INVARIANCE OF CLOSED CONVEX CONES FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

INVARIANCE OF CLOSED CONVEX CONES FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS STEFAN TAPPE Absrac. The goal of his paper is o clarify when a closed convex cone is invarian for a sochasic parial differenial equaion (SPDE) driven by a Wiener process and a Poisson random measure, and o provide condiions on he parameers of he SPDE, which are necessary and sufficien. 1. Inroducion Consider a semilinear sochasic parial differenial equaion (SPDE) of he form (1.1) { dr = (Ar + α(r ))d + σ(r )dw + E γ(r, x)(µ(d, dx) F (dx)d) r = h driven by a race class Wiener process W and a Poisson random measure µ. The sae space of he SPDE (1.1) is a separable Hilber space H, and he operaor A is he generaor of a srongly coninuous semigroup (S ) on H. Le K H be a closed convex cone of he sae space H. We say ha he cone K is invarian for he SPDE (1.1) if for each saring poin h K he soluion process r o (1.1) says in K. The goal of his paper is o clarify when he cone K is invarian for he SPDE (1.1), and o provide condiions on he parameers (A, α, σ, γ) or, equivalenly, on ((S ), α, σ, γ) of he SPDE (1.1), which are necessary and sufficien. Sochasic invariance of a given subse K H for jump-diffusion SPDEs (1.1) has already been sudied in he lieraure, mosly for diffusion SPDEs { dr = (Ar + α(r ))d + σ(r )dw (1.2) r = h wihou jumps. The classes of subses K H, for which sochasic invariance has been invesigaed, can roughly be divided as follows: For a finie dimensional submanifold K H he sochasic invariance has been sudied in [8] and [29] for diffusion SPDEs (1.2), and in [11] for jumpdiffusion SPDEs (1.1). Here a relaed problem is he exisence of a finie dimensional realizaion (FDR), which means ha for each saring poin h H a finie dimensional invarian manifold K H wih h K exiss. This problem has mosly been sudied for he so-called Heah-Jarrow- Moron-Musiela (HJMM) equaion from mahemaical finance, and we refer, for example, o [5, 4, 13, 14, 34, 38] for he exisence of FDRs for diffusion SPDEs (1.2), and, for example, o [35, 32, 37] for he exisence of FDRs for SPDEs driven by Lévy processes, which are paricular cases of jump-diffusion SPDEs (1.1). Dae: 1 March, 216. 21 Mahemaics Subjec Classificaion. 6H15, 6G17. Key words and phrases. Sochasic parial differenial equaion, closed convex cone, sochasic invariance, parallel funcion.

2 STEFAN TAPPE For an arbirary closed subse K H he sochasic invariance has been sudied for PDEs in [19], and for diffusion SPDEs (1.2) in [2] and based on he suppor heorem presened in [28] in [29]. Boh auhors obain he so-called sochasic semigroup Nagumo s condiion (SSNC) as a crierion for sochasic invariance, which is necessary and sufficien. An indispensable assumpion for he formulaion of he SSNC is ha he volailiy σ is sufficienly smooh; i mus be wo imes coninuously differeniable. For a closed convex cone K H as in our paper he sochasic invariance has been sudied in wo paricular siuaions on funcion spaces. In [26] he sae space H is an L 2 -space, K is he closed convex cone of nonnegaive funcions, and is sochasic invariance is invesigaed for diffusion SPDEs (1.2). In [1] he sae space H is a Hilber space consising of coninuous funcions, K is also he closed convex cone of nonnegaive funcions, and is sochasic invariance is invesigaed for jump-diffusion SPDEs (1.1); a paricular applicaion in [1] is he posiiviy preserving propery of ineres rae curves from he aforemenioned HJMM equaion, which appears in mahemaical finance. In his paper, we provide a general invesigaion of he sochasic invariance problem for an arbirary closed convex cone K H, conained in an arbirary separable Hilber space H, for jump-diffusion SPDEs (1.1). Taking advanage of he srucural properies of closed convex cones, we do no need smoohness of he volailiy σ, as i is required in [2] and [29], and also in [1]. In order o presen our main resul of his paper, le K H be a closed convex cone, and le K H be is dual cone (1.3) K = {h H : h, h }. h K Then he cone K has he represenaion (1.4) K = {h H : h, h }. h K We fix a generaing sysem G of he cone K; ha is, a subse G K such ha he cone admis he represenaion (1.5) K = {h H : h, h }. h G In paricular, we could simply ake G = K. However, for applicaions we will choose a generaing sysem G which is as convenien as possible. Throughou his paper, we make he following assumpions: The semigroup (S ) is pseudo-conracive; see Assumpion 2.1. The coefficiens (α, σ, γ) are locally Lipschiz and saisfy he linear growh condiion, which ensures exisence and uniqueness of mild soluions o he SPDE (1.1); see Assumpion 2.2. The cone K is invarian for he semigroup (S ) ; see Assumpion 2.12. The cone K is generaed by an uncondiional Schauder basis; see Assumpion 4.2. We refer o Secion 2 for he precise mahemaical framework. We define he se D G K as { D := (h, h) G h }, S h (1.6) K : lim inf <.

INVARIANCE OF CLOSED CONVEX CONES 3 Since he cone K is invarian for he semigroup (S ), for all (h, h) G K he limes inferior in (1.6) exiss wih value in R + = [, ]. Now, our main resul reads as follows. 1.1. Theorem. Suppose ha Assumpions 2.1, 2.2, 2.12 and 4.2 are fulfilled. Then he following saemens are equivalen: (i) The closed convex cone K is invarian for he SPDE (1.1). (ii) We have (1.7) (1.8) (1.9) h + γ(h, x) K for F -almos all x E, for all h K, and for all (h, h) D we have + h, α(h) lim inf h, S h h, σ j (h) =, j N. E h, γ(h, x) F (dx) Condiions (1.7) (1.9) are geomeric condiions on he coefficiens of he SPDE (1.1); condiion (1.7) concerns he behaviour of he soluion process in he cone, and condiions (1.8) and (1.9) concern he behaviour of he soluion process a boundary poins of he cone: Condiion (1.7) is a condiion on he jumps; i means ha he cone K is invarian for he funcions h h + γ(h, x) for F -almos all x E. Condiion (1.8) means ha he drif is inward poining a boundary poins of he cone. Condiion (1.9) means ha he volailiies are parallel a boundary poins of he cone. Figure 1 illusraes condiions (1.7) (1.9). Le us provide furher explanaions regarding he drif condiion (1.8). For his purpose, we fix an arbirary pair (h, h) D. By he definiion (1.6) of he se D, we have h, h =, indicaing ha we are a he boundary of he cone. The drif condiion (1.8) implies (1.1) h, γ(h, x) F (dx) <. E This means ha he jumps of he soluion process a boundary poins of he cone are of finie variaion, unless hey are parallel o he boundary. If h D(A), hen he drif condiion (1.8) is fulfilled if and only if (1.11) h, Ah + α(h) h, γ(h, x) F (dx). E In view of condiion (1.11), we poin ou ha K D(A) is dense in K. If h D(A ), hen he drif condiion (1.8) is fulfilled if and only if (1.12) A h, h + h, α(h) h, γ(h, x) F (dx). In paricular, if A is a local operaor, hen he drif condiion (1.8) is equivalen o (1.13) h, α(h) h, γ(h, x) F (dx). E In any case, condiion (1.13) implies he drif condiion (1.8). We refer o Secion 2 for he proofs of hese and of furher saemens. We emphasize ha for (h, h) G K wih h, h = i may happen ha (h, h) / D. In his case, condiions (1.8) and hence (1.1) and (1.9), he wo boundary condiions illusraed in Figure 1, do no need o be fulfilled. Inuiively, a such a boundary E

4 STEFAN TAPPE jump volailiy drif Figure 1. Illusraion of he invariance condiions. poin h of he cone, here is an infinie drif pulling he process in he inerior of he half space {h H : h, h }, whence we can skip condiions (1.8) and (1.9) in his siuaion. This phenomenon is ypical for SPDEs, as for norm coninuous semigroups (S ) (in paricular, if A = ) he limes inferior appearing in (1.6) is always finie. Now, le us ouline he essenial ideas for he proof of Theorem 1.1: In Theorem 3.1 we will prove ha condiions (1.7) (1.9) are necessary for invariance of he cone K, where he main idea is o perform a shor-ime analysis of he sample pahs of he soluion processes. We emphasize ha for his implicaion we do no need he assumpion ha K is generaed by an uncondiional Schauder basis; ha is, we can skip Assumpion 4.2 here. In order o show ha condiions (1.7) (1.9) are sufficien for invariance of he cone K, we perform several seps: (1) Firs, we show ha he cone K is invarian for diffusion SPDEs (1.2) wih smoohs volailiies σ j Cb 2 (H), j N; see Theorem 5.3. The essenial idea is o verify he aforemenioned SSNC. (2) Then, we show ha he cone K is invarian for diffusion SPDEs (1.2) wih Lipschiz coefficiens wihou imposing smoohness on he volailiies; see Theorem 6.1. The main idea is o approximae he volailiy σ by a sequence (σ n ) n N of smooh volailiies, and o apply a sabiliy resul (see Proposiion B.3) for SPDEs. (3) Then, we show ha he cone K is invarian for general jump-diffusion SPDEs (1.1) wih Lipschiz coefficiens; see Theorem 7.1. This is done by using he so-called mehod o swich on he jumps also used in [1] and he aforemenioned sabiliy resul for SPDEs. (4) Finally, we show ha he cone K is invarian for he SPDE (1.1) in he general siuaion, where he coefficiens are locally Lipschiz and saisfy he linear growh condiion; see Theorem 8.1. This is done by approximaing he parameers (α, σ, γ) of he SPDE (1.1) by a sequence (α n, σ n, γ n ) n N of globally Lipschiz coefficiens, and o argue by sabiliy. In order o ensure ha he modified coefficiens (α n, σ n, γ n ) also saisfy he required invariance condiions (1.7) (1.9), he srucural properies of closed convex cones are essenial. The mos challenging is he second sep, where we approximae he volailiy σ by a sequence (σ n ) n N of smooh volailiies. In paricular, for an applicaion of

INVARIANCE OF CLOSED CONVEX CONES 5 our sabiliy resul (Proposiion B.3) we mus ensure ha all σ n are Lipschiz coninuous wih a join Lipschiz consan. We can roughly divide he approximaion procedure ino he following seps: (a) Firs, we approximae σ by a sequence (σ n ) n N of bounded volailiies wih finie dimensional range; see Proposiions D.13 and D.15. We consruc similar approximaions (α n ) n N for he drif α; see Proposiions C.8 and C.11. (b) Then, we approximae a bounded volailiy σ wih finie dimensional range by a sequence (σ n ) n N from C 1,1 b. This is done by he so-called sup-inf convoluion echnique from [23]; see Proposiion D.27. Alhough we do no use i in his paper, we menion he relaed aricle [22], which shows how a Lipschiz funcion can be approximaed by uniformly Gâeaux differeniable funcions. (c) Finally, we approximae a volailiy σ from C 1,1 b by a sequence (σ n ) n N from Cb 2 ; see Proposiion D.37. This is done by a generalizaion of he mollifying echnique in infinie dimension. For his procedure, we follow he consrucion provided in [15], which consiues a generalizaion of a resul from Moulis (see [27]), whence we also refer o his mehod as Moulis mehod. Concerning smooh approximaions in infinie dimensional spaces, we also menion he relaed papers [1, 2, 17, 18]. We emphasize ha we canno direcly apply Moulis mehod in sep (b), because for a Lipschiz coninuous funcion σ his would only provide a sequence (σ n ) n N from C 2 in fac, even C bu he second order derivaives migh be unbounded. Applying he sup-inf convoluion echnique before ensures ha we obain a sequence from Cb 2. We menion ha a combinaion of he sup-inf convoluion echnique and Moulis mehod has also been used in [1] in order o prove ha every Lipschiz coninuous funcion defined on a (possibly infinie dimensional) separable Riemannian manifold can be uniformly approximaed by smooh Lipschiz funcions. Besides he aforemenioned required join Lipschiz consan, we have o ake care ha he respecive approximaions (σ n ) n N of he volailiy σ remain parallel a boundary poins of he cone; ha is, condiion (1.9) mus be preserved, which is expressed by Definiion C.3. The siuaion is similar for he approximaions (α n ) n N of he drif α. They mus remain inward poining a boundary poins of he cone; ha is, condiion (1.8) mus be preserved, which is expressed by Definiion C.2. I arises he problem ha we can generally no ensure in seps (b) and (c) ha he approximaing volailiies remain parallel. In order o illusrae he siuaion in sep (c), where we apply Moulis mehod, le us assume for he sake of simpliciy ha he sae space is H = R d. Then he consrucion of he approximaing sequence (σ n ) n N becomes simpler han in he infinie dimensional siuaion in [15], and i is given by he well-known consrucion σ n : R d R d, σ n (h) := σ(h g)ϕ n (g)dg, R d where (ϕ n ) n N C (R d, R + ) is an appropriae sequence of mollifiers. Then, for (h, h) D, which implies h, h =, we generally have h, σ n (h) = h, σ(h g) ϕ n (g)dg, R d because we only have h, σ(h) =, bu generally no h, σ(h g) = for all g R d from a neighborhood of. This problem leads o he noion of locally parallel funcions (see Definiion D.1), which have he desired propery ha h, σ(h g) = for all g R d from an appropriae neighborhood of. In order o implemen his concep, we have o show ha a parallel funcion can be approximaed by a sequence

6 STEFAN TAPPE of locally parallel funcions. The idea is o approximae a funcion σ : R d R d for ɛ > by aking σ Φ ɛ, where Φ ɛ : R d R d, Φ ɛ (h) := (φ ɛ (h 1 ),..., φ ɛ (h d )), and where he funcion φ ɛ : R R is defined as (1.14) φ ɛ (x) := (x + ɛ)1 (, ɛ] (x) + (x ɛ)1 [ɛ, ) (x), see Figure 2. We can also esablish his procedure in infinie dimension; see Proposiion D.18. y x Figure 2. Approximaion wih locally parallel funcions. The remainder of his paper is organized as follows. In Secion 2 we presen he mahemaical framework and preliminary resuls. In Secion 3 we prove ha our invariance condiions are necessary for invariance of he cone. In Secion 4 we provide he required background abou closed convex cones generaed by uncondiional Schauder basis. Aferwards, we sar wih he proof ha our invariance condiions are sufficien for invariance of in he cone. In Secion 5 we prove his for diffusion SPDEs wih smooh volailiies, in Secion 6 for diffusion SPDEs wih Lipschiz coefficiens wihou imposing smoohness on he volailiy, in Secion 7 for general jump-diffusion SPDEs wih Lipschiz coefficiens, and in Secion 8 for he general siuaion of jump-diffusion SPDEs wih coefficiens being locally Lipschiz and saisfying he linear growh condiion. In Appendix A we collec he funcion spaces which we use hroughou his paper, and in Appendix B we presen he required sabiliy resul for SPDEs. In Appendix C we provide he required resuls abou inward poining funcions, and in Appendix D abou parallel funcions. 2. Mahemaical framework and preliminary resuls In his secion, we presen he mahemaical framework and preliminary resuls. Le (Ω, F, (F ) R+, P) be a filered probabiliy space saisfying he usual condiions. Le H be a separable Hilber space and le A : D(A) H H be he infiniesimal generaor of a C -semigroup (S ) on H. 2.1. Assumpion. We assume ha he semigroup (S ) is pseudo-conracive; ha is, here exiss a consan β such ha (2.1) S e β for all.

INVARIANCE OF CLOSED CONVEX CONES 7 Le U be a separable Hilber space, and le W be an U-valued Q-Wiener process for some nuclear, self-adjoin, posiive definie linear operaor Q L(U); see [6, pages 86, 87]. There exis an orhonormal basis {e j } j N of U and a sequence (λ j ) j N (, ) wih j N λ j < such ha Qe j = λ j e j for all j N. Le (E, E) be a Blackwell space, and le µ be a homogeneous Poisson random measure wih compensaor d F (dx) for some σ-finie measure F on (E, E); see [21, Def. II.1.2]. The space U := Q 1/2 (U), equipped wih he inner produc (2.2) u, v U := Q 1/2 u, Q 1/2 v U, is anoher separable Hilber space. We denoe by L 2(H) := L 2 (U, H) he space of all Hilber-Schmid operaors from U ino H. We fix he orhonormal basis {g j } j N of U given by g j := λ j e j for each j N, and for each σ L 2(H) we se σ j := σg j for j N. Furhermore, we denoe by L 2 (F ) := L 2 (E, E, F ; H) he space of all square-inegrable funcions from E ino H. Le α : H H, σ : H L 2(H) and γ : H L 2 (F ) be measurable funcions. Concerning he upcoming noaion, we remind he reader ha in Appendix A we have colleced he funcion spaces used in his paper. 2.2. Assumpion. We suppose ha α Lip loc (H) LG(H), σ Lip loc (H, L 2(H)) LG(H, L 2(H)), γ Lip loc (H, L 2 (F )) LG(H, L 2 (F )). Assumpion 2.2 ensures ha for each h H he SPDE (1.1) has a unique mild soluion; ha is, an H-valued càdlàg adaped process r, unique up o indisinguishabiliy, such ha (2.3) r = S h + + The sequence (β j ) j N defined as (2.4) S s α(r s )ds + S s σ(r s )dw s S s σ(r s, x)(µ(ds, dx) F (dx)ds), R +. β j := 1 λj W, e j, j N is a sequence of real-valued sandard Wiener processes, and we can wrie (2.3) equivalenly as (2.5) r = S h + + S s α(r s )ds + j N S s σ j (r s )dβ j s S s σ(r s, x)(µ(ds, dx) F (dx)ds), R +. Noe ha Assumpion 2.2 is implied by he slighly sronger condiions α Lip(H), σ Lip(H, L 2(H)) and γ Lip(H, L 2 (F )). Under such global Lipschiz condiions, we refer he reader o [6, 33, 16, 24] for diffusion SPDEs, o [31] for Lévy driven SPDEs, and o [25, 9] for general jump-diffusion SPDEs. Under he local Lipschiz and linear growh condiions from Assumpion 2.2, we refer o [36].

8 STEFAN TAPPE 2.3. Definiion. A subse K H is called invarian for he SPDE (1.1) if for each h K we have r K up o an evanescen se 1, where r denoes he mild soluion o (1.1) wih r = h. 2.4. Definiion. A subse K H is called a cone if we have λh K for all λ and all h K. 2.5. Definiion. A cone K H is called a convex cone if we have h + g K for all h, g H. Noe ha a convex cone K H is indeed a convex subse of H. 2.6. Definiion. A convex cone K H is called a closed convex cone if i is closed as a subse of H. For wha follows, we fix a closed convex cone K H. Denoing by K H is dual cone (1.3), he cone K has he represenaion (1.4). 2.7. Definiion. A subse G K is called a generaing sysem of he cone K if we have he represenaion (1.5). Of course G = K is a generaing sysem of he cone K. However, for applicaions we will choose he generaing sysem G as convenien as possible. In his respec, we menion ha, by Lindelöf s lemma, he cone K admis a generaing sysem G which is a mos counable. For wha follows, we fix a generaing sysem G K. 2.8. Definiion. For a funcion f : H H we say ha K is f-invarian if f(k) K. 2.9. Definiion. The closed convex cone K is called invarian for he semigroup (S ) if K is S -invarian for all. According o [3, Cor. 1.1.6] he adjoin semigroup (S ) is a C -semigroup on H wih infiniesimal generaor A. 2.1. Lemma. The following saemens are equivalen: (i) K is invarian for he semigroup (S ). (ii) K is invarian for he adjoin semigroup (S ). Proof. For all (h, h) K K and all we have h, S h = S h, h, and hence, he represenaions (1.4) and (1.3) of K and K prove he claimed equivalence. For λ > β, where he consan β sems from he growh esimae (2.1), we define he resolven R λ := (λ A) 1. We consider he absrac Cauchy problem { dr = Ar d (2.6) r = h. 2.11. Lemma. The following saemens are equivalen: (i) K is invarian for he semigroup (S ). (ii) K is invarian for he absrac Cauchy problem (2.6). (iii) K is R λ -invarian for all λ > β. 1 A random se A Ω R+ is called evanescen if he se {ω Ω : (ω, ) A for some R + } is a P-nullse, cf. [21, 1.1.1].

INVARIANCE OF CLOSED CONVEX CONES 9 Proof. (i) (ii): This equivalence follows, because for each h K he mild soluion o he absrac Cauchy problem (2.6) is given by r = S h for. (i) (iii): For each λ > β and each h K we have R λ h = e λ S h d K. (iii) (i): Le > and h K be arbirary. By he exponenial formula (see [3, Thm. 1.8.3]) we have ( ) n n S h = lim n R n/ h K, compleing he proof. From now on, we make he following assumpion. 2.12. Assumpion. We assume ha he cone K is invarian for he semigroup (S ) ; ha is, any of he equivalen condiions from Lemma 2.11 is fulfilled. 2.13. Lemma. For all (h, h) G K we have lim inf h, S h R +. Proof. Since K is invarian for he semigroup (S ), we have h, S h for all, which esablishes he proof. 2.14. Definiion. For g, h H we wrie g K h if h g K. Recall he se D G K defined in (1.6). We define he funcion a : D R +, a(h, h) := lim inf h, S h. 2.15. Lemma. For each (h, h) D he following saemens are rue: (2.7) (2.8) (1) We have h, h =. (2) For all λ we have (h, λh) D and a(h, λh) = λa(h, h). (3) For all g K wih g K h we have (h, g) D and a(h, g) a(h, h). Proof. For each (h, h) G K wih h, h > we have and hence lim h, S h = h, h >, lim inf h, S h =, showing ha (h, h) / D. This proves he firs saemen, and we proceed wih he second saemen. Since K is a cone, we have λh K. Furhermore, we have lim inf h, S (λh) = λ lim inf h, S h <, showing (h, λh) D and he ideniy (2.7). For he proof of he hird saemen, le be arbirary. By Lemma 2.1 we have S h K. Since g K h, we obain S h, h g, and hence h, S g = S h, g S h, h = h, S h.

1 STEFAN TAPPE Consequenly, we have (2.9) h, S g h, S h for all. There exiss a sequence ( n ) n N (, ) wih n such ha he sequence (b n ) n N R + defined as b n := h, S n h n, n N converges o a(h, h) R +. Defining he sequence (a n ) n N R + as a n := h, S n g n, n N, by (2.9) we have a n b n for each n N. Hence, he sequence (a n ) n N is bounded, and by he Bolzano-Weiersrass heorem here exiss a subsequence (n k ) k N such ha (a nk ) k N converges o some a R + wih a a(h, h), which proves (h, g) D and (2.8). 2.16. Lemma. Le (h, h) G K wih h, h = be arbirary. Then he following saemens are rue: (1) If h D(A), hen we have (h, h) D and (2.1) (2.11) lim inf h, S h = h, Ah. (2) If h D(A ), hen we have (h, h) D and lim inf h, S h = A h, h. (3) If he semigroup (S ) is norm coninuous, hen we have (h, h) D as well as (2.1) and (2.11). Proof. If h D(A), hen we have h, S h = h, S h h, h = h, S h h = h, S h h h, Ah as, showing he firs saemen. Furhermore, if h D(A ), hen we obain h, S h = S h, h = S h h, h = S h, h h, h S = h h, h A h, h as, showing he second saemen. The hird saemen is an immediae consequence of he firs and he second saemen. The following definiion is inspired by [26, Lemma 5]. 2.17. Definiion. We call A a local operaor if G D(A ), and for all (h, h) D we have A h, h =. 2.18. Proposiion. Suppose ha condiion (1.7) is fulfilled. Then for all (h, h) D he following saemens are rue: (1) We have (2) We have h, γ(h, x) for F -almos all x E. E h, γ(h, x) F (dx) R +. (3) If condiion (1.8) is saisfied, hen we have (1.1).

INVARIANCE OF CLOSED CONVEX CONES 11 (4) If h D(A), hen condiions (1.8) and (1.11) are equivalen. (5) If h D(A ), hen condiions (1.8) and (1.12) are equivalen. (6) If A is a local operaor, hen condiions (1.8) and (1.13) are equivalen. (7) Condiion (1.13) implies (1.8). Proof. By (1.7), for F -almos all x E we have h, γ(h, x) = h, h + h, γ(h, x) = h, h + γ(h, x), which esablishes he firs saemen. The second saemen is an immediae consequence, and he hird saemen is obvious. The fourh and he fifh saemen follow from Lemma 2.16. Taking ino accoun Definiion 2.17, he sixh saemen is an immediae consequence of he fifh saemen. Finally, he las saemen follows from he firs saemen. In view of condiion (1.11), we emphasize ha K D(A) is dense is K, which follows from he nex resul. 2.19. Lemma. We have K = K D(A). Proof. Since K is closed, we have K D(A) K. In order o prove he converse inclusion, le h K be arbirary. For > we se h := 1 S shds. Then we have h D(A) for each >, and we have h h for. I remains o show ha h K for each >. For his purpose, le > and h G be arbirary. Since K is invarian for he semigroup (S ), we obain h, h = h, 1 S s hds = 1 h, S s h ds, showing ha h K. 3. Necessiy of he invariance condiions In his secion, we prove he necessiy of our invariance condiions. 3.1. Theorem. Suppose ha Assumpions 2.1, 2.2 and 2.12 are fulfilled. If he closed convex cone K is invarian for he SPDE (1.1), hen we have (1.7), and for all (h, h) D we have (1.8) and (1.9). Proof. Condiion (1.7) follows from [12, Lemma 2.11]. Le (h, h) D be arbirary, and denoe by r he mild soluion o (1.1) wih r = h. Since he measure space (E, E, F ) is σ-finie, here exiss an increasing sequence (B n ) n N E wih F (B n ) < for each n N such ha E = n N B n. Le n N be arbirary. According o [12, Lemma 2.2] he mapping T n : Ω R + given by T n := inf{ R + : µ([, ] B n ) = 1} is a sricly posiive sopping ime. We denoe by r n he mild soluion o he SPDE dr n = (Ar n + α(r n ) B n γ(r n, x)f (dx))d + σ(r n )dw + γ(r B, n x)(µ(d, dx) F (dx)d) n c r n = h. Since K is a closed subse of H, by [12, Prop. 2.21] we obain (r n ) Tn K up o an evanescen se. We define he sricly posiive, bounded sopping ime T := inf{ R + : r n > 1 + h } T n 1.

12 STEFAN TAPPE Furhermore, for every sopping ime R T we define he processes A n (R) and M n (R) as ( ) A n (R) := h, S R s α(rs n ) γ(rs n, x)f (dx) 1 {R s} ds, R +, B n M n (R) := + h, S R s σ(r n s ) 1 {R s} dw s B n h, S R s γ(r n s, x) 1 {R s} (µ(ds, dx) F (dx)ds), R +. Then, by he Cauchy-Schwarz inequaliy and Assumpions 2.1, 2.2 we have A n (R) A and M n (R) H 2 for each sopping ime R T, where A denoes he space of all finie variaion processes wih inegrable variaion (see [21, I.3.7]) and H 2 denoes he space of all square-inegrable maringales (see [21, Def. I.1.41]). Moreover, we have P-almos surely h, r n T = h, S T h + A n (T ) T + M n (T ) T for all R +. Le ( k ) k N (, ) be a sequence wih k such ha (3.1) lim inf h, S h h, S k h = lim. k k By Lebesgue s dominaed convergence heorem we obain lim k = lim k showing ha (3.2) 1 E[ h, rt n k ] = lim k k h, S k h k lim inf 1 E[ h, S T k h ] + lim k + h, α(h) h, γ(h, x) F (dx), B n k 1 k E[A n (T k ) T k ] 1 h, S h + h, α(h) h, γ(h, x) F (dx). B n Furhermore, by he monoone convergence heorem and Proposiion 2.18 we have (3.3) h, γ(h, x) F (dx) = lim h, γ(h, x) F (dx). E n B n Combining (3.2) and (3.3), we arrive a (1.8). Now, suppose ha condiion (1.9) is no fulfilled. Then here exis j N and (h, h) D such ha h, σ j (h). We define η, Φ R by (3.4) η := lim inf h, S h + h, α(h) and Φ := η + 1 h, σ j (h). Noe ha, by (1.8) and Proposiion 2.18 we have η R +. The sochasic exponenial Z := E(Φβ j ), where he Wiener process β j is given by (2.4), is a sricly posiive, coninuous local maringale. We define he sricly posiive, bounded sopping ime T := inf{ R + : r > 1 + h } inf{ R + : Z > 2} inf{ R + : Z, Z > 1} 1.

INVARIANCE OF CLOSED CONVEX CONES 13 For every sopping ime R T we define he processes A(R), M(R) and N(R) as A(R) := M(R) := N(R) := + h, S R s α(r s ) 1 {R s} ds, R +, h, S R s σ(r s ) 1 {R s} dw s E h, S R s γ(r s, x) 1 {R s} (µ(ds, dx) F (dx)ds), R +, (A(R) s + M(R) s )1 {R s} dz s + Z s 1 {R s} dm(r) s, R +. Then, by Assumpions 2.1, 2.2 we have A(R) A and M(R), N(R) H 2 for each sopping ime R T. Moreover, we have P-almos surely h, r T = h, S T h + A(T ) T + M(T ) T for all R +. Le R T be an arbirary sopping ime. By [21, Prop. I.4.49] we have [A(R), Z R ] =, and by [21, Thm. I.4.52] we have [M(R), Z R ] = M(R) c, Z R. Therefore, and since Z R by [21, Def. I.4.45] we obain (3.5) = 1 + Φ (A(R) + M(R) )Z R = N(R) + = N(R) + Z s 1 {R s} dβ j s, R +, Z s 1 {R s} da(r) s + M(R) c, Z R h, S R s (α(r s ) + Φσ j (r s )) Z s 1 {R s} ds, R +. Le ( k ) k N (, ) be a sequence wih k such ha we have (3.1). By (3.5), Lebesgue s dominaed convergence heorem and (3.4) we obain lim k a conradicion. + lim k = lim inf 1 E[ h, rt n k Z T k ] = lim k k 1 1 k E[ h, S T k h Z T k ] T k ] E[(A(T k ) T k + M(T k ) T k )Z T k k h, S h + h, α(h) + Φσ j (h) = η + Φ h, σ j (h) = η (η + 1) = 1, 4. Cones generaed by uncondiional Schauder bases In his secion, we provide he required background abou closed convex cones generaed by uncondiional Schauder bases. Le {e k } k N be an uncondiional Schauder basis of he Hilber space H; ha is, for each h H here is a unique sequence (h k ) k N R such ha (4.1) h = k N h k e k, and he series (4.1) converges uncondiionally. Wihou loss of generaliy, we assume ha e k = 1 for all k N. 4.1. Remark. Every orhonormal basis of he Hilber space H is an uncondiional Schauder basis. Of course, he converse saemen is no rue, bu for every uncondiional Schauder basis of he Hilber space H here is an equivalen inner produc

14 STEFAN TAPPE on H under which he uncondiional Schauder basis is an orhonormal basis; see [3]. There are unique elemens {e k } k N H such ha e k, h = h k for each h H, where we refer o he series represenaion (4.1); see [7, page 164]. Given hese coordinae funcionals {e k } k N, we also call {e k, e k} k N an uncondiional Schauder basis of H. Recall ha, hroughou his paper, we consider a closed convex cone K H wih represenaion (1.5) for some generaing sysem G K. Now, we make an addiional assumpion on he generaing sysem G of he cone. 4.2. Assumpion. We assume here is an uncondiional Schauder basis {e k, e k} k N of H such ha G {θe k : θ { 1, 1} and k N}. 4.3. Remark. Equivalenly, we could demand G k N e k. Assumpion 4.2 ensures ha he generaing sysem G becomes minimal. We define he sequence (E n ) n N of finie dimensional subspaces E n H as E n := e 1,..., e n. Furhermore, we define he sequence (Π n ) n N of projecions Π n L(H, E n ) as (4.2) n n Π n h = e k, h e k = h k e k, h H, k=1 where we refer o he series represenaion (4.1) of h. We denoe by bc({e l } l N ) := sup n N Π n he basis consan of he Schauder basis {e k } k N. Since he Schauder basis is uncondiional, by [7, Prop. 6.31] here is a consan C R + such for all m N, all λ 1,..., λ m R and all ɛ 1,..., ɛ m { 1, 1} we have m m (4.3) ɛ k λ k e k C λ k e k. k=1 The smalles possible consan C R + such ha he inequaliy (4.3) is fulfilled, is called he uncondiional basis consan, and is denoed by ubc({e l } l N ). 4.4. Lemma. The following saemens are rue: (1) We have 1 bc({e l } l N ) ubc({e l } l N ). (2) For each k N we have e k, 2bc({e l} l N ). (3) For all h H wih represenaion (4.1) and every bounded sequence (λ k ) k N we have k=1 k=1 g := k N λ k h k e k H wih norm esimae ( g ubc({e l } l N ) sup k N ) λ k h. Proof. The firs saemen follows he proof of [7, Prop. 6.31]. Noing ha e k = 1, by he Cauchy-Schwarz inequaliy, Assumpion 4.2 and he ideniy e k e k 2bc({e l } l N ) from [7, page 164], for each h H we obain e k, h e k h 2bc({e l } l N ) h. The hird saemen follows from [7, Lemma 6.33].

INVARIANCE OF CLOSED CONVEX CONES 15 4.5. Lemma. The following saemens are rue: (1) We have Π n Id H as n. (2) For all k, n N, all h e k and all h H we have h, Π n h = h, h 1 {k n}. Proof. The firs saemen follows from [7, Lemma 6.2.iii], and he second saemen follows from he definiion (4.2) of he projecion Π n. 5. Sufficiency of he invariance condiions for diffusion SPDEs wih smooh volailiies In his secion, we prove he sufficiency of our invariance condiions for diffusion SPDEs (1.2) wih smooh volailiies. Recall ha he disance funcion d K : H R + of he cone K is given by d K (h) := inf h g. g K 5.1. Lemma. The following saemens are rue: (1) For all λ and h H we have (5.1) d K (λh) = λd K (h). (2) For all h H and g K we have (5.2) d K (h + g) d K (h). Proof. Le h H be arbirary. For λ = boh sides in (5.1) are zero, and for λ >, by Definiion 2.4 we obain d K (λh) = inf λh g = inf λh λf = λ inf h f = λd K(h), g K f K f K proving he firs saemen. For he proof of he second saemen, le h H and g K be arbirary. Noe ha K K {g}. Indeed, for each f K by Definiion 2.5 we have f + g K, and hence f = (f + g) g K {g}. This gives us d K (h + g) = inf (h + g) f = inf h (f g) f K f K = inf h e inf h e = d K(h), e K {g} e K esablishing he second saemen. The following resul ensures ha he sochasic semigroup Nagumo s condiion (SSNC) is fulfilled in our siuaion. 5.2. Proposiion. Le Σ F(H) be such ha for all (h, h) D we have (5.3) lim inf Then, for each h K we have (5.4) lim inf h, S h + h, Σ(h). 1 d K(S h + Σ(h)) =. Proof. Since Σ F(H), here is an index n N such ha Σ(H) E n. Le h K be arbirary. We se N n := {1,..., n} and N 1 n := {k N n : (e k, h) D or ( e k, h) D}, N 2 n := {k N n : e k G or e k G } {k N n : (e k, h) / D and ( e k, h) / D}, N 3 n := {k N n : e k / G and e k / G }.

16 STEFAN TAPPE Then we have he decomposiion N n = N 1 n N 2 n N 3 n, for each k N 1 n here exiss θ k { 1, 1} such ha (θ k e k, h) D, and for each k N2 n here exiss θ k { 1, 1} such ha θ k e k G and (θ k e k, h) / D. Furhermore, we se θ k := 1 for each k N 3 n. There is a sequence ( m ) m N (, ) wih m such ha (5.5) where we agree on he noaion c m (k) for all m N and all k N 2 n. c m (k) := θ ke k, S m h + m Σ(h) m for all m N and all k N n. Inducively, we define he subsequences (m(k) p ) p N for k {} N 1 n as follows: (1) For k = we se m() p := p for each p N. (2) Le k N 1 n be arbirary, and suppose ha we have defined (m(l) p ) p N, where l denoes he larges ineger from {} N 1 n wih l < k. We disinguish wo cases: If lim inf p c m(l)p (k) =, hen we choose a subsequence (m(k) p ) p N of (m(l) p ) p N such ha c m(k)p (k) for all p N. Oherwise, we choose a subsequence (m(k) p ) p N of (m(l) p ) p N such ha c m(k)p (k) converges o a finie limi for p. Now, we define he subsequence (m p ) p N as m p := m(k) p for each p N, where k denoes he larges ineger from {} N 1 n. Furhermore, we define he ses N 1a n := N 1b n := Then we have he decomposiion N 1 n = N 1a n (5.6) (5.7) lim p c m p (k) R + c mp (k) { } k N 1 n : lim inf c m p (k) <, p { } k N 1 n : lim inf c m p p (k) =. N 1b n, and by (5.3) we have for all k N 1a n, for all p N and all k N 1b n. Since Σ(H) E n, and K is invarian for he semigroup (S ) and (Id Π n )- invarian, by Lemma 5.1 and (5.5), (5.7), for each p N we obain 1 d K (S mp h + mp Σ(h)) = 1 ( d K (Id Πn )S mp h +Π n (S mp h + mp Σ(h)) ) mp mp }{{} 1 ( d K Πn (S mp h + mp Σ(h)) ) ( ) S mp h + mp Σ(h) = d K Π n mp mp ( ) ( = d K c mp (k)θ k e k + c mp (k)θ k e k d K k N 1a n k N 1b n N2 n N3 n K }{{} K k N 1a n and by he coninuiy of he disance funcion d K and (5.6) we have ( ) ( ) lim d K c mp (k)θ k e k = d K lim c m p (k)θ k e k =, p p k N 1a n compleing he proof. k N 1a n }{{} K c mp (k)θ k e k ), 5.3. Theorem. Suppose ha Assumpions 2.1, 2.12 and 4.2 are fulfilled, and ha α Lip(H) F(H) B(H), σ F(H, L 2(H)) C 2 b (H, L 2(H)).

INVARIANCE OF CLOSED CONVEX CONES 17 If we have (5.8) lim inf h, S h + h, α(h) for all (h, h) D, and for all (h, h) D and each j N here exiss ɛ = ɛ(h, h, j) > such ha (5.9) h, σ j (h g) = for all g H wih g ɛ, hen he closed convex cone K is invarian for he SPDE (1.2). Proof. Condiion (5.9) jus means ha for each j N he funcion σ j : H H is weakly locally parallel in he sense of Definiion D.2, which allows us o apply Lemma D.7 in he sequel. Le ρ : H H be he funcion defined in (D.3). According o our hypoheses and Lemma D.6, all assumpions from [29] are saisfied. Le u U be arbirary, and define he funcion Σ : H H as Σ(h) := α(h) ρ(h) + σ(h)u, h H. Since α F(H) and σ F(H, L 2(H)), we have Σ F(H). Le (h, h) D be arbirary. Then, by (5.8) and Lemmas D.7, D.8 we deduce ha condiion (5.3) is fulfilled. Therefore, by Proposiion 5.2 he SSNC (5.4) is fulfilled. Consequenly, applying [29, Prop. 1.1] yields ha he closed convex cone K is invarian for he SPDE (1.2). 6. Sufficiency of he invariance condiions for diffusion SPDEs wih Lipschiz coefficiens In his secion, we prove ha our invariance condiions are sufficien for diffusion SPDEs (1.2) wih Lipschiz coefficiens, wihou imposing smoohness on he volailiy. 6.1. Theorem. Suppose ha Assumpions 2.1, 2.12 and 4.2 are fulfilled, and ha α Lip(H) and σ Lip(H, L 2(H)). If for all (h, h) D we have (5.8) and (1.9), hen he closed convex cone K is invarian for he SPDE (1.2). Proof. For he proof of his resul, we will apply he resuls from Appendices C and D. Noe ha Assumpion C.1 is fulfilled by virue of Lemma 2.15. Concerning he drif α, we use he approximaion resuls from Appendix C as follows: (1) Condiion (5.8) jus means ha (a, α) is inward poining in he sense of Definiion C.2. (2) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion C.8 we may assume ha α Lip(H) F(H). (3) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion C.11 we may assume ha α Lip(H) F(H) B(H). Furhermore, concerning he volailiy σ, we use he approximaion resuls from Appendix D as follows: (1) Condiion (1.9) jus means ha for each j N he volailiy σ j : H H is parallel in he sense of Definiion C.3. (2) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.11 we may assume ha σ Lip(H, L 2(H)) G(H, L 2(H)). This allows us o apply he remaining resuls from Appendix D (Proposiions D.13 D.37), which are all saed for volailiies of he form σ : H H.

18 STEFAN TAPPE (3) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.13 we may assume ha σ Lip(H, L 2(H)) F(H, L 2(H)). (4) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.15 we may assume ha σ Lip(H, L 2(H)) F(H, L 2(H)) B(H, L 2(H)). (5) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.18 we may assume ha for each j N he volailiy σ j : H H is locally parallel in he sense of Definiion D.1. (6) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.27 we may assume ha σ F(H, L 2(H)) C 1,1 b (H, L 2(H)), and ha σ j : H H is locally parallel for each j N. (7) By our sabiliy resul for SPDEs (Proposiion B.3) and Proposiion D.37 we may assume ha σ F(H, L 2(H)) C 2 b (H, L 2(H)), and ha for each j N he volailiy σ j : H H is weakly locally parallel in he sense of Definiion D.2. Consequenly, applying Theorem 5.3 complees he proof. 7. Sufficiency of he invariance condiions for SPDEs wih Lipschiz coefficiens In his secion, we prove ha our invariance condiions are sufficien for general jump-diffusion SPDEs (1.1) wih Lipschiz coefficiens. 7.1. Theorem. Suppose ha Assumpions 2.1, 2.12 and 4.2 are fulfilled, and ha α Lip(H), σ Lip(H, L 2(H)) and γ Lip(H, L 2 (F )). If we have (1.7), and for all (h, h) D we have (1.8) and (1.9), hen he closed convex cone K is invarian for he SPDE (1.1). Proof. Since he measure F is σ-finie, by our sabiliy resul (Proposiion B.3) i suffices o prove ha for each B E wih F (B) < he cone K is invarian for he SPDE dr = (Ar + α(r ) B γ(r, x)f (dx))d + σ(r )dw r = h. + B γ(r, x)µ(d, dx) Moreover, by he jump condiion (1.7) and [12, Lemmas 2.12 and 2.2], i suffices o prove ha he cone K is invarian for he SPDE { dr = (Ar + α B (r ))d + σ(r )dw (7.1) r = h. where α B : H H is given by α B (h) := α(h) B γ(h, x)f (dx), h H.

INVARIANCE OF CLOSED CONVEX CONES 19 Noe ha by he Cauchy-Schwarz inequaliy we have α B Lip(H). Le (h, h) D be arbirary. By (1.8) and Proposiion 2.18 we obain lim inf h, S h + h h, S h, α B (h) = lim inf + h, α(h) h, γ(h, x) F (dx) + h, γ(h, x) F (dx). E E\B Therefore, applying Theorem 6.1 yields ha he cone K is invarian for he SPDE (7.1), compleing he proof. 8. Sufficiency of he invariance condiions and proof of he main resul In his secion, we prove ha our invariance condiions are sufficien for jumpdiffusion SPDEs (1.2) wih coefficiens being locally Lipschiz and saisfying he linear growh condiion. 8.1. Theorem. Suppose ha Assumpions 2.1, 2.2, 2.12 and 4.2 are fulfilled. If we have (1.7), and for all (h, h) D we have (1.8) and (1.9), hen he closed convex cone K is invarian for he SPDE (1.1). Proof. Le h K be arbirary. Le (R n ) n N be he sequence of reracions R n : H H defined according o Definiion A.9. We define he sequences of funcions (α n ) n N, (σ n ) n N and (γ n ) n N as α n := α R n, σ n := σ R n and γ n := γ R n. Le n N be arbirary. Then, by Lemma A.1 we have α n Lip(H), σ n Lip(H, L 2(H)) and γ Lip(H, L 2 (F )), and hence, here exiss a unique mild soluion r n o he SPDE (B.1) wih r n = h. Now, we check ha condiions (1.7) (1.9) are fulfilled wih (α, σ, γ) replaced by (α n, σ n, γ n ). Following he noaion from Definiion A.9, here is a funcion λ n : H (, 1] such ha R n (h) = λ n (h)h for all h H. Le h K be arbirary. By he properies of he closed convex cone K we have λ n (h)h K and (1 λ n (h))h K, and hence, since condiion (1.7) is saisfied for γ, we obain h + γ n (h, x) = h + γ(λ n (h)h, x) = (1 λ n (h))h + λ n (h)h + γ(λ n (h)h, x) K }{{}}{{} K K for F -almos all x E, showing (1.7) wih γ replaced by γ n. Now, le h G be such ha (h, h) D. Then, by Lemma 2.15 we also have (h, λ n (h)h) D, and since condiion (1.9) is saisfied for σ, we obain h, σ j n(h) = h, σ j (λ n (h)h) =, j N,

2 STEFAN TAPPE showing (1.9) wih σ replaced by σ n. Furhermore, since condiion (1.8) is saisfied for (α, γ), we obain h, S h lim inf + h, α n (h) h, γ n (h, x) F (dx) E = lim inf h, S h + h, α(λ n (h)h) E h, γ(λ n (h)h, x) F (dx) h, S h h, S (λ n (h)h) (1 λ n (h)) lim inf + lim inf + h, α(λ n (h)h) h, γ(λ n (h)h, x) F (dx), E showing (1.8) wih (α, γ) replaced by (α n, γ n ). Consequenly, by Theorem 7.1 we have r n K up o an evanescen se. Now, we define he increasing sequence (T n ) n N of sopping imes by T := and T n := inf{ R + : r n > n} for all n N. Then we have P(T n ) = 1, and he mild soluion r o (1.1) wih r = h is given by (8.1) r = h 1 [T ] + n N r n 1 ]Tn 1,T n ], showing ha r K up o an evanescen se. Now, we are ready o provide he proof of our main resul, which concludes he paper. Proof of Theorem 1.1. (i) (ii): This implicaion follows from Theorem 3.1. (ii) (i): This implicaion follows from Theorem 8.1. Appendix A. Funcion spaces In his appendix, we collec he funcion spaces used in his paper. Le X and Y be wo normed spaces. A.1. Definiion. We inroduce he following noions: (1) For a consan L R + a funcion f : X Y is called L-Lipschiz if f(x) f(y) L x y for all x, y X. (2) For a consan L R + we define he space Lip L (X, Y ) := {f : X Y : f is L-Lipschiz}. (3) A funcion f Lip L (X, Y ) is called Lipschiz coninuous. (4) We define he space Lip(X, Y ) := L R + Lip L (X, Y ). (5) For a consan L R + we define he space Lip L (X) := Lip L (X, X). (6) We define he space Lip(X) := Lip(X, X). A.2. Definiion. We inroduce he following noions: (1) A funcion f : X Y is called locally Lipschiz if for each C R + here is a consan L(C) R + such ha f(x) f(y) L(C) x y for all x, y X wih x, y C. (2) We denoe by Lip loc (X, Y ) he space of all locally Lipschiz funcions f : X Y. (3) We define he space Lip loc (X) := Lip loc (X, X). A.3. Definiion. We inroduce he following noions:

INVARIANCE OF CLOSED CONVEX CONES 21 (1) We say ha a funcion f : X Y saisfies he linear growh condiion if here is a finie consan C R + such ha f(x) C(1 + x ) for all x X. (2) We denoe by LG(X, Y ) he space of all funcions f : X Y saisfying he linear growh condiion. (3) We define he space LG(X) := LG(X, Y ). Noe ha Lip(X, Y ) Lip loc (X, Y ) LG(X, Y ). A.4. Definiion. We inroduce he following noions: (1) A funcion f : X Y is called bounded if here is a consan M R + such ha f(x) M for all x X. (2) We denoe by B(X, Y ) he space of all bounded funcions f : X Y. (3) We define he space B(X) := B(X, X). A.5. Definiion. We inroduce he following noions: (1) A funcion f : X Y is called locally bounded if for each C R + here is a consan M(C) R + such ha f(x) M(C) for all x X wih x C. (2) We denoe by B loc (X, Y ) he space of all locally bounded funcions f : X Y. (3) We define he space B loc (X) := B loc (X, X). Noe ha LG(X, Y ) B loc (X, Y ). A.6. Definiion. We inroduce he following noions: (1) We denoe by C(X, Y ) he space of all coninuous funcions f : X Y. (2) We define he space C b (X, Y ) := C(X, Y ) B(X, Y ). (3) We define he spaces C(X) := C(X, X) and C b (X) := C b (X, X). Noe ha Lip loc (X, Y ) C(X, Y ). For he nex definiion, we agree abou he convenion N := N { }, where N = {1, 2, 3,...} denoes he naural numbers. A.7. Definiion. Le p N be arbirary. (1) We denoe by C p (X, Y ) he space of all p-imes coninuously differeniable funcions f : X Y. (2) We denoe by C p b (X, Y ) he space of all f Cp (X, Y ) such ha f is bounded and he derivaives D k f, k = 1,..., p are bounded. (3) We define he spaces C p (X) := C p (X, X) and C p b (X) := Cp b (X, X). Noe ha Cb 1 (X, Y ) Lip(X, Y ) B(X, Y ). A.8. Definiion. We inroduce he following noions: (1) We denoe by C 1,1 b (X, Y ) he space of all f Cb 1 (X, Y ) such ha Df Lip(X, L(X, Y )). (2) We define he space C 1,1 b (X) := C 1,1 b (X, X). Noe ha Cb 2 (X, Y ) C1,1 b (X, Y ) Cb 1 (X, Y ). A.9. Definiion. For each n N we define he reracion R n : X X, R n (x) := λ n (x)x, where he funcion λ n : X (, 1] is given by λ n (x) := 1 { x n} + n x 1 { x >n}, x X.

22 STEFAN TAPPE The following auxiliary resul is well-known. A.1. Lemma. The following saemens are rue: (1) We have R n Id X as n. (2) For each n N we have R n Lip 1 (X) B(X). Appendix B. Sabiliy resul for SPDEs In his appendix, we presen he required sabiliy resul for SPDEs. The mahemaical framework is ha of Secion 2. Apar from he SPDE (1.1), we consider he sequence of SPDEs given by (B.1) { dr n = (Ar n + α n (r n ))d + σ n (r n )dw + E γ n(r, n x)(µ(d, dx) F (dx)d) r n = h for each n N. B.1. Assumpion. We suppose ha he following condiions are fulfilled: (1) There exiss L R + such ha α n Lip L (H), σ n Lip L (H, L 2(H)) and γ n Lip L (H, L 2 (F )) for all n N. (2) We have α n α, σ n σ and γ n γ for n. B.2. Proposiion. Suppose ha Assumpion B.1 is fulfilled. Then, for each h H we have [ ] E sup r r n 2 for every T R +, [,T ] where r denoes he mild soluion o (1.1) wih r = h, and for each n N he process r n denoes he mild soluion o (B.1) wih r n = h. Proof. This is a consequence of [9, Prop. 9.1.2]. B.3. Proposiion. Suppose ha Assumpion B.1 is fulfilled, and ha for each n N he closed convex cone K is invarian for he SPDE (B.1). Then K is also invarian for he SPDE (1.1). Proof. Le h K be arbirary. We denoe by r he mild soluion o (1.1) wih r = h, and for each n N we denoe by r n he mild soluion o (B.1) wih r n = h. Then, for each n N here is an even Ω n F wih P( Ω n ) = 1 such ha r n (ω) K for all (ω, ) Ω n R +. Seing Ω := Ω n N n F we have P( Ω) = 1 and r n (ω) K for all (ω, ) Ω R + and all n N. Now, le N N be arbirary. By Proposiion B.2 we have [ ] E sup r r n 2, [,N] and hence, here is a subsequence (n k ) k N such ha P-almos surely sup r r n k. [,N] Since K is closed, here is an even Ω N F wih P( Ω N ) = 1 such ha r (ω) K for all (ω, ) Ω N [, N]. Therefore, seing Ω := Ω N N N F we obain P( Ω) = 1 and r (ω) K for all (ω, ) Ω R +, showing ha K is invarian for (1.1).

INVARIANCE OF CLOSED CONVEX CONES 23 Appendix C. Inward poining funcions In his appendix, we provide he required resuls abou inward poining funcions, which we need for he proof of Theorem 6.1. As in Secion 2, le H be a separable Hilber space, le K H be a closed convex cone, and le G K be a generaing sysem of he cone such ha Assumpion 4.2 is fulfilled. Le D G K be a subse, and le a : D R + be a funcion. C.1. Assumpion. We suppose ha for each (h, h) D he following condiions are fulfilled: (1) We have h, h =. (2) For all λ we have (h, λh) D and a(h, λh) = λa(h, h). (3) For all g K wih g K h we have (h, g) D and a(h, g) a(h, h). C.2. Definiion. Le α : H H be a funcion. We call he pair (a, α) inward poining a he boundary of K (in shor inward poining) if for all (h, h) D we have a(h, h) + h, α(h). C.3. Definiion. A funcion σ : H H is called parallel a he boundary of K (in shor parallel) if for all (h, h) D we have h, σ(h) =. C.4. Definiion. Le σ : H H be a funcion. Then he se D is called (Id H, σ)- invarian if (h, σ(h)) D for all (h, h) D. C.5. Remark. Le σ : H H be a funcion. If D is (Id H, σ)-invarian, hen σ is parallel. C.6. Lemma. Le α : H H be a funcion such ha (a, α) is inward poining. Then, for each n N he pair (a, Π n α) is inward poining, oo. Proof. Le (h, h) D be arbirary. By Assumpion 4.2 we have h e k for some k N. Thus, by Lemma 4.5, and since a is nonnegaive, we obain finishing he proof. a(h, h) + h, Π n (α(h)) = a(h, h) + h, α(h) 1 {k n}, C.7. Definiion. We inroduce he following spaces: (1) For each n N we denoe by F n (H) he space of all funcions α : H E n. (2) We se F(H) := n N F n(h). C.8. Proposiion. Le α Lip(H) be a funcion such ha (a, α) is inward poining. Then, here are a consan L R + and a sequence (C.1) (α n ) n N Lip L (H) F(H) such ha (a, α n ) is inward poining for each n N, and we have α n α. Proof. We se α n := Π n α for each n N. Then, by consrucion for each n N we have α n F(H). By hypohesis here exiss a consan M R + such ha α Lip M (H). Seing L := Mbc({e l } l N ), we have α n Lip L (H) for each n N, showing (C.1). Furhermore, by Lemma C.6, for each n N he pair (a, α n ) is inward poining, and by Lemma 4.5 we have α n α.

24 STEFAN TAPPE C.9. Lemma. Le α, β : H H be wo funcions such ha he following condiions are fulfilled: (1) (a, α) is inward poining. (2) D is (Id H, β)-invarian, and for all (h, h) D we have (C.2) a(h, β(h)) a(h, h). Then he pair (a, α β) is inward poining. Proof. Le (h, h) D be arbirary. Since he se D is (Id H, β)-invarian, we have (h, β(h)) D. Therefore, by (C.2), and since (a, α) is inward poining, we obain finishing he proof. a(h, h) + h, α(β(h)) a(h, β(h)) + h, α(β(h)), We denoe (R n ) n N he reracions R n : H H defined according o Definiion A.9. We will need he following auxiliary resul. C.1. Lemma. Le n N be arbirary. Then D is (Id H, R n )-invarian, and for all (h, h) D we have a(h, R n (h)) a(h, h). Proof. Le n N be arbirary. Recalling he noaion from Definiion A.9, here is a funcion λ n : H (, 1] such ha R n (h) = λ n (h)h for each h H. By Assumpion C.1 we obain (h, R n (h)) = (h, λ n (h)h) D and compleing he proof. a(h, R n (h)) = a(h, λ n (h)h) = λ n (h)a(h, h) a(h, h), C.11. Proposiion. Le α Lip(H) F(H) be a funcion such ha (a, α) is inward poining. Then here are a consan L R + and a sequence (C.3) (α n ) n N Lip L (H) F(H) B(H) such ha (a, α n ) is inward poining for each n N, and we have α n α. Proof. We se α n := α R n for each n N. Le n N be arbirary. Then we have α n F(H), because α F(H). By hypohesis here exiss a consan L R + such ha α Lip L (H), and by Lemma A.1 and he inclusion Lip L (H) B loc (H) i follows ha α n Lip L (H) B(H), showing (C.3). Combining Lemmas C.9 and C.1, we obain ha (a, α n ) is inward poining. Furhermore, by Lemma A.1 we have α n α. Appendix D. Parallel funcions In his appendix, we provide he required resuls abou parallel funcion, which we need for he proofs of Theorems 5.3 and 6.1. The general mahemaical framework is ha of Appendix C. Firs, we will exend he Definiion C.3 of a parallel funcion. D.1. Definiion. A funcion σ : H H is called locally parallel o he boundary of K (in shor locally parallel) if here exiss ɛ > such ha for all (h, h) D we have (D.1) h, σ(h g) = for all g H wih g ɛ. D.2. Definiion. A funcion σ : H H is called weakly locally parallel o he boundary of K (in shor weakly locally parallel) if for all (h, h) D here exiss ɛ = ɛ(h, h) > such ha we have (D.1).