitali convergence theorem.
Let C 2 c be the set of all twice continuously differentiable functions f : R → R with compact support.
(ii) Let f ∈ UC κ , t ≥ 0, λ ∈ Λ and x ∈ R. It follows from Lemma 6.5 that Hence, we obtain S λ (t)f κ ≤ e ωt f κ , and therefore (iii) For every f, g ∈ UC κ and t ≥ 0, In particular, we obtain I(t) : UC κ → UC κ , because I(t) : Lip b → Lip b , Lip b ⊂ UC κ is dense and I(t) is Lipschitz continuous. Third, we verify Assumption 2.4. Let f ∈ C 2 c and choose r ≥ 0 with supp(f ) ⊂ [−r, r]. Fix λ ∈ Λ, t ≥ 0 and x ∈ R. Itô's formula implies Hence, we can estimate where c := sup λ∈Λ µr f ′ ∞ + σ 2 2 r 2 f ′′ ∞ . Since I(t) : Lip b (c) → Lip b (e ωt c) for all c, t ≥ 0, it follows from Lemma 3.3 that the sequence (I(π t n )f ) n∈N is relatively compact in UC κ for all f ∈ Lip b and t ∈ T . Furthermore, by Lemma 3.4, we can choose a countable set D ⊂ C 2 c , which is dense in UC κ . By Theorem 2.5 there exists a semigroup (S(t)) t≥0 on UC κ associated to (I(t)) t≥0 .
By Lemma 6.6, there exists s 1 ∈ (0, s 0 ], independent of λ and x, such that We obtain Combining inequality (6.9) and inequality (6.10) with the definition of c yields By similar arguments, it follows from Lemma 6.6 with q = 2 that As seen before, equation (6.5) is a consequence of inequality (6.6) and inequality (6.11). Fifth, it follows from Theorem 2.5 that Assumption 4.1 is satisfied. In addition, by Lemma 4.4, condition (4.2) holds for all f, g ∈ UC κ . Hence, Theorem 4.3 implies for all f ∈ C 2 c and x ∈ R.
6.4. Ordinary differential equations. In this subsection, we obtain the well-known existence and uniqueness result for ODEs with locally Lipschitz continuous data. Let f : R d → R d be a function, which satisfies the following conditions: • There exists K ≥ 0 such that |f (x)| ≤ K(1 + |x|) for all x ∈ R d .
We define I(t)x := x + tf (x) for all t ≥ 0 and x ∈ R d .
Moreover, we have |I(t)x − I(t)y| ≤ e Lrt |x − y| for all r, t ≥ 0 and x, y ∈ B(0, r). In addition, Lemma 2.7 implies that the sequence I(π t n )x) n∈N ⊂ R d is bounded, and therefore relatively compact for all (x, t) ∈ R d × T . Theorem 2.5 yields a semigroup (S(t)) t≥0 on R d associated to (I(t)) t≥0 .
Second, we note that I(t)x − x t − f (x) = 0 for all t > 0 and x ∈ R d .
Furthermore, one can show that condition (4.2) holds for all x, y ∈ R d , for details we refer to the proofs of Lemma 6.13 and Theorem 6.14 below. Hence, it follows from Theorem 4.3 that D(A) = R d and Ax = f (x) for all x ∈ R d . Let x ∈ R d and define y(t) := S(t)x for all t ≥ 0. For every t ≥ 0, the right-derivative of y exists and is given by Ay(t) = f (y(t)). Since y and f (y(·)) are continuous, it follows from [28, Corollary 1.2 in Section 2] that y is continuously differentiable. The uniqueness follows from Theorem 4.8.
6.5. Lipschitz perturbation. Throughout this subsection, we consider vector valued functions f : R d → R m . We construct a semigroup corresponding to a perturbed linear semigroup, which adds a nonlinear zero-order coupling to the generator of the linear semigroup. The ODEs from the previous section are including in this setting, as well as reaction-diffusion equations, see Example 6.15.
Assumption 6.11. Let (S 0 (t)) t≥0 be a strongly continuous monotone linear semigroup on C 0 (R d ; R m ), which satisfies the following condition: (i) There exists ω ≥ 0 such that S 0 (t)f ∞ ≤ e ωt f ∞ for all t ≥ 0 and f ∈ C 0 . (ii) L S 0 ∩ Lip 0 ⊂ C 0 is dense.
(iv) lim |x|→∞ sup t∈[0,T ] |S 0 (t)f )(x)| = 0 for all T ≥ 0 and f ∈ C 0 . Furthermore, let Ψ : R m → R m be a continuous function with Ψ(0) = 0, which satisfies the following conditions: In particular, Assumption 6.11(ii) implies that L I ∩ Lip 0 ⊂ C 0 is dense. Hence, in order to verify Assumption 2.4, it suffices to show that the assumptions from Lemma 3.1 are satisfied for all f ∈ Lip 0 .
Let c ≥ 0, f ∈ Lip 0 (c) and t ∈ T . By induction, it follows that I(π t n )f ∈ Lip 0 (e (ω+Lr)t c) for all n ∈ N, where r := α(c, t). In particular, the sequence I(π t n ) n∈N is equicontinuous. Fifth, it remains to show that lim |x|→∞ sup n∈N |(I(π t n )f )(x)| = 0 for all f ∈ Lip 0 and t ∈ T .
To do so, for every r, t ≥ 0, we define J r (t) : C 0 → C 0 , f → S 0 (t)f + tL r f.
To determine the generator of (S(t)) t≥0 , we need the following recursion.