STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS AND SUBMANIFOLDS IN HILBERT SPACES

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1 STOCHASTIC PARTIAL DIFFRNTIAL QUATIONS AND SUBMANIFOLDS IN HILBRT SPACS DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Abstract. The goal of this appendix is to provide results about stochastic partial differential equations driven by Wiener processes and Poisson measures and results about submanifolds in Hilbert spaces. It should serve as a reference for auxiliary results that we require in [7]. 1. Introduction In [7], we have provided necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds with boundary in Hilbert spaces for stochastic partial differential equations SPDs) of the form 1.1) { drt = Ar t + αr t ))dt + σr t )dw t + γr t, x)µdt, dx) F dx)dt) r = h driven by Wiener processes and Poisson random measures. The goal of this appendix is to serve as a reference for auxiliary results that we require for the proofs in [7]. In Section we provide results about SPDs driven by Wiener processes and Poisson random measures, and in Section 3 we provide results about finite dimensional submanifolds with boundary in Hilbert spaces.. SPDs driven by Wiener processes and Poisson random measures In this section, we provide results about SPDs driven by Wiener processes and Poisson random measures. References on this topic are, e.g., [, 13, 6]. Furthermore, we mention [14] regarding SPDs driven by Lévy processes, and [3, 15, 8] regarding SPDs driven by Wiener processes. In the sequel, Ω, F, F t ) t, P) denotes a filtered probability space satisfying the usual conditions. Let H be a separable Hilbert space and let S t ) t be a C - semigroup on H with infinitesimal generator A : DA) H H. We denote by A : DA ) H H the adjoint operator of A. Recall that the domains DA) and DA ) are dense in H, see, e.g., [16, Theorems c and 13.1]. Let H be another separable Hilbert space and let Q LH) be a nuclear, selfadjoint, positive definite linear operator. Then, there exist an orthonormal basis e j ) of H and a sequence λ j ), ) with λ j < such that Qu = λ j u, e j H e j, u H, Date: June Mathematics Subject Classification. 6H15, 6G17. Key words and phrases. Stochastic partial differential equation, jump-diffusion, submanifold with boundary, tangent space. 1

2 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN namely, the λ j are the eigenvalues of Q, and each e j is an eigenvector corresponding to λ j. The space H := Q 1/ H), equipped with the inner product u, v H := Q 1/ u, Q 1/ v H, is another separable Hilbert space and λ j e j ) is an orthonormal basis. Let W be a H-valued Q-Wiener process, see [3, pages 86, 87]. We denote by L H) := L H, H) the space of Hilbert-Schmidt operators from H into H, which, endowed with the Hilbert-Schmidt norm ) 1/ Φ L H) := λ j Φe j, Φ L H) itself is a separable Hilbert space. According to [3, Proposition 4.1], the sequence of stochastic processes β j ) defined as β j := 1 λj W, e j, j N is a sequence of real-valued independent standard Wiener processes and we have the expansion Note that L H) = l H), because.1) W = λj β j e j. L H) l H), Φ Φ j ) with Φ j := λ j Φe j, j N is an isometric isomorphism. According to [3, Theorem 4.3], for every predictable process Φ : Ω R + L H) satisfying t ) P Φ s L H)ds < = 1 for all t we have the identity.) t Φ s dw s = t Φ j sdβ j s, t. Let, ) be a measurable space which we assume to be a Blackwell space see [4, 9]). We remark that every Polish space with its Borel σ-field is a Blackwell space. Furthermore, let µ be a time-homogeneous Poisson random measure on R +, see [1, Definition II.1.]. Then its compensator is of the form dt F dx), where F is a σ-finite measure on, ). We shall now focus on SPDs of the type 1.1). Let α : H H, σ : H L H) and γ : H H be measurable mappings..1. Definition. Let h : Ω H be a F -measurable random variable. Furthermore, let r = r h) be a H-valued càdlàg adapted process and let τ > a stopping time such that for all t we have t τ ) ) P r s + αr s ) + σr s ) LH) + γr s, x) F dx) ds < = 1. Then the process r is called a local strong solution to 1.1), if.3).4) r t τ DA) for almost all t R +, P-almost surely, t τ ) P Ar s ds < = 1 for all t

3 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 3.5) and we have P-almost surely r t τ = h + + t τ t τ Ars + αr s ) ) t τ ds + σr s )dw s γr s, x)µds, dx) F dx)ds), t. a local weak solution to 1.1), if for all ζ DA ) we have P-almost surely t τ ζ, r t τ = ζ, h + A ζ, r s + ζ, αr s ) ) t τ ds + ζ, σr s ) dw s t τ + ζ, γr s, x) µds, dx) F dx)ds), t. a local mild solution to 1.1), if we have P-almost surely t τ t τ r t τ = S t τ h + S t τ) s αr s )ds + S t τ) s σr s )dw s t τ + S t τ) s γr s, x)µds, dx) F dx)ds), t. We call τ the lifetime of r. If τ =, then we call r a strong, weak or mild solution to 1.1), respectively... Remark. Since the process r is càdlàg, we have r t = r t for almost all t R +, P-almost surely, and hence, relation.3) implies r t τ) DA) for almost all t R +, P-almost surely. According to [5, Lemma.4.], the process f defined by { Ar t, if r t DA).6) f t :=, otherwise is predictable. By slight abuse of notation, we have written Ar instead of f in.4) and.5)..3. Remark. The following results are well-known: very local) strong solution to 1.1) is also a local) weak solution to 1.1). very local) weak solution to 1.1) is also a local) mild solution to 1.1). If A is bounded, i.e. generates a norm-continuous semigroup S t ) t, then the concepts of local) strong, weak and mild solutions to 1.1) are equivalent..4. Definition. The semigroup S t ) t is called pseudo-contractive, if for some constant ω R. S t e ωt, t.5. Definition. The mappings α, σ, γ) are called locally) Lipschitz continuous, if: α : H H is locally) Lipschitz continuous. σ : H L H) is locally) Lipschitz continuous. For each h H we have γh, ) L F ) and γ : H L F ) is locally) Lipschitz continuous, where we use the notation L F ) := L,, F ; H)..6. Remark. The following results are well-known:

4 4 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN If S t ) t is pseudo-contractive and α, σ, γ) are Lipschitz continuous, then we have existence and uniqueness of mild and weak solutions to 1.1). If S t ) t is pseudo-contractive and α, σ, γ) are locally Lipschitz continuous, then we have existence and uniqueness of local mild and weak solutions to 1.1). If α, σ, γ) are locally Lipschitz continuous, then we have uniqueness of local mild solutions to 1.1)..7. Lemma. Let M DA) be a subset such that A is continuous on M, and let r = r h) be a local weak solution to 1.1) with lifetime τ > for some F - measurable random variable h : Ω H such that r τ ) M up to an evanescent set. Then r is also a local strong solution to 1.1) with lifetime τ. Proof. Condition.3) is satisfied, because r τ ) M DA) up to an evanescent set, and condition.4) is satisfied due to the continuity of A on M. Taking into account Remark., we obtain for each ζ DA ) that P-almost surely ζ, r t τ = ζ, h + = t τ + ζ, h + t τ t τ A ζ, r s + ζ, αr s ) )ds + t τ ζ, γr s, x) µds, dx) F dx)ds) t τ Ar s + αr s ))ds + σr s )dw s t τ + γr s, x)µds, dx) F dx)ds) Since DA ) is dense in H, we get P-almost surely r t τ = h + + t τ t τ ζ, σr s ) dw s, t. Ars + αr s ) ) t τ ds + σr s )dw s γr s, x)µds, dx) F dx)ds), t, showing that r is a local strong solution to 1.1) with lifetime τ..8. Remark. According to [1, Proposition II.1.14], there exist a sequence τ n ) n N of finite stopping times with [τ n ] [τ m ] = for n m and an -valued optional process ξ such that for every optional process δ : Ω R + H with t ).7) P δs, x) µds, dx) < = 1 for all t we have.8) t δs, x)µds, dx) = n N δτ n, ξ τn )1 {τn t}, t. Furthermore, for every predictable process δ : Ω R + H with t ) P δs, x) F dx)ds < = 1 for all t the jumps of the integral process t X t := δs, x)µds, dx) F dx)ds), t are given by.9) X t = δt, ξ t ) n N 1 {τn=t}, t,

5 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 5 see [1, Section II.1.d]..9. Lemma. Let r = r h) be a local weak solution to 1.1) with lifetime τ > for some F -measurable random variable h : Ω H. Then the following statements are true: 1) The jumps of the stopped process r τ are given by r t τ = γr t τ), ξ t τ ) n N 1 {τn=t τ}, t. ) For each n N we have r τn 1 {τn τ} = γr τn, ξ τn )1 {τn τ}. Proof. Let X be the process t τ X t := ζ, γr s, x) µds, dx) F dx)ds), t. Since r is a local weak solution to 1.1), for every ζ DA ) we have, by using.9), ζ, r t τ = ζ, r t τ = X t τ = ζ, γr t τ), ξ t τ ) n N 1 {τn=t τ} = ζ, γr t τ), ξ t τ ) 1 {τn=t τ}, t. n N Taking into account that DA ) is dense in H, the first statement follows. Since [τ n ] [τ m ] = for n m, we deduce that r τn 1 {τn τ} = r τn τ 1 {τn τ} = γr τn τ), ξ τn τ ) ) 1 {τm=τ n τ} 1 {τn τ} m N = γr τn, ξ τn ) ) 1 {τm=τ n} 1 {τn τ} = γr τn, ξ τn )1 {τn τ} m N for each n N, establishing the second statement. From now on, we fix mappings α : H H, σ j : H H, j N, γ : H H satisfying the following regularity conditions: The mapping α : H H is locally Lipschitz continuous, that is, for each n N there is a constant L n such that.1).11).1) αh 1 ) αh ) L n h 1 h, h 1, h H with h 1, h n. For each n N there exists a sequence κ j n) R + with κj n) < such that for all j N the mapping σ j : H H satisfies σ j h 1 ) σ j h ) κ j n h 1 h, h 1, h H with h 1, h n, σ j h) κ j n, h H with h n. Consequently, for each j N the mapping σ j is locally Lipschitz continuous. The mapping γ : H H is measurable, and for each n N there exists a measurable function ρ n : R + with.13) ρ n x) F dx) <..14).15) such that for all x the mapping γ, x) : H H satisfies γh 1, x) γh, x) ρ n x) h 1 h, h 1, h H with h 1, h n, γh, x) ρ n x), h H with h n.

6 6 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Consequently, for each x the mapping γ, x) is locally Lipschitz continuous. We assume that for each j N the mapping σ j : H H is continuously differentiable, that is.16) σ j C 1 H) for all j N. Using the identification L H) = l H), which holds true by the isometric isomorphism defined in.1), we can identify the sequence σ j ) of mappings σ j : H H with a locally Lipschitz continuous mapping σ : H L H), and, in view of.), equation 1.1) can be rewritten equivalently.17) dr t = Ar t + αr t ))dt + σj r t )dβ j t + γr t, x)µdt, dx) F dx)dt) r = h..1. Definition. Let B 1 B H be two nonempty Borel sets. B 1 is called prelocally invariant in B for.17), if for all h B 1 there exists a local mild solution r = r h) to.17) with lifetime τ > such that r τ ) B 1 and r τ B up to an evanescent set..11. Lemma. Let B 1 B H be two Borel sets such that B 1 is prelocally invariant in B for 1.1). Then we have Proof. We denote by h + γh, x) B for F -almost all x, for all h B 1. the distance function of the set B. Since d B : H R +, d B h) := inf g B h g d B h 1 ) d B h ) h 1 h for all h 1, h H, by the linear growth condition.15), for all n N, all h B with h n and all x we have.18) d B h + γh, x)) = d B h + γh, x)) d B h) γh, x) ρ n x). Thus, by.14) and Lebesgue s dominated convergence theorem, the mapping.19) B R, h d B h + γh, x)) F dx) is continuous. Now, let h B 1 be arbitrary. Since B 1 is prelocally invariant in B for 1.1), there exists a local mild solution r = r h) to 1.1) with lifetime τ > such that r τ ) B 1 and r τ B up to an evanescent set. Taking into account [1, Theorem II.1.8], identity.8) and Lemma.9, we obtain [ τ ] d B r s + γr s, x)) F dx)ds [ τ ] = d B r s + γr s, x)) µds, dx) [ ] = d B r τn + γr τn, ξ τn )) 1 {τn τ} n N [ ] = d B r τn + r τn ) 1 {τn τ} n N [ = n N ] d B r τn ) 1 {τn τ} =.

7 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 7 Therefore, we have P-almost surely τ ).) d B r s + γr s, x)) F dx) ds =, t. Since the process r is càdlàg with r τ ) B 1 up to an evanescent set and the mapping.19) is continuous, the integrand appearing in.) is continuous in s =. Thus, we deduce that d B h + γh, x)) F dx) =. This provides and hence completing the proof. d B h + γh, x)) = for F -almost all x, h + γh, x) B for F -almost all x,.1. Lemma. Let B 1 B H be two Borel sets such that.1) h + γh, x) B for F -almost all x, for all h B 1. Let h : Ω H be a F -measurable random variable and let r = r h) be a local mild solution to 1.1) with lifetime τ > such that r τ ) B 1 and r τ 1 [,τ [ B up to an evanescent set. Then we have r τ B up to an evanescent set. Proof. Since r τ 1 [,τ [ B up to an evanescent set, it suffices to prove that.) Pr τ 1 {τ< } B ) = 1. By.8), [1, Theorem II.1.8] and.1) we obtain [ ] [ ] = 1 {rs +γr s,x) / B }µds, dx) n N [ = which yields 1 {rτn +γr τn,ξ τn ) / B } ] 1 {rs +γr s,x) / B }F dx)ds =, Pr τn + γr τn, ξ τn ) B for all n N) = 1. Therefore, by Lemma.9 we obtain P-almost surely r τ 1 {τ< } = r τ + r τ )1 {τ< } = r τ + γr τ, ξ τ ) ) 1 {τn=τ} 1 {τ< } B, n N proving.)..13. Lemma. Let B C H be two Borel sets such that C is closed in H and h + γh, x) C for F -almost all x, for all h B. Let h : Ω H be a F -measurable random variable and let r = r h) be a local mild solution to 1.1) with lifetime τ > such that r τ ) B up to an evanescent set. Then we have r τ C up to an evanescent set. Proof. By the closedness of C in H, we have r τ 1 [,τ [ C up to an evanescent set. Thus, the statement follows from Lemma.1.

8 8 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN.14. Lemma. Let G 1, G be metric spaces such that G 1 is separable. Let B G 1 be a Borel set, let C G be a closed set and let δ : G 1 G be a measurable mapping such that δ, x) : G 1 G is continuous for all x. Suppose that.3) Then we even have.4) δh, x) C for F -almost all x, for all h B. δh, x) C for all h B, for F -almost all x. Proof. By separability of G 1 there exists a countable set D, which is dense in B. By.3), for each h D there exists a F -nullset N h such that δh, x) C for all x N c h. The set N := h D N h is also a F -nullset. Now, let h B be arbitrary. Then there exists a sequence h n ) n N D with h n h, and we obtain δh n, x) C for all n N and x N c. Since δ, x) is continuous for all x and the set C is closed in G, we deduce providing.4). δh, x) = lim n δh n, x) C for all x N c, Recall that a closed, convex cone C is a nonempty, closed subset C H such that h + g C for all h, g C and λh C for all λ and h C..15. Lemma. Let G, G, ν) be a σ-finite measure space, let C H be a closed, convex cone and let f L 1 G; H) be such that fx) C for ν-almost all x G. Then we have fdν C. G Proof. First, we assume that f L 1 G; H) is a simple function of the form m.5) f = c k 1 Ak k=1 with c k C and A k G satisfying νa k ) < for k = 1,..., m. Then we have m fdν = c k νa k ) C. G k=1 Now, let f L 1 G; H) be an arbitrary function such that fx) C for ν-almost all x G. Arguing as in the proof of [3, Lemma 1.1], there exists a a sequence f n ) n N of simple functions of the form.5) such that f n f in L 1 G; H). Therefore, we get fdν = lim f n dν C, G n G finishing the proof..16. Lemma. Let C H be a closed, convex cone and let δ : Ω R + H be an optional process satisfying.7) such that δ, x) C up to an evanescent set, for F -almost all x. Then we have X C up to an evanescent set, where X denotes the integral process t X t := δs, x)µds, dx), t.

9 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 9 Proof. By assumption, there is a F -nullset N such that δ, x) C up to an evanescent set, for all x N c. Using identity.8) and [1, Theorem II.1.8] we obtain [ ] [ ] = 1 {x N} µds, dx) n N which gives us Pξ τn 1 {ξτn N} [ = ] 1 {x N} F dx)ds =, / N for all n N) = 1. Using.8) we obtain P-almost surely X t = n N δτ n, ξ τn )1 {τn t} C for all t, finishing the proof..17. Lemma. The following statements are true: 1) For each h H we have.6) Dσ j h)σ j h) <. ) The mapping.7) H H, h Dσ j h)σ j h) is continuous. Proof. Let j N be arbitrary. Furthermore, let h H be arbitrary. There exists n N such that h n. By estimates.11),.1) we have.8) Dσ j h)σ j h) Dσ j h) σ j h) κ j n). Since κj n) <, we have.6), showing that the first statement holds true. For each j N the mapping H H, Dσ j h)σ j h) is continuous, because for all h 1, h H we have Dσ j h 1 )σ j h 1 ) Dσ j h )σ j h ) Dσ j h 1 )σ j h 1 ) Dσ j h 1 )σ j h ) + Dσ j h 1 )σ j h ) Dσ j h )σ j h ) Dσ j h 1 ) σ j h 1 ) σ j h ) + Dσ j h 1 ) Dσ j h ) σ j h ). Denoting by ν the counting measure on N, PN)) given by ν{j}) = 1 for all j N, we can express the mapping.7) as Dσ j h)σ j h) = Dσ j h)σ j h)νdj). Taking into account estimate.8), Lebesgue s dominated convergence theorem yields the continuity of the mapping.7)..18. Lemma. Let B be a set with F B c ) <. 1) For each h H we have.9) γh, x) F dx) <. B c N

10 1 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN ) The mappings α B : H H and γ B : H H defined as.3) α B h) := αh) γh, x)f dx), B c.31) γ B h, x) := γh, x)1 B x) also satisfy the regularity conditions.1),.14),.15). Proof. Let h H be arbitrary. There exists n N with h n. By the Cauchy- Schwarz inequality and.15),.13) we have ) 1/ γh, x) F dx) F B c ) 1/ γh, x) F dx) B c 1/ F B c ) 1/ ρ n x) F dx)) <, showing.9). Now, let n N and h 1, h H with h 1, h n be arbitrary. By the Cauchy-Schwarz inequality and.14) we obtain γh 1, x)f dx) γh, x)f dx) γh 1, x) γh, x) F dx) B c B B c c ) 1/ F B c ) 1/ γh 1, x) γh, x) F dx) F B c ) 1/ ρ n x) F dx)) 1/ h 1 h, which, in view of.13), proves that α B also satisfies.1). Furthermore, the mapping γ B also satisfies.14),.15), which directly follows from its Definition.31)..19. Lemma. For every set B with F B c ) < the process N t := µ[, t] B c ), t is a càdlàg, adapted process with N =, N N and N {, 1} up to an evanescent set, and we have the representation.3) N t = n N 1 {ξτn / B}1 {τn t}, t. Proof. We have N =, because µω; {} ) = for all ω Ω by the definition of a random measure, see [1, Definition II.1.3]. By.8) we have t N t = µ[, t] B c ) = 1 {x / B} µds, dx) = n N 1 {ξτn / B}1 {τn t}, t which provides the representation.3) and shows that N N. Since [N t ] = [µ[, t] B c )] = tf B c ) < for all t, we deduce that PN t < ) = 1 for all t. Therefore, the representation.3) shows that the process N is càdlàg, adapted with N N up to an evanescent set. Since µω; {t} ) 1 for all ω, t) Ω R + by the definition of an integer-valued random measure, see [1, Definition II.1.13], we obtain N {, 1}.

11 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 11 For any set B we define the mapping ϱ B : Ω R + as ϱ B := inf{t : µ[, t] B c ) = 1}. For the representation.34) below we recall that for any stopping time τ and any set A F τ the mapping τ A : Ω R + given by.33) is also a stopping time. τ A ω) := { τω), ω A, ω / A.. Lemma. For every set B with F B c ) < the mapping ϱ B is a strictly positive stopping time and we have the representation.34) ϱ B {ξτn / B} = min τ n. n N Proof. This is a direct consequence of Lemma.19. We shall now consider the SPD drt B = Art B + α B rt B ))dt + σrt B )dw t +.35) γb rt, B x)µdt, dx) F dx)dt) r B = h, where B is a set with F B c ) <, and the mappings α B γ B : H H are given by.3),.31). : H H and.1. Proposition. Let h : Ω H be a F -measurable random variable, let B be a set with F B c ) <, and let < τ ϱ B be a stopping time. Then the following statements are true: 1) If there exists a local mild solution r to 1.1) with lifetime τ, then there also exists a local mild solution r B to.35) with lifetime τ such that.36) r τ 1 [,τ [ = r B ) τ 1 [,τ [. ) If there exists a local mild solution r B to.35) with lifetime τ, then there also exists a local mild solution r to 1.1) with lifetime τ such that.36) is satisfied. In particular, in either case we have r τ ) = r B ) τ ). Proof. Let r be a local mild solution to 1.1) with lifetime τ. We define the process r B by r B := r γr ϱb, ξ ϱ B)1 [ϱ B, [. Then r B is càdlàg and adapted, because γr B ϱ B, ξ ϱ B) is F ϱb-measurable, and, since τ ϱ B, we have r B ) τ = r τ γr ϱ B, ξ ϱ B)1 {τ=ϱ B }1 [τ, [ = r τ γr τ, ξ τ )1 {τ=ϱ B }1 [τ, [.

12 1 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Therefore, we have.36), and hence r τ ) = r B ) τ ). Since r is a local mild solution to 1.1) with lifetime τ, we have P-almost surely t τ t τ rt τ B = S t τ h + S t τ) s αr s )ds + S t τ) s σr s )dw s t τ + S t τ) s γr s, x)µds, dx) F dx)ds) γr τ, ξ τ )1 {τ=ϱb }1 {τ t} t τ t τ = S t τ h + S t τ) s αrs B )ds + S t τ) s σrs B )dw s t τ + S t τ) s γrs, B x)µds, dx) F dx)ds) γrτ, B ξ τ )1 {τ=ϱb }1 {τ t}, t. Hence, by the Definitions.3),.31) of α B, γ B we get P-almost surely t τ t τ rt τ B = S t τ h + S t τ) s α B rs B )ds + S t τ) s σrs B )dw s t τ + S t τ) s γ B rs, B x)µds, dx) F dx)ds) t τ γrτ, B ξ τ )1 {τ=ϱb }1 {τ t} + S t τ) s γrs, B x)f dx)ds B c t τ + S t τ) s γrs, B x)µds, dx) F dx)ds), t. B c By.8) and the representation.34) from Lemma. we have P-almost surely.37) t τ S t τ) s γrs, B x)µds, dx) B c = n N S t τ) τn γr B τ n, ξ τn )1 {ξτn / B}1 {τn t τ} = S t τ) ϱ Bγr B ϱ B, ξ ϱ B)1 {ϱ B t τ} = S t τ) ϱ Bγr B ϱ B, ξ ϱ B)1 {τ=ϱ B }1 {τ t} = γr B τ, ξ τ )1 {τ=ϱ B }1 {τ t}, t. Therefore, we obtain P-almost surely t τ t τ rt τ B = S t τ h + S t τ) s α B rs B )ds + S t τ) s σrs B )dw s t τ + S t τ) s γ B rs, B x)µds, dx) F dx)ds), t showing that r B is a local mild solution to.35) with lifetime τ. This proves the first statement. Now, let r B be a local mild solution to.35) with lifetime τ. We define the process r by r := r B + γr B ϱ B, ξ ϱ B)1 [ϱ B, [. Then r is càdlàg and adapted, because γr B ϱ B, ξ ϱ B) is F ϱb-measurable, and, since τ ϱ B, we have r τ = r B ) τ + γr B ϱ B, ξ ϱ B)1 {τ=ϱ B }1 [τ, [ = r B ) τ + γr B τ, ξ τ )1 {τ=ϱb }1 [τ, [.

13 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 13 Therefore, we have.36), and hence r τ ) = r B ) τ ). Since r B is a local mild solution to.35) with lifetime τ, we have P-almost surely t τ r t τ = S t τ h + t τ S t τ) s α B rs B )ds + S t τ) s σrs B )dw s t τ + S t τ) s γ B rs, B x)µds, dx) F dx)ds) + γrτ, B ξ τ )1 {τ=ϱ }1 B {τ t} t τ t τ = S t τ h + S t τ) s α B r s )ds + S t τ) s σr s )dw s t τ + S t τ) s γ B r s, x)µds, dx) F dx)ds) + γr τ, ξ τ )1 {τ=ϱb }1 {τ t}, t. Hence, by the Definitions.3),.31) of α B, γ B we get P-almost surely t τ r t τ = S t τ h + t τ S t τ) s αr s )ds + S t τ) s σr s )dw s t τ + S t τ) s γr s, x)µds, dx) F dx)ds) B + γr τ, ξ τ )1 {τ=ϱb }1 {τ t} t τ Arguing as in.37), we have P-almost surely t τ B c S t τ) s γr s, x)f dx)ds, t. B c S t τ) s γr s, x)µds, dx) = γr τ, ξ τ )1 {τ=ϱ B }1 {τ t}, t. Therefore, we obtain P-almost surely t τ r t τ = S t τ h + t τ S t τ) s αr s )ds + S t τ) s σr s )dw s t τ + S t τ) s γr s, x)µds, dx) F dx)ds), t showing that r is a local mild solution to 1.1) with lifetime τ. This proves the second statement. Now, let G be another separable Hilbert space. For any k N we denote by C k b G; H) the linear space consisting of all f Ck G; H) such that D i f is bounded for all i = 1,..., k. In particular, for each f C k b G; H) the mappings Di f, i =,..., k 1 are Lipschitz continuous. We do not demand that f itself is bounded, as this would exclude continuous linear operators f LG; H)... Definition. Let α : H H, σ j : H H, j N and γ : H H be mappings satisfying.38) σ j h) <, h H,.39) γh, x) F dx) <, h H,

14 14 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN and let f : G H and g Cb H; G) be mappings. We define the mappings f, g) λ α : G G, f, g) W σj : G G, j N and f, g) µγ : G G as.4).41).4) where h = fz). f, g) λα)z) := Dgh)αh) + 1 D gh)σ j h), σ j h)) ) + gh + γh, x)) gh) Dgh)γh, x) F dx), f, g) W σ j )z) := Dgh)σ j h), f, g) µγ)z, x) := gh + γh, x)) gh),.3. Remark. Note that the mapping f, g) λα is well-defined. Indeed, for any h H, by.38) we have σ j h) <, D gh)σ j h), σ j h)) D gh) and by.39) and Taylor s theorem we have gh + γh, x)) gh) Dgh)γh, x) F dx) 1 D g γh, x) F dx) <..4. Lemma. Let α : H H, σ j : H H, j N and γ : H H be mappings satisfying the regularity conditions.1).1) and.14).16) such that the mappings ρ n : R +, n N appearing in.14),.15) even satisfy.43) ρn x) ρ n x) 4) F dx) <. Furthermore, let f Cb 1G; H) and g C3 b H; G) be arbitrary. Then the following statements are true: 1) The mappings f, g) λ α, f, g) W σj ) and f, g) µγ also fulfill the regularity conditions.1).1) and.14).16) with the mappings ρ n : R +, n N appearing in.14),.15) satisfying.13). ) If g LH; G), then the mappings ρ n : R +, n N appearing in.14),.15) even satisfy.43). Proof. We define the mappings â : H G, ˆb j : H G, j N and ĉ : H G as âh) := â 1 h) + â h) + â 3 h), ˆbj h) := Dgh)σ j h), where â 1, â, â 3 : H G are given by ĉh, x) := gh + γh, x)) gh), â 1 h) := Dgh)αh), â h) := 1 D gh)σ j h), σ j h)), ) â 3 h) := gh + γh, x)) gh) Dgh)γh, x) F dx).

15 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 15 Then we have ˆb j C 1 H; G) for all j N. By Taylor s theorem, we have the representations 1.44) â 3 h) = 1 t)d gh + γh, x))γh, x), γh, x))dtf dx), h H.45) ĉh, x) = 1 Dgh + tγh, x))γh, x)dt, h, x) H. Let n N be arbitrary. Furthermore, let h H with h n be arbitrary. By.1), for all j N we have ˆb j h) Dg σ j h) Dg κ j n, and by.15) and the representation.45), for all x we have ĉh, x) 1 Dg γh, x) dt Dg ρ n x). Now, let h 1, h H with h 1, h n be arbitrary. Using estimate.1), we obtain â 1 h 1 ) â 1 h ) = Dgh 1 )αh 1 ) Dgh )αh ) Dgh 1 )αh 1 ) Dgh )αh 1 ) + Dgh )αh 1 ) Dgh )αh ) D g L n n + α) ) + Dg L n ) h1 h. Moreover, we have â h 1 ) â h ) 1 D gh 1 )σ j h 1 ), σ j h 1 )) D gh )σ j h ), σ j h )) 1 D gh 1 ) σ j h 1 ) σ j h 1 ) σ j h ) + 1 D gh 1 ) D gh ) σ j h 1 ) σ j h ) + 1 D gh ) σ j h 1 ) σ j h ) σ j h ). By estimates.11),.1) we obtain â h 1 ) â h ) D g + 1 ) D3 g κ j n) ) h 1 h. Furthermore, we have â 3 h 1 ) â 3 h ) 1 D gh 1 + tγh 1, x))γh 1, x), γh 1, x)) D gh + tγh, x))γh, x), γh, x)) dtf dx) 1 D gh 1 + tγh 1, x)) γh 1, x) γh 1, x) γh, x) dtf dx) D gh 1 + tγh 1, x)) D gh + tγh, x)) γh 1, x) γh, x) dtf dx) D gh + tγh, x)) γh 1, x) γh, x) γh, x) dtf dx).

16 16 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Noting that, by.14), for all x, t) [, 1] we have.46) D gh 1 + tγh 1, x)) D gh + tγh, x)) D 3 g h 1 + tγh 1, x) h tγh, x) D 3 g h 1 h + γh 1, x) γh, x) ) D 3 g 1 + ρ n x)) h 1 h, using estimates.14),.15) we get â 3 h 1 ) â 3 h ) D g ρ n x) F dx) + D 3 g By estimates.11),.1), for all j N we obtain ˆb j h 1 ) ˆb j h ) = Dgh 1 )σ j h 1 ) Dgh )σ j h ) ρn x) + ρ n x) 3) ) F dx) h 1 h. Dgh 1 ) Dgh ) σ j h 1 ) + Dgh ) σ j h 1 ) σ j h ) D g + Dg ) κ j n h 1 h. For all x we obtain ĉh 1, x) ĉh, x) Dgh 1 + tγh 1, x))γh 1, x) Dgh + tγh, x))γh, x) dt Dgh 1 + tγh 1, x)) Dgh + tγh, x)) γh 1, x) dt Dgh + tγh, x)) γh 1, x) γh, x) dt. Arguing as in.46), for all x, t) [, 1] we have Dgh 1 + tγh 1, x)) Dgh + tγh, x)) D g 1 + ρ n x)) h 1 h. Using estimates.14),.15), we obtain.47) ĉh 1, x) ĉh, x) D g ρ n x) + ρ n x) ) + Dg ρ n x) ) h 1 h. Since f, g) λ α = â f, f, g) W σj = ˆb j f, j N and f, g) µγ), x) = ĉ, x) f, x as well as f Cb 1 G; H), we deduce that conditions.1).1) and.14).16) are satisfied with the mappings ρ n : R +, n N appearing in.14),.15) satisfying.13), which proves the first statement. If g LH; G), then we have D g, and hence, estimate.47) shows that the mappings ρ n : R +, n N appearing in.14),.15) even satisfy.43), establishing the second statement. The following result is a version of Itô s formula for jump-diffusions in infinite dimension..5. Proposition. Let α : Ω R + G, σ : Ω R + L G) and γ : Ω R + G be predictable processes such that for all t we have t ) ) P α s + σ s LG) + γs, x) F dx) ds < = 1. Furthermore, let Y : Ω G be a F -measurable random variable, let Y be the G-valued Itô process t Y t = Y + α s ds + t t σsdβ j s j + γs, x)µds, dx) F dx)ds), t

17 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 17 and let φ Cb G; H) be arbitrary. Then we have P-almost surely t φy t ) = φy ) + DφY s )α s + 1 D φy s )σ s, j σs) j + φys + γs, x)) φy s ) DφY s )γs, x) ) ) F dx) ds t + t + DφY s )σ j sdβ j s φys + γs, x)) φy s ) ) µds, dx) F dx)ds), t where σ j := λ j σe j for each j N. Proof. For the following particular cases, this version of Itô s formula is known: For γ it follows by applying [8, Theorem.9] to the function h, φy ) for each h H. For σ it follows from [1, Theorem 3.6]. The general result follows by executing the proofs of the above-mentioned results simultaneously. 3. Finite dimensional submanifolds with boundary in Hilbert spaces In this section, we provide results about finite dimensional submanifolds with boundary in Hilbert spaces. For more details, we refer to any textbook about manifolds, e.g., [1], [11] or [17]. Let H be a Hilbert space and let m N be a positive integer. We denote by R m + the set of m-tuples y R m with non-negative first coordinate y 1, that is R m + = R + R m 1 = {y R m : y 1 }. We consider the relative topology on R m +. Let V be an open subset in R m +, i.e., there exists an open set Ṽ Rm such that Ṽ Rm + = V. A boundary point of V is by definition any point y V with vanishing first coordinate y 1 =. The set of all boundary points of V is denoted by V, i.e. Let k N be arbitrary. V = {y V : y 1 = } Definition. A map φ : V R m + H is called a C k -map, if there is an open set Ṽ Rm together with a C k -map φ : Ṽ H such that Ṽ Rm + = V and φ V = φ. For a C k -map φ : V R m + H and y V we define the derivative Dφy) := D φy). Note that this definition does not depend on the choice of φ. 3.. Definition. A map φ : V R m + W R m + is called a C k -diffeomorphism, if φ is bijective and both, φ and φ 1, are C k -maps. The following lemma is a standard result, whence we omit the proof Lemma. Let φ : V R m + W R m + be a C k -diffeomorphism for some k N. Then the following statements are true: 1) We have φ V ) = W. ) For each y V we have Dφy)R m + R m +. Hence, boundary points of V are mapped to boundary points of W under a C k -diffeomorphism.

18 18 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN 3.4. Definition. Let M H be a nonempty subset. 1) M is a m-dimensional C k -submanifold with boundary of H, if for all h M there exist an open neighborhood U H of h, an open set V R m + and a map φ C k V ; H) such that a) φ : V U M is a homeomorphism; b) Dφy) is one to one for all y V. The map φ is called a parametrization of M around h. ) The boundary of M is defined as the set of all points h M such that φ 1 h) V for some parametrization φ : V H around h. The set of all boundary points is denoted by M and is a submanifold without boundary of dimension m 1 of H. Parametrizations of M are provided by restricting parametrizations φ : V H of M to the boundary V. Notice that any submanifold is a submanifold with empty boundary. In what follows, let M be a m-dimensional C k -submanifold with boundary of H Definition. Let h M be arbitrary and let φ : V R m + U M be a parametrization around h. 1) The tangent space to M at h is the subspace 3.1) 3.) T h M := Dφy)R m, y = φ 1 h) V. ) For h M we can distinguish a half space in T h M, namely the set of all inward pointing directions in M, given by T h M) + := Dφy)R m +, y = φ 1 h) V Remark. By [5, Lemma 6.1.1] and Lemma 3.3, the Definitions 3.1), 3.) of the tangent spaces T h M and T h M) + are independent of the choice of the parametrization. Since parametrizations of M are provided by restricting parametrizations φ : V R m + U M of M to the boundary V, for any h M we have 3.3) In particular, we see that 3.4) For a subset A H we define T h M = Dφy) R m +, y = φ 1 h) V. T h M = T h M) + T h M) + T h M) + T h M) + T h M) + = T h M, A := {h H : h, g = for all g A}, A + := {h H : h, g for all g A}. h M. In order to introduce the inward pointing normal vectors at boundary points of the submanifold M, we require the following auxiliary result. The proof is elementary and therefore omitted Lemma. For each h M there exists a unique vector η h T h M) + T h M) with η h = 1 such that 3.5) Moreover, for each h M we have 3.6) 3.7) T h M = T h M span{η h }. T h M = T h M {η h }, T h M) + = T h M {η h } Definition. For each h M we call η h the inward pointing normal vector to M at h.

19 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 19 In the sequel, the vector e 1 R m denotes the first unit vector e 1 = 1,,..., ) Lemma. Let φ : V R m + U M be a parametrization. Then, for every h U M there exists a unique number λ > such that 3.8) where y = φ 1 h). η h, Dφy)v = λ e 1, v for all v R m, Proof. Let h U M be arbitrary. We define the continuous linear functional l : R m R, There is a unique z R m such that 3.9) lv) := η h, Dφy)v. lv) = z, v for all v R m. In order to complete the proof, we shall show that z = λe 1 for some λ >. By identity 3.6) from Lemma 3.7, for any v R m we have lv) = if and only if Dφy)v T h M, which, in view of 3.3), means that v R m +. This shows kerl) = R m +, and hence, there exists a unique λ R such that z = λe 1. Consequently, identity 3.8) is valid. By 3.), 3.3) we have Dφy)e 1 T h M) + \ T h M, and hence, inserting v = e 1 into 3.8), by 3.6), 3.7) we obtain finishing the proof. λ = λ e 1, e 1 = η h, Dφy)e 1 >, In the sequel, for h H and ɛ > we denote by B ɛ h ) the open ball B ɛ h ) = {h H : h h < ɛ} Lemma. For each h M there exists ɛ > such that for all < ɛ ɛ the following statements are true: 1) The set B ɛ h ) M is compact. ) We have B ɛ h ) M B ɛ h ) M. Proof. Let h M be arbitrary, let φ : V R m + U M be a parametrization around h and set y := φ 1 h ) V. Since V is open in R m +, there exist X K V such that X is open in R m + and K is compact. Since φ : V U M is a homeomorphism, φx) is open in U M and φk) is compact. Therefore, and since U is an open neighborhood of h, there exists ɛ > such that B ɛ h ) U and B ɛ h ) U M) φx). Let < ɛ ɛ be arbitrary. Since φx) φk) U M, we have the identity B ɛ h ) M = B ɛ h ) φk), showing that B ɛ h ) M is closed in φk). Since φk) is compact, we deduce that B ɛ h ) M is compact, establishing the first statement. For the proof of the second statement, let h B ɛ h ) M be arbitrary. Since h M, there exists a sequence h n ) n N M with h n h. Therefore, and since h B ɛ h ), there exists an index n N such that h n B ɛ h ) for all n n. Consequently, we have h n B ɛ h ) M for all n n. By the closedness of B ɛ h ) M we deduce that h B ɛ h ) M, completing the proof Proposition. Let M H be a m-dimensional C k -submanifold with boundary of H, let h M be arbitrary and let D H be a dense subset. Then there exist a constant ɛ > such that M := B ɛ h ) M is a m-dimensional C k - submanifold with boundary of H, a m-dimensional C k -submanifold N with boundary of R m,

20 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN 3.1) parametrizations φ : V M and ψ : V N, and elements ζ 1,..., ζ m D such that the mapping f := φ ψ 1 : N M has the inverse f 1 : M N, In other words, the diagram f 1 h) = ζ, h := ζ 1, h,..., ζ m, h ). f N R m M H ζ, ψ φ V R m + commutes. Furthermore, the mappings φ, ψ, Φ := φ 1, Ψ := ψ 1 have extensions φ C k b Rm ; H), ψ C k b Rm ), Φ C k b H; Rm ), Ψ C k b Rm ). Proof. Taking into account [5, Proposition 6.1.], there exist a constant ɛ >, a m-dimensional C k -submanifold M of H without boundary, a parametrization φ : Ṽ R m M and such that φv ) = M, where V := Ṽ Rm + and M := B ɛ h ) M, elements ζ 1,..., ζ m D and a parametrization f : Ñ R m M with inverse f 1 : M Ñ, f 1 h) = ζ, h := ζ 1, h,..., ζ m, h ). We set φ := φ V, N := f 1 M), f := f N and ψ := f 1 φ. Then φ : V R m + M is a parametrization, N is a m-dimensional C k -submanifold with boundary of R m and ψ : V R m + N is a parametrization. By the inverse mapping theorem, see [1, Theorem.5.], the parametrization ψ is a local diffeomorphism. Hence, arguing as in [5, Remark 6.1.1], we may assume that the mappings φ, ψ, Φ := φ 1, Ψ := ψ 1 after restricting to smaller neighborhoods, if necessary) have the desired extensions Lemma. Let h M be arbitrary and let φ : V R m + U M be a parametrization around h such that Φ := φ 1 has an extension Φ C k H; R m ). Then we have where y = φ 1 h). Dφy) 1 w = DΦh)w for all w T h M, Proof. The identity DΦh)Dφy) = DΦ φ)y) = Id R m yields the assertion. In what follows, let M be a m-dimensional C 3 -submanifold with boundary of H Lemma. Let φ : V R m + U M be a parametrization and let σ C 1 H) be a mapping such that 3.11) We define the mapping 3.1) 3.13) σh) T h M, h U M. θ : V R m, θy) := Dφy) 1 σh), where h := φy) U M. 1) For each h U M we have the decomposition Dσh)σh) = Dφy)Dθy)θy)) + D φy)θy), θy)), where y = φ 1 h) V.

21 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 1 ) If, moreover, we have 3.14) σh) T h M, then for each h U M we have h U M, 3.15) η h, Dσh)σh) = η h, D φy)θy), θy)), where y = φ 1 h) V. Proof. Let h U M be arbitrary and set y := φ 1 h) V. There exist ɛ > and Λ { 1, 1} such that 3.16) y + Λtθy) V Consequently, the curve c : [, ɛ) U M, for all t [, ɛ). ct) := φy + Λtθy)) is well-defined and we have c C 1 [, ɛ); H). Note that c) = h and by the Definition 3.1) of θ. Therefore, we have On the other hand, by 3.1), d dt ct) t= = ΛDφy)θy) = Λσh) d dt σct)) t= = ΛDσh)σh). d dt σct)) t= = d dt Dφy + Λtθy))θy + Λtθy)) t= = Λ Dφy)Dθy)θy)) + D φy)θy), θy)) ). Combining the latter two identities yields 3.13), proving the first statement. Now, suppose that 3.14) is satisfied. Then we have θy) R m + for all y V, and therefore 3.17) e 1, θy) = for all y V. Let h U M be arbitrary and set y := φ 1 h) V. There exist ɛ > and Λ { 1, 1} such that 3.16) is satisfied. Moreover, we have which gives us e 1, y + Λtθy) = e 1, y + Λt e 1, θy) = for all t [, ɛ), y + Λtθy) V for all t [, ɛ). Consequently, using Lemma 3.9 and 3.17), for some λ > we obtain η h, Dφy)Dθy)θy)) = λ e 1, Dθy)θy) e 1, θy + Λtθy)) e 1, θy) = λ lim =. t t In view of 3.13), identity 3.15) follows, establishing the second statement. Let γ : H H be a mapping fulfilling conditions.14),.15) with the mappings ρ n : R +, n N satisfying.13) Definition. We introduce the following notions:

22 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN 1) Let h M be arbitrary. We say that γ satisfies the ɛ-δ-jump condition in h, if there exists ɛ > such that for every < ɛ ɛ the set B ɛ h ) M is compact, and there are < δ < ɛ and a set B with F B c ) < such that 3.18) h + γh, x) B ɛ h ) M for F -almost all x B, for all h B δ h ) M. ) We say that γ satisfies the ɛ-δ-jump condition on M, if γ satisfies the ɛ-δ-jump condition in h for each h M Lemma. Let h M be such that for some neighborhood U of h we have 3.19) h + γh, x) M for F -almost all x, for all h U M. Then γ satisfies the ɛ-δ-jump condition in h. Proof. By Lemma 3.1 there exists ɛ > such that for every < ɛ ɛ the set B ɛ h ) M is compact and we have B ɛ h ) M B ɛ h ) M. Let < ɛ ɛ be arbitrary. There exists < δ < ɛ/ such that B δ h ) U. Moreover, there is n N such that h n for all h B δ h ) M. Setting B := {ρ n < δ}, by.13) and Chebyshev s inequality we obtain F B c ) 1 δ ρ n x) F dx) <. Let h B δ h ) M be arbitrary. By.15) we have Taking into account 3.19), we deduce γh, x) ρ n x) < δ for all x B. h + γh, x) B ɛ h ) M B ɛ h ) M for F -almost all x, showing that γ satisfies the ɛ-δ-jump condition in h Lemma. Let h M be such that γ satisfies the ɛ-δ-jump condition in h. Let φ : V R m + U M be a parametrization around h such that φ and Φ := φ 1 have extensions φ C b Rm ; H) and Φ C 1 b H; Rm ). Then there exist δ >, a set B with F B c ) < and a measurable mapping ρ : R + satisfying ρx) F dx) < such that 3.) γh, x) Dφy)Φh + γh, x)) Φh)) ρx) for F -almost all x B, where y = φ 1 h). for all h B δ h ) M, Proof. Since γ satisfies the ɛ-δ-jump condition in h, there exist δ > and a set B with F B c ) < such that h + γh, x) U M for F -almost all x B, for all h B δ h ) M. Furthermore, there exists n N such that h n for all h B δ h ) M. Let h B δ h ) M be arbitrary and set y := φ 1 h). With M := D φ and N := DΦ, by Taylor s theorem and.15), for F -almost all B we obtain γh, x) Dφy)Φh + γh, x)) Φh)) φφh + γh, x))) φφh)) Dφy)Φh + γh, x)) Φh)) 1 M Φh + γh, x)) Φh) 1 MN γh, x) 1 MNρ nx), proving 3.).

23 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 3 For a closed subspace K H we denote by Π K : H K the orthogonal projection on K, that is, for each h H the vector Π K h is the unique element from K such that Π K h h = inf g h. g K Lemma. Suppose that γ satisfies the ɛ-δ-jump condition on M. Then the following statements are true: 1) For each h M we have 3.1) Π Th M) γh, x) F dx) <. 3.) ) The mapping is continuous. M H, h Π Th M) γh, x)f dx) Proof. Let h M be arbitrary. By Proposition 3.11 there exists a parametrization φ : V R m + U M around h such that φ and Φ := φ 1 have extensions φ Cb Rm ; H) and Φ Cb 1H; Rm ). According to Lemma 3.16 there exist δ >, a set B with F B c ) < and a measurable mapping ρ : R + satisfying ρx) F dx) < such that 3.) is satisfied. Let h B δ h ) M be arbitrary. Then, for F -almost all x B we obtain Π Th M) γh, x) = γh, x) Π T h Mγh, x) γh, x) Dφy)Φh + γh, x)) Φh)) ρx). Moreover, by.14), for each x the mapping H H, h Π Th M) γh, x) is continuous. Thus, by Lebesgue s dominated convergence theorem and Lemma.18 we deduce 3.1) and the continuity of the mapping 3.) Lemma. Suppose that γ satisfies the ɛ-δ-jump condition on M and let φ : V R m + U M be a parametrization such that φ and Φ := φ 1 have extensions φ C b Rm ; H) and Φ C 1 b H; Rm ). Then the following statements are equivalent: 1) We have η h, γh, x) F dx) <, h U M. ) We have η h, Dφy)Φh + γh, x)) Φh)) F dx) <, where y = φ 1 h). h U M Proof. Let h U M be arbitrary and set y := φ 1 h). By Lemma 3.16 there exists set B with F B c ) < such that η h, γh, x) Dφy)Φh + γh, x)) Φh)) F dx) <. B Setting M := Dφ and N := DΦ, by using Lemma.18 we have η h, γh, x) F dx) η h γh, x) F dx) < B c B c

24 4 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN as well as η h, Dφy)Φh + γh, x)) Φh)) F dx) B c η h MN γh, x) F dx) <. B c Therefore, the claimed equivalence follows. Let β : H H and γ : H H be mappings such that conditions.14),.15) are fulfilled with the mappings ρ n : R +, n N satisfying.13). Let B be a set with F B c ) < and define the mappings β B : H H and γ B : H H as β B h) := βh) γh, x)f dx), B c γ B h, x) := γh, x)1 B x). Note that β B is well-defined according to Lemma Proposition. Suppose that γ satisfies the ɛ-δ-jump condition on M. Then the following statements are true: 1) We have 3.3) 3.4) 3.5) 3.6) 3.7) 3.8) η h, γh, x) F dx), h M βh) Π Th M) γh, x)f dx) T hm, h M η h, βh) η h, γh, x) F dx), h M if and only if η h, γ B h, x) F dx), h M β B h) Π Th M) γb h, x)f dx) T h M, h M η h, β B h) η h, γ B h, x) F dx), h M. ) The mapping in 3.4) is continuous on M if and only if the mapping in 3.7) is continuous on M. Proof. This is a consequence of Lemmas below. 3.. Lemma. Conditions 3.3) and 3.6) are equivalent. Proof. Let h M be arbitrary. Then we have η h, γh, x) F dx) = η h, γh, x) F dx) + η h, γh, x) F dx) B c B = η h, γh, x) F dx) + η h, γ B h, x) F dx). B c Taking into account Lemma.18, the claimed equivalence 3.3) 3.6) follows Lemma. Suppose that γ satisfies the ɛ-δ-jump condition on M. Then the following statements are true: 1) Conditions 3.4) and 3.7) are equivalent.

25 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 5 ) The mapping in 3.4) is continuous on M if and only if the mapping in 3.7) is continuous on M. Proof. Let h M be arbitrary. The calculation β B h) Π Th M) γb h, x)f dx) = βh) γh, x)f dx) Π Th M) γh, x)f dx) B c B = βh) Π Th M) γh, x)f dx) γh, x)f dx) B c Π Th M) γh, x)f dx) + Π Th M) γh, x)f dx) B = βh) Π Th M) γh, x)f dx) Π T h M γh, x)f dx), B c together with Lemma.18, proves the claimed equivalences. 3.. Lemma. Suppose that 3.3) is satisfied. Then conditions 3.5) and 3.8) are equivalent. Proof. According to Lemma 3., condition 3.6) is satisfied, too. Let h M be arbitrary. Then we have η h, β B h) η h, γ B h, x) F dx) = η h, βh) γh, x)f dx) η h, γh, x) F dx) B c B = η h, βh) η h, γh, x) F dx), proving the claimed equivalence 3.5) 3.8). Let G be another separable Hilbert space and let N a m-dimensional C 3 -submanifold with boundary of G. We assume there exist parametrizations φ : V R m + M and ψ : V R m + N. Defining f := φ ψ 1 : N M and g := ψ φ 1 : M N, the diagram f N G M H g ψ φ V R m + commutes. We assume that φ, ψ, Φ := φ 1, Ψ := ψ 1 have extensions φ Cb 3Rm ; H), ψ Cb 3Rm ; G), Φ Cb 3H; Rm ), Ψ Cb 3G; Rm ). Consequently, the mappings f, g have extensions f Cb 3G; H), g C3 b H; G). Let O M C M M be subsets such that O M is open in M. We define the subsets O N C N N by O N := go M ), C N := gc M ) and the subsets O V C V V by O V := ψ 1 O N ), C V := ψ 1 C N ). Since f : N M and ψ : V N are homeomorphisms, O N is open in N and O V is open in V. Let β : O M H, σ j : H H, j N, γ : H H and a : O N G, b j : G G, j N, c : G G be mappings satisfying the regularity conditions.11),.1) and.14).16). The mappings fλ β : O N G, fw σj : O N G,

26 6 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN j N and f µγ : O N G are defined as f λβ)z) := f, g) λβ)z), f W σ j )z) := f, g) W σ j )z), f µγ)z, x) := f, g) µγ)z, x) according to.4).4). In the sequel, for z N the vector ξ z denotes the inward pointing normal vector to N at z Proposition. Suppose that 3.9) 3.3) 3.31) az) = f λβ)z), z O N, b j z) = f W σ j )z), j N and z O N, cz, x) = f µγ)z, x) for F -almost all x, for all z O N, and that the following conditions are satisfied: 3.3) 3.33) 3.34) 3.35) 3.36) 3.37) σ j h) T h M, h O M, j N, σ j h) T h M, h O M M, j N, h + γh, x) C M for F -almost all x, for all h O M, η h, γh, x) F dx) <, h O M M, βh) 1 Dσ j h)σ j h) Π Th M) γh, x)f dx) T hm, h O M, η h, βh) 1 η h, Dσ j h)σ j h) η h, γh, x) F dx), h O M M. Then the following conditions also hold true: 3.38) 3.39) 3.4) 3.41) 3.4) 3.43) b j z) T z N, z O N, j N, b j z) T z N, z O N N, j N, z + cz, x) C N for F -almost all x, for all z O N, ξ z, cz, x) F dx) <, z O N N, az) 1 Db j z)b j z) Π TzN ) cz, x)f dx) T zn, z O N, ξ z, az) 1 ξ z, Db j z)b j z) ξ z, cz, x) F dx), z O N N.

27 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 7 For the proof of Proposition 3.3 we prepare several auxiliary results. Note that, under conditions 3.9) 3.31), for all z O N we have az) = Dgh)βh) ) D gh)σ j h), σ j h)) ) + gh + γh, x)) gh) Dgh)γh, x) F dx), 3.45) b j z) = Dgh)σ j h) for all j N, 3.46) where h = fz) O M. cz, x) = gh + γh, x)) gh) for F -almost all x, 3.4. Lemma. Let h M be arbitrary and set z := gh) N. 1) For each w T h M we have Dgh)w T z N. ) For each w T h M) + we have Dgh)w T z N ) +. 3) For each w T h M we have Dgh)w T z N. Proof. Let w T h M be arbitrary and set y := φ 1 h) V. By Lemma 3.1 we have Dgh)w = Dψ Φ)h)w = Dψy)DΦh)w = Dψy)Dφy) 1 w), proving the three assertions Lemma. Suppose that 3.3) is satisfied. Then the following statements are true: 1) Condition 3.3) implies 3.38). ) Condition 3.33) implies 3.39). Proof. This follows from 3.45) and Lemma Lemma. Suppose that 3.31) is satisfied. Then condition 3.34) implies 3.4). Proof. Let z O N be arbitrary and set h := fz) O M. Then, by 3.46) and 3.34), for F -almost all x we obtain showing 3.4). z + cz, x) = z + gh + γh, x)) gh) = gh + γh, x)) C N, 3.7. Lemma. Suppose that h + γh, x) M for F -almost all x, for all h O M. Then γ satisfies the ɛ-δ-jump condition on O M. Proof. Let h O M be arbitrary. By Lemma 3.15, and since O M is open in M, there exists ɛ > such that for every < ɛ ɛ there are < δ < ɛ and a set B with F B c ) < such that B ɛ h ) M is compact, B ɛ h ) M O M and 3.18) is satisfied. Noting that we deduce that B ɛ h ) M = B ɛ h ) O M, h + γh, x) B ɛ h ) O M for F -almost all x B, for all h B δ h ) O M, finishing the proof Corollary. Suppose that condition 3.34) is satisfied. Then the following statements are true:

28 8 DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN 1) For each h O M we have Π Th M) γh, x) F dx) <. ) The mapping is continuous. O M H, h Π Th M) γh, x)f dx) Proof. This is a direct consequence of Lemmas 3.7 and Lemma. For every h M there exists a unique number λ > such that 3.47) ξ z, Dψy)v = λ η h, Dφy)v for all v R m, where y = φ 1 h) and z = ψy). Moreover, we have 3.48) ξ z, Dgh)w = λ η h, w for all w T h M. Proof. Identity 3.47) is a direct consequence of Lemma 3.9. Using Lemma 3.1 and 3.47), for all w T h M we obtain ξ z, Dgh)w = ξ z, Dψ Φ)h)w = ξ z, Dψy)DΦh)w = ξ z, Dψy)Dφ 1 y)w = λ η h, w, which proves 3.48) Lemma. Suppose that 3.31), 3.34) are satisfied. Then condition 3.35) implies 3.41). Proof. According to Lemma 3.6, condition 3.4) is satisfied, too. Let z O N N be arbitrary. We set h := fz) O M M and y := φ 1 h) O V V. By Lemma 3.18 we have η h, Dφy)Φh + γh, x)) Φh)) F dx) <. Using 3.46) and Lemma 3.9, for some λ > we obtain ξ z, Dψy)Ψz + cz, x)) Ψz)) F dx) = ξ z, Dψy)Ψgh + γh, x))) Ψz)) F dx) = ξ z, Dψy)Φh + γh, x)) Φh)) F dx) = λ η h, Dφy)Φh + γh, x)) Φh)) F dx) <. Applying Lemma 3.18 yields condition 3.41) Lemma. Suppose that 3.3), 3.3) are satisfied and let j N be arbitrary. For each z O N we have the decomposition Db j z)b j z) = Dgh)Dσ j h)σ j h)) + D gh)σ j h), σ j h)), where h = fz) O M.

29 SPDS AND SUBMANIFOLDS IN HILBRT SPACS 9 Proof. According to Lemma 3.5, condition 3.38) is satisfied, too. Note that, by Lemma 3.1 and 3.45), for all y O V we have Dφy) 1 σ j h) = DΦh)σ j h) = DΨ g)h)σ j h) = DΨz)Dgh)σ j h) = DΨz)b j z) = Dψy) 1 b j z), where h := φy) O M and z := ψy) O N. We define the mapping θ j : O V R m, θ j y) := Dφy) 1 σ j h), where h := φy). Let z O N be arbitrary. We set h := fz) O M and y := φ 1 h) O V. Using Lemma 3.13 we obtain the decompositions 3.49) 3.5) Dσ j h)σ j h) = Dφy)Dθ j y)θ j y)) + D φy)θ j y), θ j y)), Db j z)b j z) = Dψy)Dθ j y)θ j y)) + D ψy)θ j y), θ j y)). Note that we have 3.51) Dψy)Dθ j y)θ j y)) = Dg φ)y)dθ j y)θ j y)) = Dgh)Dφy)Dθ j y)θ j y)). By the second order chain rule we obtain 3.5) D ψy)θ j y), θ j y)) = D g φ)y)θ j y), θ j y)) = D gh)dφy)θ j y), Dφy)θ j y)) + Dgh)D φy)θ j y), θ j y))) = D gh)σ j h), σ j h)) + Dgh)D φy)θ j y), θ j y))). Moreover, by 3.49) we have 3.53) Dgh)D φy)θ j y), θ j y))) = Dgh)Dσ j h)dσ j h)) Dgh)Dφy)Dθ j y)θ j y)). Inserting 3.51) 3.53) into 3.5) we arrive at Db j z)b j z) = Dgh)Dφy)Dθ j y)θ j y)) + D gh)σ j h), σ j h)) completing the proof. + Dgh)D φy)θ j y), θ j y))) = Dgh)Dφy)Dθ j y)θ j y)) + D gh)σ j h), σ j h)) + Dgh)Dσ j h)σ j h)) Dgh)Dφy)Dθ j y)θ j y)) = Dgh)Dσ j h)σ j h)) + D gh)σ j h), σ j h)), 3.3. Lemma. Suppose that 3.9) 3.31) and 3.3), 3.34) are satisfied. Then condition 3.36) implies 3.4). Proof. According to Lemma 3.6, condition 3.4) is satisfied, too. Let z O N be arbitrary and set h := fz) O M. By 3.44), 3.46) we obtain az) 1 Db j z)b j z) Π TzN ) cz, x)f dx) = Dgh)βh) + 1 D gh)σ j h), σ j h)) ) + gh + γh, x)) gh) Dgh)γh, x) F dx) 1 Db j z)b j ) z) Π TzN ) gh + γh, x)) gh) F dx).