Anticipating Quantum Stochastic Integrals for Basic Quantum Martingales (Mathematical aspects of quantum fields and related topics)

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(1)29. Anticipating Quantum Stochastic Integrals for Basic Quantum Martingales Un Cig Ji Department of Mathematics, Chungbuk National University. 1. Introduction. Since the quantum stochastic integrals of adapted quantum stochastic processes have been been introduced introduced by by Hudson Hudson and and Parthasarathy Parthasarathy [10] [10] as as aa quantum quantum extension extension of of the the Itˆ Itôo (stochastic) (stochastic) integral, integral, the the quantum quantum stochastic stochastic calculus calculus has has been been studied studied extensively extensively with with wide wide applications applications (see (see [28, [28, 32]). 32]). The Hudson-Parthasarathy Hudson‐Parthasarathy quantum stochastic integrals has been extended to the quantum stochastic integrals of nonadapted quantum stochastic processes by Belavkin [3], [3], Lindsay Lindsay [24] [24] and and Attal Attal & & Lindsay Lindsay [2]. [2]. Since Since then then the the nonadapted nonadapted quantum quantum stochastic stochastic integral has been studied systematically in terms of quantum stochastic gradients by Ji & & Obata Obata [16, [16, 18]. 18]. Based Based on on the the quantum quantum white white noise noise theory theory [12], [12], the the notion notion of of quantum quantum white noise derivatives has been introduced by Ji & Obata (see [14, 15, 16, 17, white noise derivatives has been introduced by Ji & Obata (see [14, 15, 16, 17, 19, 19, 20]). 20]). The explicit formulas [16] of integrands for quantum stochastic integral representation The explicit formulas [16] of integrands for quantum stochastic integral representation of of quantum quantum martingales martingales [11] [11] has has been been derived derived in in terms terms of of the the quantum quantum white white noise noise derivatives. Also, the notion of quantum stochastic gradients [18] has been derivatives. Also, the notion of quantum stochastic gradients [18] has been introduced introduced based based on on the the notion notion of of the the quantum quantum white white noise noise derivatives. derivatives. Recently, Recently, Ji Ji & & Sinha Sinha [21] [21] studied the quantum stochastic integrals for quadratic quantum noises. On On the the other other hand, hand, based based on on the the white white noise noise theory theory [8, [8, 22, 22, 30] 30] introduced introduced by by Hida, Hida, Kuo & Russek [23] studied anticipating (classical) stochastic integrals by applying Kuo & Russek [23] studied anticipating (classical) stochastic integrals by applying the the quantum decomposition of a Brownian motion. In this paper, we study some regularity properties of the quantum Hitsuda–Skorohod Hitsuda‐Skorohod integrals as anticipating quantum stochastic integrals. Also, motivated by the results in in [23], [23], we we discuss discuss new new types types of of anticipating anticipating quantum quantum stochastic stochastic integrals integrals in in terms terms of of pointwisely defined defined quantum white noise derivatives.. 2 2.1. Admissible Generalized Operators Admissible Rigging of Fock Space. We now review a construction of admissible rigging of Fock space which provides the basic = L2 (R+ , dt) basic structure structure of of this this paper. paper. Let Let H H=L^{2}(\mathbb{R}_{+}, dt) be be the the Hilbert Hilbert space space of of complex complex valued valued. 1.

(2) 30 dt and square ∞)) with + = [0, \infty square integrable integrable functions functions on on R \mathbb{R}_{+}=[0, with respect respect to to the the Lebesque Lebesque measure measure dt and H let Γ(H) defined by \Gamma(H) be the Fock space over H defined ∞ ∞ ⊗n 2 n!|fn | < ∞ , Γ(H) = φ = (fn )n=0 ; fn ∈ H ,. \Gamma(H)=\{\phi=(f_{n})_{n=0}^{\infty};f_{n}\in H^{\otimes n}\wedge, \sum_{n= 0}^{\infty}n!|f_{n}|^{2}<\infty\}, n=0. H^{\otime⊗n sn}\wedge is the n-fold n ‐fold symmetric tensor product of H H and || ·. || is the Hilbertian norm where H ⊗n H and H H^{\otimes n}\wedge . For pp\geq on H ≥ 0, 0 , we set ∞ 2 ∞ 2pn 2 n!e |fn | < ∞ Gp = φ = (fn )n=0 ∈ Γ(H) ; ||| φ |||p =. \mathcal{G}_{p}=\{\phi=(f_{n})_{n=0}^{\infty}\in\Gamma(H);\Vert|\phi\Vert|_{p} ^{2}=\sum_{n=0}^{\infty}n!e^{2pn}|f_{n}|^{2}<\infty\} n=0. and G\math−p defined by \Gamma(H) with respect to the norm |||\Ver·t|\cdot|||\Ver−p cal{G}_{-p} to be the completion of Γ(H) t|_{-p} defined ∞ . \Vert|\phi\Vert|_{-p}^{2}=\sum_{n=0}^{\infty}n!e^{-2pn}|f_{n}|^{2}. ||| φ |||2−p =. n=0. n!e−2pn |fn |2 .. Then {G \{\mathpcal{;G}_{pp};p\i∈ n \mathR} bb{R}\} forms a chain of weighted Fock spaces and so we have ∗ G\mathcal= = ind lim G−p {G}= proj lim Gp ⊂ Gp ⊂ G0 = Γ(H) ⊂ G−p ⊂ G ∼. \lim \mathcal{G}_{p}\subset \mathcal{G}_{p}\subset \mathcal{G}_{0}=\Gamma(H) par ow\infty\subset \mathcal{G}_{-p}\subset \mathcal{G}^{*}\cong ind\lim \mathcal{G}_{-p}par ow\infty p→∞. p→∞. ∗ for pp\geq ≥ 0, 0 , where the strong dual space Γ(H) identified with Γ(H), \Gamma(H)^{*} of Γ(H) \Gamma(H) is identified \Gamma(H) , and ∗ the strong dual space G \mathcal{G}^{*} of G \mathcal{G} is topologically isomorphic to the inductive limit space ∗ ind lim form: \mathb {C} ‐bilinear form ·, \cros \matG hcal{G} takes the form: p→∞ \langle\langle\cdot, · \cdot\rangle\rangle on G\mathcal{G}^{*}× \lim_{parrow\i nfty}\matG hcal{−p G}_{-p} .. The canonical C-bilinear ∞ . \langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle f_{n}, g_{n}\rangle, \Phi=(f_{n})\in \mathcal{G}^{*}, \phi=(g_{n})\in \mathcal{G},. Φ, φ =. n=0. n! fn , gn ,. Φ = (fn ) ∈ G ∗ ,. φ = (gn ) ∈ G,. ⊗nn}\cross H^{\otimes H^{\otimes \mathb {C}‐bilinear form on H where f ×H ⊗nn} . Note that G \mathcal{G} is a countable \langlenf_{n,}, gg_{nn}\rangle is the canonical C-bilinear Hilbert space but not necessarily a nuclear space. An element in G \mathcal{G} is said to be admissible admissible or regular. ∗ Remark 2.1 Let E E_{\mathR bb{R} is a nuclear space, where E_{\matRhbb{R}⊂ \subset H_{H \mathbb{RR} \subset⊂E_{\matE hbb{RR} ^{*} be a Gelfand triple, i.e. E 2 H \mathbb{+ R}_{+} H_{\matRhbb{= R} =L_{\L mathRbb{R(R } ^{2}(\mat+hbb{, Rdt) }_{+}, dt) is the Hilbert space of real valued square integrable functions on R ∗ dt . Then for the standard Gaussian measure µ with respect to dt. \mu on E E_{\mathbR{R} ^{*} characterized by 1 2 ξ ∈ ER , eix, ξ dµ(x) = e− 2 |ξ| ,. \int_{E_{\mathb {R} ^{*} e^{i\langle x,\xi\rangle}d\mu(x)=e^{-\frac{1}{2} |\xi|^{2} \xi\in E_{\mathb {R} , ER∗. ∗ o-Segal where ·, Wiener‐Itô‐Segal \langle\cdot, · \cdot\rangle is the canonical bilinear form on E E_{\matRhbb{R} ^× {*}\cros E_E {\mathbb{RR} again, by the Wiener-Itˆ 2 ∗ isomorphism, Γ(H) is unitarily equivalent with the Hilbert space L (E , µ) of complex \Gamma(H) L^{2}(E_{\mathbb{RR} ^{*}, \mu) ∗ valued square integrable functions on E \mu . In E_{\mathbR{R} ^{*} with respect to the Gaussian measure µ. this sense, the elements of G \mathcal{G} are considered as admissible Gaussian functionals. The spaces \mathcal{G} and \mathcal{G∗}^{*} were spaces G and G were introduced introduced by by Belavkin Belavkin [3] [3] and and have have appeared appeared along along with with classical classical and quantum stochastic analysis, see e.g., [1, 4, 7, 11, 13, 14, 24, 25, 26, 33, and quantum stochastic analysis, see e.g., [1, 4, 7, 11, 13, 14, 24, 25, 26, 33, 34]. 34].. 2.

(3) 31 31. 2.2. Multiplications of Admissible Gaussian Functionals. Let φ\phi=(f_{n}), = (fn ), ψ\psi=(g_{n})\in = (gn\mathcal{G} ) ∈ G be given. Suppose that ff_{n}=0 finite n = 0 and gg_{m}=0 m = 0 except for finite n and m. m numbers of n Then the Wiener product (or pointwise multiplication) \phi\psi\in∈ \mathcalG{G} of numbers of and . Then the Wiener product (or pointwise multiplication) φψ of φ\phi and ψ \psi is defined defined by ∞ m+k l+k }\kotimges_{m+k ff_{ll+k k! hn = +k^{\wedge}⊗ k}g_{m+k} ,, k k l+m=n k=0. h_{n}= \sum_{l+m=n}\sum_{k=0}^{\infty}k! (\begin{aray}{l l+ k k \end{aray})(\begin{aray}{l m+k k \end{aray}). φψ = (hn ),, \phi\psi=(h_{n}). (2.1) (2.1). \kotimges_{m+k k ‐contraction of ⊗ where }+k^{\wedge} k}g_{m+k} is l+k and m+k ,, see where ff_{ll+k is the the k-contraction of ff_{l+k} and gg_{m+k} see [30]. [30]. The following lemma is useful to study the continuities of Wiener product of admisadmis‐ sible sible Gaussian Gaussian functionals functionals and and similar similar estimates estimates can can be be found found in in [25] [25] (see (see also also [30, [30, 33]). 33]). = (gn\mathcal{G} ) ∈ G be Lemma 2.2 Let φ\phi=(f_{n}), = (fn ), ψ\psi=(g_{n})\in be given. Suppose that ff_{n}=0 and gg_{m}=0 n = 0 and m = 0 m . Then s\in ∈ \mathbb{R} except finite numbers of R with rr+s>0 + s > 0 and p, r, r, s except for aa finite of nn and and m. Then for any any p, and n (s−3r)/2 + e(r−3s)/2 e ≤ cn (2.2) (n + 1) (2.2) e−2p (r + s). (n+1)( \frac{e^{(s-3r)/2}+e^{(r-3s)/2} {e^{-2p}(r+s)})^{n}\leq c^{n}. < c < 1,, it holds that for some 00<c<1 1. \Vert|\phi\psi\Vert|_{p}^{2}\leq\f1rac{1}−{1-c}\Vecrt|\ph|||i\Verφt|_{r}^{|||2} \Ver|||t|\psiψ\Vert|_|||{s}^{2} ... ||| φψ |||2p ≤. 2 r. 2 s. (2.3) (2.3). Proof. PROOF. For For given given hh_{n}n as as in in (2.1), (2.1), we we obtain obtain that that. 2. ∞ l+k m+k 2 k! |fl+k ||gm+k | n!|hn | = n! k k l+m=n k=0. 2 ∞ . r(l+k) s(m+k) ≤ n! Ml,m,k (l + k)!e |fl+k | (m + k)!e |gm+k | , (2.4) (2.4). n!|h_{n}|^{2}=n!(\sum_{l+m=n}\sum_{k=0}^{\infty}k!(\begin{ar ay}{l} l+ k k \end{ar ay})(\begin{ar ay}{l m+k k \end{ar ay})|f_{l+k}|g_{m+k}|)^{2} \leq n!(\sum_{l+m=n}\sum_{k=0}^{\infty}M_{l,m,k}\sqrt{(l+k)!}e^{r(l+k)}|f_{l+ k}|\sqrt{(m+k)!}e^{s(m+k)}|g_{m+k}|)^{2} l+m=n k=0. where. (l + k)!(m + k)! −(r+s)k e−rl−sm e Cl,m;r+s , ≤ k! l!m!. M_{l,m,k}= \frac{e^{-rl-sm}}{l!m!}\frac{\sqrt{(l+k)!(m+k)!} {k!}e^{-(r+s)k}\leq \frac{e^{-rl-sm}}{l!m!}\sqrt{C_{lm,r+s}},. Ml,m,k where. e−rl−sm = l!m!

(4). . l+m. C_{l,m,q}= \sup_{n\geq 0}\{\frac{(l+n)!}{n!}(n+m)!_{e^{-2qn}\} n!\leq e^{q}l^{ \iota}m^{m}(\frac{e^{q/2} {eq})^{l+m}<\infty. Cl,m;q = sup n≥0. (l + n)! (n + m)! −2qn e n! n!. q l. ≤e lm. m. eq/2 eq. <∞. (2.5) (2.5). \mathbb{R} \mathbb{R} for > 0 (see R and R with + s > 0, for qq>0 (see e.g., e.g., [30]: [30]: Section Section 4.1). 4.1). Therefore, Therefore, for for any any rr\in ∈ and ss\in∈ with rr+s>0, from from (2.4), (2.4), by by applying applying Cauchy-Schwarz Cauchy‐Schwarz inequality inequality we we obtain obtain that that. 2 e−rl−sm n!|hn | ≤ n! Cl,m;r+s ||| φ |||2r ||| ψ |||2s l!m! l+m=n. 2. √ n! −rl−sm e Cl,m;r+s ||| φ |||2r ||| ψ |||2s .. ≤ l!m! l+m=n. n!|h_{n}|^{2} \leq n!(\sum_{l+m=n}\frac{e^{-rl-sm} {l!m!}\sqrt{C_{lmr+s} )^{2} \Vert|\phi\Vert|_{r}^{2}\Vert|\psi\Vert|_{s}^{2} \leq(\sum_{l+m=n}\frac{\sqrt{n!} {l!m!}e^{-rl-sm}\sqrt{C_{l,mr+s} )^{2} \Vert|\phi\Vert|_{r}^{2}\Vert|\psi\Vert|_{s}^{2} 2. 3. (2.6) (2.6).

(5) 32 n n By ≤ ee^{n}n! n!,, from By applying applying aa simple simple inequality inequality nn^{n}\leq from (2.5) (2.5) we we see see that that. (r+s)/2 (l+m)/2 (r+s)/2 (l+m)/2 √ √ e e ≤ e(r+s)/2 l!m! Cl,m;r+s ≤ e(r+s)/2 ll mm e(r + s) r+s. \sqrt{C_{lmr+s} \leq e^{(r+s)/2\sqrt{l^{l}m^{m} }(\frac{e^{(r+s)/2} {e(r+s)})^ {(l+m)/2}\leq e^{(r+s)/2}\sqrt{l!m!}(\frac{e^{(r+s)/2} {r+s})^{(l+m)/2}. r+s \mathbb{R} \mathbb{R} Therefore, R and R with ≥ 2, 2 , from Therefore, for for any any rr\in ∈ and ss\in∈ with ee^{r+s}\geq from (2.6) (2.6) we we obtain obtain that that. √ 2 n! −rl−sm 2 n!|hn | ≤ Cl,m;r+s ||| φ |||2r ||| ψ |||2s e l!m! l+m=n. √ (r+s)/2 (l+m)/2 2 n! e √ ||| φ |||2r ||| ψ |||2s e−rl−sm ≤ er+s r + s l!m! l+m=n. n! e(s−3r)/2 l e(r−3s)/2 m ≤ (n + 1) ||| φ |||2r ||| ψ |||2s l!m! r + s r + s l+m=n (s−3r)/2 n + e(r−3s)/2 e ||| φ |||2r ||| ψ |||2s .. = (n + 1) r+s. n!|h_{n}|^{2} \leq(\sum_{l+m=n}\frac{\sqrt{n!} {l!m!}e^{-rl-sm}\sqrt{C_{lmr+s} )^{2}\Vert|\phi\Vert|_{r}^{2}\Vert|\psi\Vert|_{s}^{2} \leqe^{r+s}(\sum_{l+m=n}\frac{\sqrt{n!} {\sqrt{l!m!} e^{-rl-sm}(\frac{e^{(r+ s)/2} {r+s})^{(l+m)/2})^{2}\Vert|\phi\Vert|_{r}^{2}\Vert|\psi\Vert|_{\mathcal{S} }^{2}. \leq(n+1)(\sum_{l+m=n}\frac{n!}{l!m!}(\frac{e^{(s-3r)/2} {r+s})^{l} (\frac{e^{(r-3s)/2} {r+s})^{m})\Vert|\phi\Vert|_{r}^{2}\Vert|\psi\Vert|^{2}. =(n+1)( \frac{e^{(s-3r)/2}+e^{(r-3s)/2} {r+s})^{n}\Vert|\phi\Vert|^{2}. \Vert|\psi\Vert|^{2}. (2.7) (2.7). Therefore, Therefore, from from (2.1) (2.1) and and (2.7) (2.7) we we obtain obtain that that n (s−3r)/2 ∞ ∞ + e(r−3s)/2 e 2 2pn 2 ||| φ |||2r ||| ψ |||2s n!e |hn | ≤ (n + 1) ||| φψ |||p = −2p (r + s) e n=0 n=0. \Vert|\phi\psi\Vert 1_{p}^{2}=\sum_{n=0}^{\infty}n!e^{2pn}|h_{n}|^{2} \leq[\sum_{n=0}^{\infty}(n+1)(\frac{e^{(s-3r)/2}+e^{(r-3s)/2} {e^{-2p}(r+s)} ^{n}]1 \phi\Vert 1_{r}^{2}1 \psi\Vert 1_{s}^{2} 1. \leq\fra1c{1}{1−-c}\Verct 1\ph|||i\Veφrt 1_{r|||}^{2}\Ve|||rt 1\pψsi\Ver|||t 1^{2}. ≤. 2 r. 2 s. (2.8) (2.S). for < c < 1 satisfying for some some 00<c<1 satisfying (2.3), (2.3), which which gives gives the the proof. proof.. \square . The following two theorem are obvious consequences of Lemma 2.2. Theorem functionals is Theorem 2.3 2.3 ([33]) ([33]) The The Wiener Wiener product product of of admissible admissible Gaussian Gaussian functionals is concon‐ tinuous from G × G (equipped with the product topology) onto G. In particular, G \mat h cal { G } \ cr o s \mat h cal { G } \ m a t h c a l { G } \ m a t tinuous from (equipped with the product topology) onto . In particular, hcal{G} is is an an algebra algebra with respect to the Wiener Wiener product. Theorem 2.4 The The Wiener Wiener product of of admissible admissible white noise functionals is continuous continuous ∗ ∗ from G \mathcal{G}^{*}× \cros \matG hcal{G} (equipped \mathcal{G}^{*} . from (equipped with with the the product product topology) topology) onto onto G = (gn\mathcal{G} ) ∈ G be given. Suppose that ff_{n}=0 Let φ\phi=(f_{n}), = (fn ), ψ\psi=(g_{n})\in n = 0 and gg_{m}=0 m = 0 except for m finite ψ finite numbers numbers of of nn and and m. . Then Then the the Wick Wick product product (or (or normal-ordered normal‐ordered product) product) φ\phi⋄0\psi of φ\phi and ψ \psi is defined defined by m ,, fl ⊗g (2.9) kn = φ ⋄ ψ = (kn ), (2.9). \phi 0\psi=(k_{n}) , k_{n}=\sum_{l+m=n}f_{l}\otimes g_{m}\wedge l+m=n. see see [6, [6, 8, 8, 22] 22] The following lemma is useful to study the continuities of Wick product of admissible Gaussian Gaussian functionals functionals and and similar similar estimates estimates can can be be found found in in [33]. [33].. 4.

(6) 33 Lemma 2.5 Let φ\phi=(f_{n}), = (fn ), ψ\psi=(g_{n})\in = (gn\mathcal{G} ) ∈ G be be given. Suppose that ff_{n}=0 and gg_{m}=0 n = 0 and m = 0 m . Then s\in ∈ \mathbb{R} except finite numbers of R satisfying that p, r, r, s except for aa finite of nn and and m. Then for any any p,. it holds that. 2(p−r) + e2(p−s) < 1,, ee^{2(p-r)}+e^{2(p-s)}<1. (2.10) (2.10). 2 2 2 |||\Verφt|\phi ⋄0\psi\ψ Vert|_{|||p}^{2p}\leq\Ver≤t|\phi|||\Verφt|_{r}^|||{2}\rVert||||\psi\ψVert|_{s|||}^ s{2} ... (2.11) (2.11). Proof. PROOF. For For given given kk_{n}n as as in in (2.9), (2.9), we we obtain obtain that that. 2. |fl ||gm | n!|kn |2 = n!. n!|k_{n}|^{2}=n!( \sum_{l+m=n}|f_{l}| g_{m}|)^{2} l+m=n. e−2rl−2sm 2rl 2 2sm 2 ≤ n! l!e |fl | m!e |gm | l!m! l+m=n l+m=n. −2r −2s n 2rl 2 2sm 2 ≤ e +e l!e |fl | m!e |gm | ,,. \leq n!(\sum_{l+m=n}\frac{e^{-2rl-2sm} {l!m!})(\sum_{l+m=n}l!e^{2rl}|f_{l} |^{2}m!e^{2sm}|g_{m}|^{2}) \leq(e^{-2r}+e^{-2s})^{n}(\sum_{l+m=n}l!e^{2rl}|f_{l}|^{2}m!e^{2sm}|g_{m}|^{2} ). (2.12) (2.12). l+m=n. s\in ∈ \mathbb{R} Therefore, R satisfying p, r, r, s Therefore, for for any any p, satisfying (2.10), (2.10), from from (2.12) (2.12) we we obtain obtain that that. ∞ ∞ 2(p−r) 2 2pn 2 2rl 2 2sm 2 2(p−s) n ||| φ ⋄ ψ |||p = e n!e |kn | ≤ l!e |fl | m!e |gm | +e. \Vert|\phi 0\psi\Vert|_{p}^{2}=\sum_{n=0}^{\infty}n!e^{2pn}|k_{n}|^{2} \leq\sum_{n=0}^{\infty}(e^{2(p-r)}+e^{2(p-s)} ^{n}(\sum_{l+m=n}l!e^{2rl}|f_{l}|^ {2}m!e^{2sm}|g_{m}|^{2}) n=0. ||| φ |||2r. n=0. l+m=n. |||2s. ≤ \leq\Vert|\phi\Vert|_{r}^{2}\|||Vert|\ψpsi\Vert|_{s}^{2,},. which gives the proof.. \square . The following theorem is an obvious consequence of Lemma 2.5. Theorem from G\mathcal{G× }\cros \matG hcal{G} (equipped Theorem 2.6 2.6 ([33]) ([33]) The The Wick Wick product product is is continuous continuous from (equipped with with ∗ ∗ ∗ × G onto G . In particular, product topology) onto G, and from G \mathcal{G} and \mathcal{G∗}^{*} product topology) onto \mathcal{G} , and from \mathcal{G}^{*}\cross \mathcal{G}^{*} onto \mathcal{G}^{*} In particular, G and G algebras algebras under the Wick Wick product.. 3. the the are are. Admissible Generalized Operators. We denote by L(X, \mathcal{L}(\mathfrak{X}, Y) \mathfrak{Y}) the space of all continuous linear operators from a locally convex \mathfrak{X} into another locally convex space Y space X \mathfrak{Y} equipped with the topology of bounded ∗ ) is called convergence. An operator in L(G, G convergence. An operator in \mathcal{L}(\mathcal{G}, \mathcal{G}^{*}) is called an an admissible admissible generalized generalized operator operator [14] [14] or simply admissible operator. admissible operator.. 3.1. Integral Kernel Operators. ⊗m m be non-negative Let l,l, m ∈ L(H H ⊗l (fn )n∞ ∈ {GG}^{*∗} . For non‐negative integers. Let K \Phi== (f_{n})_{n=0}^{\i fty}\in \mathcal K_{l,l,m m}\in \mathcal {L}(H^{\otimes m},,H^{\oti mes l)}) and Φ n=0 each nn\geq ≥ 0, 0 , we put (n + m)! (Kl,m ⊗ I ⊗n fn+m )sym .. (3.1) gl+n = (3.1) n!. g_{l+n}= \frac{(n+m)!}{n!}(K_{l,m}\otimes I^{\otimes n}f_{n+m})_{sym} 5.

(7) 34 Then R and > 0,, we \mathbb{R} Then from from Lemma Lemma 4.1 4.1 in in [11], [11], for for any any pp\in ∈ and qq>0 we obtain obtain that that ∞ . ∞ . (l + n)! (n + m)! 2p(l+n) e |fn+m |2 n! n!. \sum_{n=0}^{\infty}(l+n)!e^{2p(l+n)}|g_{l+n}|^{2}\leq\Vert K_{l,m}\Vert^{2} \sum_{n=0}^{\infty}(n+m)!\frac{(l+n)!}{n!}e^{2p(l+n)}(n+m)!n!|f_{n+m}|^{2}. n=0. (l + n)!e2p(l+n) |gl+n |2 ≤ Kl,m 2. (n + m)!. n=0. 2 2 2(pl−(p+q)m) ≤ KK_{l,m \leq\Vert l,m} \Verte^{2}e^{2(pl-(p+q)m)}C_{l,mC,q}\l,m;q Vert|\phi|||\Vertφ|_|||{p+q}p+q ^{2} ,,. (3.2) (3.2). where l,m;q is where K is the the operator operator norm norm and and C is given given as as in in (2.5). (2.5). Therefore, Therefore, we we define define \Vert K_{l,m l,m}\Vert is C_{l,m;q} an linear operator Ξ\Xi_{l,m}(K_{l,m}) \mathcal{G∗}^{*} by l,m (Kl,m ) on G Ξ\Xi_{l, )Φ = (gl+n )∞ , Φ = (fn )∞ ∈ G ∗ ,, l,mm(K }(K_{l,l,m m})\Phi=(g_{l+n})_{n=0}^{\infty} \mathcal{G}^{*} n=0 n=0, \Phi=(f_{n})_{n=0}^{\infty} \in where R and > 0 it \mathbb{R} l+n is where gg_{l+n} is given given as as in in (3.1). (3.1). Then Then for for any any pp\in ∈ and qq>0 it holds holds that that. (pl−(p+q)m) |||\Vert| Ξl,m )Φ \|||Vert| Cl,m;q Φ ∈ {G}^{* G ∗} ,, p _≤ _{-l,m(K }^{-}-(K_{ll,m,m})\Phi {p}\le K q\Vertl,m K_{l, e m}\Vert e^{(pl-(p+q)m)} \sqrt{C_{l m,q}}\Vert|||| \ΦPhi\|||Vert|p+q_{p+q},, \Phi\in \mathcal. (3.3) (3.3). (3.4) (3.4). ) {L}(\mathcal ∈ L(G , G{G}_{p}) which implies that Ξ p ).. The operator Ξ l,m (Kl,m ) is called the \Xi_{l,m}(K_{l,m}) \Xil,m _{l,m}(K_{l(K ,m})\il,m n \mathcal {G}_{p+q},p+q \mathcal integral l,m (see integral kernel kernel operator operator with with kernel kernel K (see [13, [13, 9, 9, 22, 22, 30]). 30]). K_{l,m} Now the following theorem is obvious. ⊗m m be non-negative integers and let Kl,m ∈ L(H H m⊗les l). Theorem Theorem 3.1 3.1 ([11]) ([11]) Let Let l,l, m be non‐negative integers and let K_{l,m}\in \mathcal{L}(H^{\otimes m}, ,H^{\oti }) . Then it holds that Then it holds that ∗ ∗ Ξ \Xi_l,m {l,m}(K_{l(K ,m})\in \matl,m hcal{L)}(\mat∈hcal{GL(G, }, \mathcal{G})\capG) \mathcal∩{L} (\L(G mathcal{G}^{*}, \mat, hGcal{G}^{). *}) .. ∈ H. Let η\eta\in ∈ HH and let K defined by K H . For η (f ) = η, ff\ran gle for any ff\in K_{\etηa}\i∈ n \matL(H, hcal{L}(H, \mathC) bb{C}) be defined K_{\eta}(f)=\langle\eta, ∗ simple notation, we identify η\eta=K_{\eta}=K_{\eta}^{*} = Kη = Kη∗ , where K is the adjoint operator of K K_{\etηa} with K_{\etaη}∗^{*} a\in \mathbb{C} (a) = aη for all a ∈ C. Then the respect to the canonical bilinear form ·, ·, i.e., K . \langle\cdot, \cdot\rangle , η K_{\eta}^{*}(a)=a\eta ∗ (η) associated with η are defined annihilation operator a(η) and the creation operator a annihilation operator a(\eta) creation operator a^{*}(\eta) defined \eta ∗ (η) = Ξ1,0 (η),, respectively, and then from Theorem 3.1, it holds by a(η) = Ξ0,1 (η) and aa^{*}(\eta)=--1,0-(\eta) a(\eta)=--0,1-(\eta) that ∗ ∗ ∗ a(η), a(\eta), a^{*}(\eta)a\in \mat(η) hcal{L}(\mat∈hcal{GL(G, }, \mathcal{G})\capG) \mathcal∩{L} (\L(G mathcal{G}^{*}, \mat, G hcal{G}^{). *}) .. It is straightforward to verify the canonical commutation relation: ∗ ∗ ∗ ξ(t)η(t)dt = ξ, η (ξ), a (η)] = 0, [a(ξ), a (η)] = [a(ξ), a(η)] = 0, [a [a(\xi), a(\eta)]=0, [a^{*}(\xi), a^{*}(\eta)]=0, \eta\rangle. [a( \xi), a^{*}(\eta)]=\int_{\mathbb{R}_{+} \xi(t)\eta(t)dt=\langle\xi, R+. (3.5) (3.5). for ξ, ∈ H. \xi, η\eta\in H. ∞ ⊗n ∈ HH is defined /n!) The exponential vector φ\phi_{ξ\xi} associated with ξ\xi\in defined by φ\phiξ_{\xi= }=(\xi(ξ ^{\otimes n}/n! )_{n=0}^{\i nfty}.. n=0 Then {φ \mathcal{G}. ξ ; ξ ∈ H} \{\phi_{\xi};\xi\in H\} spans a dense subspace of G. Proposition ∈ HH be Proposition 3.2 3.2 ([14, ([14, 5]) 5]) Let Let ζ\zeta\in be given. given. Then Then it it holds holds that that ∗ a(ζ)(Φψ) = (a(ζ)Φ) ψ + Φ (a(ζ)ψ) Φ ∈ G\psi\in , \mathcal{G} ψ ∈ G,, a(\zeta)(\Phi\psi)=(a(\zeta)\Phi)\psi+\Phi(a(\zeta)\psi) , \Phi\in, \mathcal{G}^ {*}, a(ζ)(Φ ⋄ Ψ) = (a(ζ)Φ) ⋄ Ψ + Φ ⋄ (a(ζ)Ψ) , Φ, Ψ ∈ G ∗. a(\zeta)(\Phi 0\Psi)=(a(\zeta)\Phi)0\Psi+\Phi o(a(\zeta)\Psi) , \Phi, \Psi\in \mathcal{G}^{*}. Proof. ∈ H, \xi, η\eta\in H , we PROOF. (i) (i) For For any any ξ, we obtain obtain that that ξ, η ξ, η a(ζ)(φ a(\zeta)(\phi_{\xi}\ξphiφ_{\ηeta)})=a(\=zeta)(\a(ζ)(φ phi_{\xi+\eta})e^{\lξ+η angle\xi, )e\eta\rangle}=\langl=e\zetaζ, , \xi+\eta\rξangl+ e\phi_η {\xi+\etφa}e^{ξ+η \langle\xie, \eta\rangle}==\langleζ, \zeta, \xi+ξ\eta\r+angleη \phi_{\φxi}\phiξ φ_{\etηa} = (a(ζ)φξ ) φη + φξ (a(ζ)φη ) . =(a(\zeta)\phi_{\xi})\phi_{\eta}+\phi_{\xi}(a(\zeta)\phi_{\eta}) .. 6. (3.6) (3.6) (3.7) (3.7).

(8) 35 ∗ ∗ Therefore, by the continuity property a(ζ) ,G a(\zeta)\in \mathcal∈{L}(\mathL(G, cal{G}, \mathcal{GG) })\cap \mat∩ hcal{L}(\L(G mathcal{G}^{ *}, \mat hcal{G}^{)*}) and the fact that ∗ exponential vectors span a dense subspace of G \mathcal{G} and G \mathcal{G}^{*} , we complete the proof. (ii) (ii) The The proof proof is is similar similar to to the the proof proof of of (i). (i). In In fact, fact, we we obtain obtain that that. )e\zeta, = \xi+ \eta\rangl ζ, ξ +e\phi_{\xi+\eta}=\l η φξ+η angl=e\zeta, ζ, ξ + eη φξ ⋄ φ η a(ζ)(φ ξ ⋄ φη ) = a(ζ)(φξ+ηangl a(\zeta)(\phi_{\xi}0\phi_{\eta})=a(\zeta)(\phi_{\xi+\eta})=\l \xi+\eta\rangl \phi_{\xi}0\phi_{\eta} = (a(ζ)φξ ) ⋄ φη + φξ ⋄ (a(ζ)φη ) . =(a(\zeta)\phi_{\xi})0\phi_{\eta}+\phi_{\xi}o(a(\zeta)\phi_{\eta}) .. \square . Let K L(H, H). K\in∈\mathcal{L}(H, H) . Then from Theorem 3.1, it holds that ∗ ∗ G)hcal{∩ Λ(K) \Lambda(K):=\Xi_:= {1,1}(K)\inΞ\mat1,1 hcal{(K) L}(\mathcal{G}∈ , \mathL(G, cal{G})\cap \mat L}(\matL(G hcal{G}^{*}, \mat,hG cal{G}^{). *}) .. The operator Λ(K) ∈ R and qq>0 > 0,, from conservation operator. operator. for any pp\in \mathbb{R} \Lambda(K) is called the conservation (3.4) (3.4) we we obtain obtain that that. −q ∗ K Φ|\|||Phip+q (3.8) |||\Vert Λ(K)Φ (3.8) |\Lambda(K)\Phi|||\pVert≤|_{p}\eleq e^{-q}\sqrtC{C_{1,1;q 11,q} \Vert. K\Vert|||\Vert \Vert|_{,p+q}, \Phi\inΦ\mat∈hcal{GG}^{*} .. 3.2. Multiplication Operators. Theorem 3.3 For any \Phi\in∈ \mathcalG{G}^{*∗} and any Φ and φ, \phi, ψ \psi\in∈ \mathcalG, {G} , it holds that Φφ, \langle\langle\Phi\phi, \psψ i\rangle\rangl= e=\langleΦ, \langle\Phi, \phi\φψ psi\rangle\rangle ... (3.9) (3.9). ) ∈ G ∗} and ∈ H, Proof. = (Fn\mathcal{G}^{* \xi, η\eta\in H , from PROOF. For For given given Φ \Phi=(F_{n})\in and any any ξ, from (2.1) (2.1) we we obtain obtain that that. ∞ ∞ ξ ⊗m l+k k ξ ⊗k ⊗ Fl+k ⊗ Φφξ = k m! l+m=n k=0. \Phi\phi_{\xi}=(\sum_{l+m=n}\sum_{k=0}^{\infty}(\begin{ar ay}{l l+ k k \end{ar ay})(F_{l+k^{\wedge} \otimes_{k}\xi^{\otimesk}) \otimes\frac{\xi^{\otimesm}{m!})_{n=0}^{\infty}. n=0. and. ∞ . . ∞ n n−m+k . . \langle\langle\Phi\phi_{\xi},\phi_{\eta}\rangle\rangle=\sum_{n=0}^{\infty} \{ sum_{m=0}^{n}\sum_{k=0}^{\infty}(\begin{ar ay}{l n-m+k k \end{ar ay})(F_{n-m+k^{\wedge}\otimes_{k}\xi^{\otimesk}) \otimes\frac{\xi^{\otimesm}{m!},\eta^{\otimesn}\ =\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\{ sum_{k=0}^{\infty}(\begin{ar ay}{l n+k k \end{ar ay})(F_{n+k^{\wedge}\otimes_{k}\xi^{\otimesk}) \otimes\frac{\xi^{\otimesm}{m!},\eta^{\otimes(n+m)}\ =e^{\langle\xi,\eta\rangle}\sum_{l=0}^{\infty}\{F_{l},\sum_{n+k=l} (\begin{ar ay}{l n+k k \end{ar ay})\eta^{\otimesn_{\otimes}^{\wedge}\xi^{\otimesk}\ ⊗k. . ξ ⊗m ⊗n ,η ⊗ m! . kξ Fn−m+k ⊗ k ξ ⊗m ⊗(n+m) n+k k ξ ⊗k ⊗ Fn+k ⊗ ,η = k m! m=0 n=0 k=0 ∞ n + k ⊗n ⊗k ξ, η η ⊗ξ Fl , =e k n+k=l l=0. Φφξ , φη =. n=0 m=0 k=0 ∞ ∞ ∞ . ξ, η = =e^{\langlee\xi,\eta\rangle}\langlΦ, e\langle\Phi, \phi_{φ\xi+\eξ+η ta}\rangle\rangle = Φ, φ φ . =\langle\langle\Phi, \phi_{\eta}\phi_η{\eta}\ranglηe\rangle .. Since the exponential vectors span a dense subspace of G, \mathcal{G} , by the continuity of the Wiener \square product product (see (see Theorems Theorems 2.3 2.3 and and 2.4), 2.4), the the proof proof is is immediate. immediate. ∗ Let Φ \Phi\in∈ \mathcalG{G}^{*} be given. Then we consider the Wiener multiplication operator M M_{\Phi} Φ : ∗ then from (3.9), φ, ψ ∈ G, we obtain that it holds that G\mathcal{→ G}ar ow \matG hcal{G}^{*} and \phi , \psi \ i n \mat h cal { G } and then from (3.9), , we obtain that it holds that. M \langle\langlΦe M_{φ, \Phi}\phi, ψ \psi\rangle\ra= ngle=\langlΦφ, e\langle\Phi\phi, \psiψ \rangle\rangl= e=\langlΦ, e\langle\Phi, \phiφψ \psi\rangle\rangl.e. 7.

(9) 36 Theorem ∈ H, 0, · · ·\mathcal{G} ) ∈ G as H, X ζ = (0, Theorem 3.4 3.4 ([30]) ([30]) For For each each ζ\zeta\in as aa Wiener Wiener multiplicamultiplica‐ X_{\zeta}=(0, \zeta,ζ,0, \cdots)\in ∗ (ζ), i.e, tion operator i.e, operator is represented as as the sum of of a(ζ) and aa^{*}(\zeta), a(\zeta) and ∗ X ζ = a(ζ) + a (ζ),, X_{\zeta}=a(\zeta)+a^{*}(\zeta). (3.10) (3.10). which is called called the quantum quantum decomposition decomposition of of X η. X_{\eta}. Proof. 0, · · ·\mathcal{G} ) ∈ G,, from ζ = (0, PROOF. Since Since X from (3.9) (3.9) we we obtain obtain that that X_{\zeta}=(0, \zeta,ζ,0, \cdots)\in ξ, η ξ, η X \langle X_ζ{\zφeta}\ξphi,_{\xiφ}, \phiη_{\eta}\r= angle=\lX angle X_{\ζzeta,}, \phiφ_{\xiξ} φ\phiη_{\eta}\ra= ngle=\laX ngle X_{\zeζta},, \phiφ_{\xiξ+η +\eta}\rangle eê{\langle\xi, \eta\ra= ngle}=\laζ, ngle\zeta, \xiξ+\eta+ \rangleη e^{\langlee\xi,\eta\rangle} = (a(ζ) + a∗ (ζ)) \phi_{\eta}\rangle, φξ , φ η , =\langle(a(\zeta)+a^{* }(\zeta))\phi_{\xi},. which gives the proof.. \square . For each tt\geq ≥ 0, 0 , put B t = X1[0,t] . Then {B t }t≥0 \{B_{t}\}_{t\geq 0} becomes a Brownian motion which is B_{t}=X_{1_{[0,t]}} called a realization of Brownian motion and so from Theorem 3.4 we have the following of quantum decomposition of Brownian motion: ∗ t ≥ 00.. BB_{t}=a(1_{[0,t]})+a^{*}(1_{[0,t]}) t = a(1[0,t] ) + a (1[0,t] ), , t\geq. (3.11) (3.11). ∗ Remark 3.5 The operators L(G, essen‐ \mathcal{L}(\mathcal{G},G \mathcal{G}^){*}) on admissible Gaussian functionals play an essential tial role role in in the the study study of of quantum quantum martingales martingales and and integral integral representations representations [11, [11, 14, 14, 16, 16, 17]. 17].. 4. Quantum White Noise Derivatives. In this section, we briefly briefly review some basic properties of quantum white noise derivatives [14, 15, 16, 17, 18, [14, 15, 16, 17, 18, 20]. 20].. 4.1. Annihilation and Creation Derivatives. ∗ For any admissible operator Ξ ∈ H, H , from Theorem 3.1 the commutators \Xi\in \mat∈hcal{L(G, L}(\mathcal{G}, \matGhcal{G}^{)*}) and ζ\zeta\in. [a(ζ), = a(ζ)Ξ − Ξa(ζ), −[a∗ (ζ), Ξ] = Ξa∗ (ζ) − a∗ (ζ)Ξ [a(\zeta),Ξ]\Xi]=a(\zeta)\Xi-\Xi a(\zeta) , -[a^{*}(\zeta), \Xi]=\Xi a^{*} (\zeta)-a^{*}(\zeta)\Xi ∗ are well defined defined as compositions of admissible operators, i.e., belong to L(G, \mathcal{L}(\mathcal{G},G \mathcal{G}^). {*}) . We define define + Dζ− Ξ = −[a∗ (ζ), \Xi]. Ξ]. D ζ Ξ = [a(ζ), Ξ], \Xi], D_{\zeta}^{-}\Xi=-[a^{*}(\zeta), D_{\zeta}^{+}\Xi=[a(\zeta),. --- , respectively. These are called the creation creation derivative derivative and annihilation annihilation derivative derivative of Ξ, Both together are referred to as the quantum white noise derivatives (qwn-derivatives quantum derivatives (qwn‐derivatives \Xi . By the definitions, for brevity) brevity) of Ξ. definitions, it is obvious that + ∗ ∗ ∗ ∗ = ([a(ζ), Ξ---*])^{*}=(a(\zeta)_{--}^{-*-*}---a(\zeta)) ^{*-}=--a^{*}(\zeta)-a^{*}(\zeta)_{-}^{-}]) = (a(ζ)Ξ∗ − Ξ∗ a(ζ))∗ = Ξa∗ (ζ) − a∗ (ζ)Ξ Dζ Ξ (D_{\zeta-}^{+-*}-)^{*}=([a(\zeta),. = Dζ−}Ξ. =D_{\zeta}^{\Xi .. (4.1) (4.1). \Xi is denoted by For each admissible operator Ξ \Xi\in \mat∈hcal{L}L(G (\mathcal{G}_{pp}, \mat, G hcal{Gq}_{q), }) , we operator norm of Ξ . Ξ \Vert\Xi\Vertp;q_{p,q}. ± Theorem ∈ HH be Theorem 4.1 4.1 ([14]) ([14]) Let Let ζ\zeta\in be given. given. Then Then D are continuous continuous linear linear operators operators D_{\zeta}ζ^{\pm} are ∗ from L(G, \mathcal{L}(\mathcal{G},G \mathcal{G}^){*}) itself.. 8.

(10) 37 Proof. PROOF.. obtain that. Suppose > 0,, by Suppose that that Ξ Then for for any any rr>0 by applying applying (3.4), (3.4), we we \Xi\in \mat∈hcal{LL(G }(\mathcal{G}_{pp}, \mat, G hcal{Gq}_{q). }) . Then + D Ξ . a(ζ)Ξ − Ξa(ζ) p−r;q+r \VertζD_{\zetap−r;q+r }^{+}\Xi\Vert_={p-r,q [a(ζ), +r}=\Vert[a(\zetΞ] a), \Xip−r;q+r ]\Vert_{p-r,q=+r}=\Vert. a(\zeta)\Xi-\Xi a(\zeta)\Vert_{p-r,q+r} ≤ a(ζ) Ξ \Vert_{p, Ξ a(ζ) q;q+r p;q q+ p;qq}\Vert p−r;p \leq\Vert a(\zeta)\Vert_{q, q+r}\Vert\Xi }+\Vert\Xi \Vert_{p,. a(\zeta)\Vert_{pr,p,},. + ∗ on L(G, which implies that D \mathcal{L}(\mathcal{G},G \mathcal{G}^). {*}) . Similarly, we see D_{\zetζa}^{+} is a continuous linear operator − ∗ \square is a continuous linear operator on L(G, G ). that D . \mathcal{L}(\mathcal{G}, \mathcal{G}^{*}) D_{\zetζa}^{-}. Proposition 4.2 For each ∈ HH and \Phi\in∈ \mathcalG{G}^{*∗} , it holds that each ζ\zeta\in and Φ − + a(ζ)Φ.. Dζ M}M_{\Phi Dζ MΦ φ}0)\phi=_{0}=(D_{\zeta}^{0 = Φ φ})\phi (D_{\zeta}^{+}M_{\Phi _{0}=a(\zeta)\Phi. PROOF. We obtain that Proof. + Dζ MΦ φ0 = (a(ζ)MΦ − MΦ a(ζ)) φ0 = a(ζ)Φ, (D_{\zeta}^{+}M_{\Phi})\phi_{0}=(a(\zeta)M_{\Phi}-M_{\Phi}a(\zeta))\phi_{0}= a(\zeta)\Phi, − Dζ MΦ φ0 = (MΦ a∗ (ζ)}(\zeta)-a^{* − a∗}(ζ)M − a∗ (ζ)Φ = a(ζ)Φ, Φ ) φ0 = ΦXζX_{\zeta}-a^{* (D_{\zeta}^{-}M_{\Phi})\phi_{0}=(M_{\Phi}a^{* (\zeta)M_{\Phi}) \phi_{0}=\Phi }(\zeta)\Phi=a(\zeta)\Phi,. ∗ where we used the quantum decomposition as ΦX ζ = Xζ Φ = (a(ζ) + a (ζ))Φ. \Phi X_{\zeta}=X_{\zeta}\Phi=(a(\zeta)+a^{*}(\zeta))\Phi.. 4.2. \square . Pointwise QWN-Derivatives QWN‐Derivatives. Let φ\phi=(f_{n})\in = (fn\mathcal{G} ) ∈ G and tt\in∈ R+ be given. Suppose that ff_{n}=0 finite number \mathbb{R}_{+} n = 0 except for a finite n . We define define of n. ∞ D t φ := (nf n (t, ·))n=1 , D_{t}\phi:=(nf_{n}(t, \cdot))_{n=1}^{\infty},. where ff_n{n}(t,(t,\cdot)\i ·) n ∈ classical stochastic gradient. The H^{\otiHmes(n-⊗(n−1) 1)}\wedge , and then DD is called the classical \nabla in classical classical stochastic stochastic gradient gradient is is denoted denoted by by ∇ in some some literatures literatures see see [8, [8, 16, 16, 18, 18, 22, 22, 29]. 29]. D to the space G We now extend the domain of D \mathcal{G∗}^{*} .. Lemma R and > 0 we \mathbb{R} Lemma 4.3 4.3 ([16]) ([16]) For For any any pp\in ∈ and rr>0 we have have Dφ 2L2 (R+ ,G−p−r ) = ||| Dφ(t) |||2−p−r dt ≤ K(p, r) ||| φ |||2−p ,. \Vert D\phi\Vert_{L^{2}(\mathb {R}_{+},\mathcal{G}_{-p-r}) ^{2}= \int_{\mathb {R}_{+} \Vert|D\phi(t)\Vert|_{-p-r}^{2}dt\leq K(p, r) \Vert|\phi\Vert|_{-p}^{2}, \phi\φin \m∈athcaG,l{G} , R+. (4.2) (4.2). where K(p, r) = supn (n + 1)e2p−2rn < ∞.. In particular, the classical classical stochastic gradient K(p, r)= \sup_{n}(n+1)e^{2p-2rn}<\infty 2 2 )∼ D = D:\mat:hcalG{G}−p _{-p}arrow→L^{2}(L \mathbb{(R R}_{++}, \mat, Ghcal−p−r {G}_{-p-r})\cong L^{2} L(\math(R bb{R}_+{+})\ot) im⊗es \matGhcal−p−r {G}_{-p-r}. (4.3) (4.3). is aa continuous continuous linear map. n Proof. (fn )n∞ ∈ G{G} consisting of continuous functions ff_{n}n on R PROOF. For each φ\phi== (f_{n})_{n=0}^{\i fty}\in \mathcal \mathb {R+ }_{+}^{n} , we n=0 ∞ (t, ·)) , where the right-hand side has a pointwise meaning. have Dφ(t) = ((n + 1)f right‐hand n+1 D\phi(t)=((n+1)f_{n+1}(t, \cdot))_{n=0}^{\infty} n=0 Then we obtain that ∞ ||| Dφ(t) |||2−p−r dt = n!e−2(p+r)n |(n + 1)fn+1 (t, ·)|20 dt. \int_{\mathb {R}_{+} \Vert|D\phi(t)\Vert|_{-p-r}^{2}dt=\sum_{n=0}^{\infty}n!e^ {-2(p+r)n}\int_{\mathb {R}_{+} |(n+1)f_{n+1}(t, \cdot)|_{0}^{2}dt = \sum_{n=0}^{\infty}(n+1)e^{2p-2rn}\cros (n+1)!e^{-2p(n+1)}|f_{n+1}|_{0}^{2} R+. R+. n=0. =. ∞ n=0. (n + 1)e2p−2rn × (n + 1)!e−2p(n+1) |fn+1 |20. ≤ φ |||_{-2−p \leqK(p, K(p, r)\Vert|r)\phi|||\Vert| p}^{2},, 9.

(11) 38 which which implies implies the the proof proof of of (4.2). (4.2). Put. \square . 2 2 L = proj lim L2 (R) ⊗ Gp , L^{2}(\(R, mathbb{R}G) , \mathcal:= {G}):= proj lim L (R, Gp ) ∼. \lim L^{2}(\mathb {R}, \mathcal{G}_{p})par ow\infty\cong \lim L^{2}(\mathb {R})\otimes \mathcal{G}_{p}par ow\infty,. p→∞. p→∞. L2 (R+ , G ∗ ) := ind lim L2 (R+ , G−p ) ∼ = ind lim L2 (R) ⊗ G−p .. L^{2}( \mathbb{R}_{+}, \mathcal{G}^{*}) :=ind\lim p→∞ L^{2}(\mathbb{R}_{+}, \mathcal{G}_{-p})par ow\infty\cong ind\lim_{ap→∞ r ow p\infty}L^{2}(\mathbb{R}) \otimes \mathcal{G}_{-p}.. D is a continuous linear map from Then by Lemma 4.3, the classical stochastic gradient D 2 ∗ into 2 ∗ . into and from G L G L \mathcal{G} \mathcal{G}^{*} L^{2}(\mat(R hbb{R}_+{+}, \mat, G) hcal{G}) L^{2}(\mat(R hbb{R}_+{+}, \mat, Ghcal{G}^{). *}) L^{2}2 -function We R+ . \mathbb{R}_{+}. We see see from from (4.3) (4.3) that that DΦ(t) D\Phi(t) has has aa meaning meaning as as G\mat−p−r ‐valued L ‐function in in tt\in∈ hcal{G}_{-p-r} -valued 2 Given ζ\zeta∈ L (R + ),, the linear map G \in L^{2}(\mat hbb{R}_{+}) imes\psi\rψ angle\rangle is continuous. Therefore \matp+r hcal{G}_{p∋ +r}\ni\ψ psi\mapst→ o\langlDΦ, e\langle D\Phi, ζ\zeta\ot⊗ there exists a unique Ψ G−p−r such that \Psi\in∈\mathcal{G}_{-p-r}. DΦ, \langle\langle D\Phiζ, \zet⊗ a\otimψ es\psi\rangl= e\ranglΨ, e=\langle\langlψ, e\Psi, \psi\rangle\ranglψe, \psi∈ \in \matGhcalp+r {G}_{p+r}.. It is reasonable to write. . \Psi=\int_{\mathbb{ζ(t)DΦ(t) R}_{+} \zeta(t)D\Phidt.(t)dt.. Ψ=. R+. As is easily seen, the Schwarz inequality holds: ≤ |ζ|0 ||| DΦ |||L2 (R+ ,G−p−r ) ,, ζ(t)DΦ(t) dt . \Vert|\int_{\mathb {R}_{+} \zeta(t)D\Phi(t)dt\Vert|_{-p-r}\leq|\zeta|_{0} \Vert|D\Phi\Vert|_{L^{2}(\mathb {R}_{+},\mathcal{G}_{-p-r}) R+. (4.4) (4.4). −p−r. which implies that the map. . \mathcal{G}_{-p}\ni\Phi\mapsto\int_{\mζ(t)DΦ(t) athb {R}_{+} \zeta(t)D\Phidt(t)dt\in∈ \mGathcal{G}_{-(p+r)}. G−p ∋ Φ →. −(p+r). R+. is continuous. On the other hand, for any ξ\xi\in ∈ H, H , we obtain that ζ(t)ξ(t)φξ dt = ζ, ξφξ ζ(t)at φξ dt = ζ(t)Dφξ (t) dt =. \int_{\mathb {R}_{+} \zeta(t)D\phi_{\xi}(t)dt=\int_{\mathb {R}_{+} \zeta(t) a_{t}\phi_{\xi}dt=\int_{\mathb {R}_{+} \zeta(t)\xi(t)\phi_{\xi}dt=\langle\zeta, \xi\rangle\phi_{\xi} R+. R+. R+. = a(ζ)φξ . =a(\zeta)\phi_{\xi}.. Therefore, we obtain that . \int_{\maζ(t)DΦ(t) thbb{R}_{+} \zeta(t)dtD\Phi=(t)dt=a(ζ)Φ, a(\zeta)\Phi, \Phi\in \maΦth∈cal{GG}^{*} ,, ∗. R+. (4.5) (4.5). see see [16]. [16]. D appeared in Aase– Remark 4.4 The space G \mathcal{G∗}^{*} as a domain of the classical gradient D Aase‐ Øksendal–Privault–Ubøe [1].. For o-Ub[1] For aa standard standard domain domain see see e.g., e.g., Kuo Kuo [22], [22], Malliavin Malliavin [27], [27], Nualart Nualart [29]. [29].. R.. Then > 0 and Let p, qq\in ∈ \mathbb{R} {G} , from Let Ξ for some some p, Then for for any any rr>0 and φ\phi\in∈\mathcalG, from (4.2) (4.2) we we \Xi\in \mat∈hcal{LL(G }(\mathcal{G}_{pp}, \mat, G hcal{Gq}_{q)}) for obtain that 2 2 ||| Ξ |||2p;q |||I I D ||| ΞDt φ |||q dt ≤ D_{tt}\φphi II_{|||pp}^dt {2}dt. \int_{\mathb {R}_{+} \Vert|\XiD_{t}\phi\Vert|_{q}^{2}dt\leq\int_{\mathb {R}_{ +} \VertI^{\Xi}I _{p,q}^{2} R+. R+. 2 φ |||1_{2p+r ≤ \leq K(−p, K(-p,r)\Vert 1\Xir) \Vert|||1_{Ξp,q}|||^{2p;q }11\phi|||\Vert p+r}^{2},,. 10.

(12) 39 which implies that. and so the map. . \int_{\mathb {R}_{+} \Vert|_{-}^{-} D_{t}\Vert|_{p+r,q}^{2}dt\leq K(-p, r) \Vert|_{-}^{-} \Vert|_{p,q}^{2}, R+. ||| ΞDt |||2p+r;q dt ≤ K(−p, r) ||| Ξ |||2p;q ,. 2 L(G \mathcal{L}(p\mat,hG cal{Gq}_{)p}, \mat∋hcal{Ξ G}_{q})− \ni\Xi→ \mapstoΞD \Xi D\in L^{2∈ } (\matLhbb{R}_(R {+}, \mat+hcal,{L}(L(G \mathcal{G}_p+r {p+r}, \mat,hcalG{Gq}_{q)}). is continuous. Similarly, we obtain that ||| Dt Ξφ |||2q−r dt ≤ K(−q, r) ||| Ξφ |||2q ≤ K(−q, r) ||| Ξ |||2p;q ||| φ |||2p ,. \int_{\mathb {R}_{+} \Vert|D_{t}\Xi\phi\Vert|_{q-r}^{2}dt\leq K(-q, r) \Vert|\Xi\phi\Vert|_{q}^{2}\leq K(-q, r)\Vert|\Xi\Vert|_{p,q}^{2} \Vert|\phi\Vert|_{p}^{2}, R+. which implies that. and so the map. . \int_{\m|||athbΞD{R}_{+} \V|||ert|\Xi D_{t}\Vdtert|_{p+≤r,q}^{K(−p, 2}dt\leq K(-p, r)\Vr)ert|\X|||i \ΞVert||||_{p,q}^{2},, t. R+. 2 p+r;q. 2 p;q. 2 L(G \mathcal{L}(p\mat,hGcal{Gq}_{)p}, \mat∋hcalΞ {G}_{q})− \ni\→ Xi\mapstoDΞ D\Xi\in L^{2∈ } (\matLhbb{R}(R _{+}, \mat+hcal,{L}L(G (\mathcal{G}p_{p,}, \matGhq−r cal{G}_{q-r)}). + is continuous. Therefore, the pointwise creation creation derivative derivative D defined by D_{t}t^{+} is defined + − \Xi\in ΞD\mathcal{L}(\mathcal{G}, Ξ ∈ L(G, G ∗ )}) D t, D_{t}^{+}\Xi=D_{t}\Xi-\Xi \mathcal{G} ^{* t Ξ = Dt ΞD_{t}, + ∗ L^{2}2 -function R+ .. Motivated and \mathbb{R}_{+} D_{t-}^{+-}and D is an an L(G, \mathcal{L}(\mathcal{G},G \mathcal{G}^)-valued {*}) ‐valued L ‐function in in tt\in∈ Motivated by by (4.1), (4.1), the the pointwise pointwise t Ξ is − annihilation annihilation derivative derivative D defined by D_{t}^{-} t is defined + ∗ ∗ − Dt^{*})^{*Ξ} \Xi\in \mat , hcal{L}(\mat Ξhcal∈{G},L(G, G ∗ ), D D_{tt}^{-}Ξ\Xi== (D_{t}^{+}\Xi \mathcal{G} ^{* }) ,. ∗ ∈ H, see H , from see [16, [16, 18]. 18]. In In fact, fact, for for given given Ξ \Xi\in \mat∈hcal{L(G, L}(\mathcal{G}, \matGhcal{G}^{)*}) and and ζ\zeta\in from (4.5) (4.5) we we obtain obtain that that ζ(t)Dt+ Ξdt = Dζ+ Ξ. \int_{\mathb {R}_{+} \zeta(t)D_{t}^{+}\Xi dt=D_{\zeta}^{+}\Xi R+. and. ∗ Dζ− Ξ = Dζ+ Ξ∗ =. . ∗ ζ(t) Dt+ Ξ∗ dt.. D_{\zeta}^{-} \Xi=(D_{\zeta}^{+}\Xi^{*})^{*}=\int_{\mathb {R}_{+} \zeta(t) (D_{t}^{+}\Xi^{*})^{*}dt. R+. Proposition 4.5 For each ≥ 00 and \Phi\in∈ \mathcalG{G}^{*∗} , it holds that each tt\geq and Φ + Dt MΦ φ0 = Dt− MΦ φ0 = Dt Φ. (D_{t}^{+}M_{\Phi})\phi_{0}=(D_{t}^{-}M_{\Phi})\phi_{0}=D_{t}\Phi.. PROOF. We obtain that Proof. + Dt MΦ φ0 = (Dt MΦ − MΦ Dt ) φ0 = Dt Φ, (D_{t}^{+}M_{\Phi})\phi_{0}=(D_{t}M_{\Phi}-M_{\Phi}D_{t})\phi_{0}=D_{t}\Phi, − ∗ ∗ + Dt MΦ φ0 = Dt+ MΦ∗ })^{*φ}\phi_{0}=(D_{t}^{+} M_{\Phi})^{* φ}0\phi_{0}=(M_{D_{t}\Phi})^{* = (MDt Φ )∗ φ}\phi_{0}=D_{t}\Phi, 0 = Dt MΦ 0 = Dt Φ, (D_{t}^{-}M_{\Phi})\phi_{0}=(D_{t}^{+}M_{\Phi}^{*. which gives the proof.. 11. \square .

(13) 40. 5. Anticipating Quantum Stochastic Integrals. \sigma ‐field generated by {Bs ; 0 ≤ s ≤ t}.. A one-parameter For each tt\geq ≥ 0, 0 , let F \mathcal{Ft}_{t} be the σ-field one‐parameter \{B_{s};0\leq s\leq t\} ∗ } ⊂ G is called a generalized stochastic process [4, 11, 31] family Φ = {Φ t 0}\subset t≥0 \mathcal{G}^{*} is called a generalized stochastic process [4, 11, 31] if family \Phi=\{\Phi_{t}\}_{t\geq if there there exists exists aa pp\geq ≥ 00 (independent of t ≥ 0) such that Φ ∈ G for all t ≥ 0 and the map t → ∈ G{G}_{t\geq 0 t\geq 0 −pp} (independent of ) such that \Phit_{t}\in \mathcal{−p and the map t\mapsto\PhiΦ_{t}\itn \mathcal G}_{-p} for all is Borel measurable on R t = (Ft;n )}t≥0 \mathbb{+ R}_{+} . A generalized stochastic process {Φ \{\Phi_{t}=(F_{t,n})\}_{t\geq 0} is said to ≥ 00 and ≥ 0, t]n . be \mathcal{F}t_{t} )) if 0, suppF t;n ⊂ [0,t]^{n}. be adapted adapted (w.r.t. (w.r. t. F if for for all all tt\geq and nn\geq suppF_{t,n}\subset[0, ∗ A one-parameter one‐parameter family {Ξ quantum stochastic process. \{\Xi_{t}\}t_{t\}in \matt∈R hbb{R}_{++} \subse⊂ t \mathcalL(G, {L}(\mathcal{G}, \matG hcal{G}^{ )*}) is called a quantum Our approach covers a wide class of classical and quantum stochastic processes in the ∗ ≥ 0, sense that G \mathcal{G∗}^{*} and L(G, 0 , we put \mathcal{L}(\mathcal{G},G \mathcal{G}^){*}) involve distributions. As examples, for each tt\geq. A ), A∗t = a∗ (1[0,t] ), Λt = Ξ1,1 (1[0,t] ). t = a(1[0,t], A_{t}^{*}=a^{*}(1_{[0,t]}) A_{t}=a(1_{[0,t]}) , \Lambda_{t}=\Xi_{1,1}(1_{[0, t]}) .. For the definition \Lambdat_{t} , the indicator function 11_{[0, definition of Λ [0,t] t]} is considered as a multiplication H , i.e., 11_{[0,t]}(\xi)=1_{[0,t]}\xi=:\xi_{[0,t]} (ξ) = 1 ξ =: ξ for any ξ\xi\in ∈ H. ≥ 0, operator on H, 0, H . Then for each tt\geq [0,t] [0,t] [0,t] ∗ ∗ ∗ ∗ } and {Λ } are called , A , Λ ∈ L(G, G) ∩ L(G , G ). The processes {A } , {A A A_{t}, t A_{t}^{* t {t}\in \mathcal{L}(\mathcal{G}, \mathcal{G})\cap \mathcal{L} (\mathcal{G}^{*}, \mathcal{G}^{*}) . t t≥00}, \{A_{t}^{*t}\}_{tt≥0 \{A_{t}\}_{t\geq \geq 0} \{\Lambda_{t t}\}t≥0 _{t\geq 0} t }, \Lambda_ the the annihilation, annihilation, creation creation and and conservation conservation (or (or gauge) gauge) processes, processes, respectively. respectively.. 5.1. Quantum Hitsuda–Skorohod Hitsuda‐Skorohod Integrals. In this section, we study the Hitsuda–Skorohod Hitsuda‐Skorohod type quantum stochastic integrals with their regular properties. 2 Theorem 5.1 Let p, R be p, q q\in∈ \mathbb{R} be given and and Ξ be aa quantum quantum stochastic \Xi\in∈ L^{2}(\matLhbb{R(R }_{+}, \mat+hcal,{L}(L(G \mathcal{G}_{pp}, ,\matGhcalq{G})) _{q}) be − (Ξ), process. Then there exists an admissible operator, denoted by δ Then exists an admissible operator, denoted by \delta^{-}(---) , in L(G \mathcal{L}(\matp+r hcal{G}_{p+r},, \matGhcal{qG}_{)q}) for r>0 any r > 0 such that any . \delta^{-}(\Xi)\phi=\int_{\mathΞ(t)(D b {R}_{+} \Xi(t)(Dφ)_{t}\phdti)dt. δ − (Ξ)φ =. (5.1) (5.1). t. R+. for any any φ \phi\i∈ n \mathG. cal{G}.. Proof. > 0,, by PROOF. For For any any φ\phi\in∈\mathcalG{G} and and rr>0 by applying applying (4.2), (4.2), we we obtain obtain that that Ξ(t)(Dt φ) dt ≤ ||| Ξ(t) |||p;q ||| Dt φ |||p dt R+ R+ q 1/2 1/2 2 2 ||| Ξ(t) |||p;q dt ||| Dt φ |||p dt ≤. \Vert|\int_{\mathb {R}_{+} \Xi(t)(D_{t}\phi)dt\Vert|_{q}\leq\int_{\mathb {R}_{ +} \Vert|\Xi(t)\Vert|_{p,q}\Vert|D_{t}\phi\Vert|_{p}dt \leq(\int_{\mathb {R}_{+} \Vert|\Xi(t)\Vert|_{p,q}^{2}dt)^{1/2} (\int_{\mathb {R}_{+} \Vert|D_{t}\phi\Vert|_{p}^{2}dt)^{1/2} \leq\sqrt{K(-p,r)}(\int_{\mathb {R}_{+} \Vert|\Xi(t)\Vert|_{p,q}^{2}dt)^{1/2} \Vert|\phi\Vert|_{p+r}^{2}, R+. ≤. K(−p, r). . R+. R+. ||| Ξ(t) |||2p;q. dt. 1/2. ||| φ |||2p+r ,. which implies that the linear operator . \mathcal{G}_{p+r}\ni\phi\mapsto\int_{\mathb Ξ(t)(D {R}_{+} \Xi(t)(D_{t}\pφ)hi)dt\indt \ma∈thcal{GG}_{q}. Gp+r ∋ φ −→. t. q. R+. is continuous.. \square . 2 − For (Ξ) satisfying For given given Ξ the admissible admissible operator operator δ\delta^{-}(---) satisfying (5.1) (5.1) is is - -\in L^{∈2}(\matL hbb{R}_{+(R }, \math+cal{L,}(\matL(G hcal{G}_{pp}, \mat, Ghcalq{G})), _{q}) , the \Xi , see called called the the annihilation annihilation integral integral of of Ξ, see [3, [3, 24, 24, 16, 16, 18]. 18].. 12.

(14) 41 41 2 Remark 5.2 Let p, R be given and Ξ p, qq\in∈ \mathbb{R} \Xi\in∈ L^{2}(\matLhbb{R(R }_{+}, \mat+hcal,{L}(L(G \mathcal{G}_{pp}, ,\matGhcalq{G})) _{q}) be a quantum stochastic process. Then for any ξ\xi\in ∈ H, H , we obtain that − ξ(t)Ξ(t)φξ dt = Ξ(t) dAt φξ , Ξ(t)(Dt φξ ) dt = δ (Ξ)φξ =. \delta^{-}(\Xi)\phi_{\xi}=\int_{\mathb {R}_{+} \Xi(t)(D_{t}\phi_{\xi})dt=\int_ {\mathb {R}_{+} \xi(t)\Xi(t)\phi_{\xi}dt=(\int_{\mathb {R}_{+} \Xi(t)dA_{t}) \phi_{\xi}, R+. R+. which implies that. . R+. \delta^{-}(\Xi)=\int_{\mathb Ξ(t) {R}_{+} \XidA (t)dA_{t}. δ − (Ξ) =. t. R+. − \Xi is adapted, then δ\del on a certain domain. Furthermore, if Ξ ta^{-(Ξ) }(\Xi) annihilation integral of Hudson-Parthasarathy. For the definition Hudson‐Parthasarathy. definition of of quantum quantum stochastic stochastic processes, processes, we we refer refer to to [11]. [11]. Also, Also, for for more more Hitsuda–Skorohod integrals, we refer to [3, 24, 18]. Hitsuda‐Skorohod integrals, we refer to [3, 24, 18].. coincides with the of the adaptedness study study on on quantum quantum. As for a criterion for δ\del−ta^{-(Ξ) }(\Xi) being a bounded operator on Γ(H), \Gamma(H) , we have the following corollary. A similar result can be found in [18]. corollary. A similar result can be found in [18]. (R,\matL(G Corollary 5.3 For any > 0 and L2hbb{R}, any rr>0 and Ξ annihilation integral \Xi\in∈L^{2}(\mat hcal{L}(\mat−rhcal,{G}_{-Γ(H))), r}, \Gamma(H))) , the annihilation − δ\delta^{-(Ξ) bounded operator operator on on Γ(H). }(\Xi) is aa bounded \Gamma(H) . Proof. PROOF. The proof is immediate from Theorem 5.1.. \square . 2 Theorem 5.4 Let p, R be p, q q\in∈ \mathbb{R} be given and and Ξ be aa quantum quantum stochastic \Xi\in∈ L^{2}(\matLhbb{R(R }_{+}, \mat+hcal,{L}(L(G \mathcal{G}_{pp}, ,\matGhcalq{G})) _{q}) be + (Ξ), for process. Then there exists an admissible operator, denoted by δ process. Then there exists an admissible operator, denoted by \delta^{+}(_{-}^{-}- ), in in L(G \mathcal{L}(\matphcal,{G}_G {p}, \matq−r hcal{G}_{q-)r}) for r>0 any r > 0 such that any + Ξ(t)φ, Dt ψ dt (5.2) δ (Ξ)φ, ψ = (5.2). \langle\langle\delta^{+}(\Xi)\phi,\psi\rangle\rangle=\int_{\mathb {R}_{+} \langle\langle\Xi(t)\phi,D_{t}\psi\rangle\rangle dt R+. for φ, \phi, ψ \psi\in∈ \mathG. cal{G}.. Proof. > 0,, by \phi, ψ \psi\in∈ \mathcalG{G} and PROOF. For For any any φ, and rr>0 by applying applying (4.2), (4.2), we we obtain obtain that that Ξ(t)φ, Dt ψ dt ≤ ||| Ξ(t)φ |||Ill q |||Ill D dt t ψ |||−q1_{-q}dt D_{t}\psi\Vert . |\int_{\mathb {R}_{+}\langle\langle\Xi(t)\phi, D_{t} \psi\rangle\rangle dt|\leq\int_{\mathb {R}_{+} \Vert 1^{\Xi(t)\phi} R+. R+. . 1/2 . 1/2. \leq(\int_{\mathb {R}_{+} \Vert|\Xi(t)\Vert|_{p,q}^{2}dt)^{1/2} (\int_{\mathb {R}_{+} \Vert|D_{t}\psi\Vert|_{-q}^{2}dt)^{1/2} \Vert|\phi\Vert|_{p} \leq\sqrt{K(qr)}(\int_{\mathb {R}_{+} \Vert|\Xi(t)\Vert|_{p,q}^{2}dt)^{1/2} \Vert|\phi\Vert|_{p}\Vert|\psi\Vert|_{-q+r},. ≤. ≤. R+. ||| Ξ(t) |||2p;q. K(q, r). . which implies that the bilinear form. R+. . dt. R+. ||| Ξ(t) |||2p;q dt. ||| Dt ψ. 1/2. |||2−q. dt. ||| φ |||p. ||| φ |||p ||| ψ |||−q+r ,. \mathcal{G}_{p}\cros \mathcal{G}_{-q+r}\ni(\phi, \psi)\mapsto\int_{\mathb {R} _{+} \Ξ(t)φ, langle\langle\Xi(t)\phi, DD_{t}\psi\ψrangle\rangdtle dt\in∈\mathbC{C}. Gp × G−q+r ∋ (φ, ψ) −→. t. R+. Ghp−r is continuous. Therefore, there exists a unique admissible operator δ\del+ta^{(Ξ) +}(\Xi)\in \mat∈hcal{LL(G }(\mathcal{Gp}_{p,}, \mat cal{G}_{p-r)}) \square such such that that (5.2) (5.2) holds. holds. 2 + For , Ghcalq{G})), (Ξ)) satisfying For given given Ξ the admissible admissible operator operator δ\delta^{+}(\Xi satisfying (5.2) (5.2) is is \Xi\in L^{∈2}(\matLhbb{R}_(R {+}, \mat+hcal{,L}(\L(G mathcal{G}_{pp}, \mat _{q}) , the \Xi called called the the creation creation integral integral of of Ξ,, see see [3, [3, 24, 24, 16, 16, 18]. 18]. + (Ξ) being a bounded operator on Γ(H), As for a criterion for δ\delta^{+}(---) \Gamma(H) , we have the following corollary. corollary. A A similar similar result result can can be be found found in in [18]. [18]. 13.

(15) 42 2 + Corollary 5.5 For any > 0 and (Ξ)) any rr>0 and Ξ creation integral δ\delta^{+}(\Xi \Xi\in∈L^{2}(L \mathbb{(R, R}, \mathL(Γ(H), cal{L}(\Gamma(H), \mathGcal{rG}_)), {r})) , the creation is aa bounded bounded operator operator on on Γ(H). \Gamma(H) .. Proof. PROOF. The proof is immediate from Theorem 5.4.. \square . Remark 5.6 The classical classical Hitsuda–Skorohod Hitsuda‐Skorohod integral δ\delta is defined defined as the adjoint map 2 ∗ D of the classical stochastic gradient D (see [8, 18, 22, 29]), i.e., of the classical stochastic gradient (see [8, 18, 22, 29]), i.e., for for given given Ψ \Psi\in∈L^{2}(L \mathbb{(R R}_{+}+, \mat,hG cal{G}^{), *}) , ∗ \Psi of Ψ is defined by the classical Hitsuda–Skorohod integral δ(Ψ) ∈ G Hitsuda‐Skorohod defined \delta(\Psi)\in \mathcal{G}^{*} δ(Ψ), φ = Ψ(t), Dt φ dt, φ ∈ G.. (5.3) (5.3). \langle\langle\delta(\Psi), \phi\rangle\rangle=\int_{\mathb {R}_{+} \langle\langle\Psi(t), D_{t}\phi\rangle\rangle dt, \phi\in \mathcal{G} R+. Therefore, = Ξ(t)φ,, from Therefore, by by denoting denoting (Ξφ)(t) (\Xi\phi)(t)=\Xi(t)\phi from (5.2) (5.2) we we have have + (Ξ)φ = δ (Ξφ) , , \phi\in \mathcal{G} φ ∈ G,, δ\delta^{+}(\Xi)\phi=\delta(\Xi\phi). (5.4) (5.4). see see [2, [2, 24, 24, 18]. 18]. The creation and annihilation integrals are related directly. The following corollary gives a relation between creation and annihilation integrals. 2 , Ghcalq{G})) Corollary R be p, qq\in ∈ \mathbb{R} Corollary 5.7 5.7 ([18]) ([18]) Let Let p, be given given and and Ξ be aa quantum quantum \Xi\in L^{∈2}(\matLhbb{R}_(R {+}, \mat+hcal{,L}(\L(G mathcal{G}_{pp}, \mat _{q}) be stochastic process. Then Then it holds that. (δ − (Ξ))∗ = δ + (Ξ∗ ). Proof. PROOF. For any φ, \phi, ψ \psi\in∈ \mathcalG, {G} , we obtain that − Ξ∗ (t)ψ, (Dt φ) dt Ξ(t)(Dt φ), ψ dt = δ (Ξ)φ, ψ = R+ R ++ ∗ = δ (Ξ )ψ, φ , =\langle\langle\delta^{+}(\Xi^{*})\psi, \phi\rangle\rangle,. (5.5) (5.5). \langle\langle\delta^{-}(\Xi)\phi, \psi\rangle\rangle=\int_{\mathb {R}_{+} \langle\langle\Xi(t)(D_{t}\phi), \psi\rangle\rangle dt=\int_{\mathb {R}_{+} \langle\{\Xi^{*}(t)\psi, (D_{t}\phi)\rangle\rangle dt. which which proves proves (5.5). (5.5).. \square . ∞ Theorem 5.8 Let p, R be p, qq\in∈ \mathbb{R} be given and and Ξ be aa quantum quantum stochastic \Xi\in ∈ L^{\inftyL }(\mathbb{R(R }_{+}, \mat+hcal,{L}(\L(G mathcal{G}_{pp}, \mat, hGcal{Gq}_{)) q}) ) be 0 (Ξ), in L(G process. Then there exists an admissible operator, denoted by δ Then exists an admissible operator, denoted by \delta^{0}(\Xi) , \mathcal{L}(\matp+r hcal{G}_{,p+r}G , \matq−r hcal{G}_{q-)r}) for r>0 any r > 0 such that any 0 Ξ(t)Dt φ, Dt ψ dt (5.6) δ (Ξ)φ, ψ = (5.6). \langle\langle\delta^{0}(\Xi)\phi, \psi\rangle\rangle=\int_{\mathb {R}_{+} \langle\langle\Xi(t)D_{t}\phi, D_{t}\psi\rangle\rangle dt R+. for φ, \phi, ψ \psi\in∈ \mathG. cal{G}. Proof. PROOF. 5.4.. The proof is a simple modification modification of the proofs of Theorems 5.1 and \square . 2 0 , Ghcalq{G})), For For given given Ξ the admissible admissible operator operator δ\delta(Ξ) ^{0}(\Xi) satisfying satisfying (5.6) (5.6) is is \Xi\in L^{∈2}(\matL hbb{R}_(R {+}, \math+cal{L,}(\L(G mathcal{G}_{pp}, \mat _{q}) , the \Xi called the conservation integral of Ξ, see [3, 24, 16, 18]. called the conservation integral of , see [3, 24, 16, 18]. As for a criterion for δ\del0ta(Ξ) ^{0}(\Xi) being a bounded operator on Γ(H), \Gamma(H) , we have the following corollary. A similar result can be found in [18]. corollary. A similar result can be found in [18]. 2 Corollary 5.9 For any > 0 and any rr>0 and Ξ conservation integral \Xi\in L^{∈2}(\matLhbb{R},(R, \mathcal{LL(G }(\mathcal{G−r }_{-r}, \mat, Ghcalr{G})), _{r}) , the conservation 0 is a bounded operator on Γ(H). δ\delta(Ξ) a bounded operator on . ^{0}(\Xi) \Gamma(H). Proof. PROOF. The proof is immediate from Theorem 5.8. 14. \square .

(16) 43 5.2. Extensions of Anticipating Quantum Stochastic Integrals. In In this this section, section, motivated motivated by by the the results results in in [23], [23], we we discuss discuss extensions extensions of of the the quantum quantum Hitsuda-Skorohod integrals studied in Section 5.1. Based on the quantum white noise Hitsuda‐Skorohod calculus calculus [12], [12], we we have have the the following following integral integral representations: representations: t t t as ds, A∗t = a∗s ds, Λt = a∗s as ds, At =. A_{t}= \int_{0}^{t}a_{s}ds, A_{t}^{*}=\int_{0}^{t}a_{s}^{*}ds, \Lambda_{t}= \int_{0}^{t}a_{s}^{*}a_{s}ds, 0. 0. 0. aa_{t}∗t^{*}. are the pointwisely defined where aa_{t}t and defined annihilation and creation operators. On the other hand, the pointwisely defined annihilation operator aa_{t}t and the stochastic gradient defined D D_{t}t coincide on a certain domain. Hence, the following informal computations gives motimoti‐ vations for extensions of the quantum Hitsuda-Skorohod Hitsuda‐Skorohod integrals: for a given quantum ∗ stochastic process {Ξ \Xi_{tt} , we may write as \{\Xi_{t}\t}_{}t\geqt≥0 0}\subset \mat⊂hcal{LL(G, }(\mathcal{G}, \matGhcal{G}^{)*}) of enough regular operators Ξ t t Ξs dAs = Ξs Ds ds = δ − (1[0,t] Ξ), 0 t t t t0 t (dAs ) Ξs = Ds Ξs ds = Ξs Ds ds + Ds+ Ξs ds = δ − (1[0,t] Ξ) + Ds+ Ξs ds, 0 0 0 0 t 0 t ∗ ∗ + (dAs ) Ξs = Ds Ξs ds = δ (1[0,t] Ξ), 0 0 t t t Ξs dA∗s = Ξs Ds∗ ds = δ + (1[0,t] Ξ) + Ds− Ξs ds, 0 0 0 t t Ξs dΛs = Ξs Ds∗ Ds ds = δ 0 (1[0,t] Ξ) + δ − (1[0,t] D·− Ξ), 0 0 t t (dΛs ) Ξs = Ds∗ Ds Ξs ds = δ 0 (1[0,t] Ξ) + δ + (1[0,t] D·+ Ξ).. (5.7) (5.7). \int_{0}^{t}\Xi_{s}dA_{S}=\int_{0}^{t}\Xi_{s}D_{s}ds=\delta^{-}(1_{[0,t]}\Xi) \int_{0}^{t}(dA_{s})\Xi_{s}=\int_{0}^{t}D_{s}\Xi_{s}ds=\int_{0}^{t}\Xi_{s} D_{s}ds+\int_{0}^{t}D_{s}^{+}\Xi_{s}ds=\delta^{-}(1_{[0,t]}\Xi)+\int_{0}^{t} D_{s}^{+}\Xi_{s}ds, \int_{0}^{t}(dA_{s}^{*})\Xi_{s}=\int_{0}^{t}D_{s}^{*}\Xi_{s}ds=\delta^{+} (1_{[0,t]}\Xi) \int_{0}^{t}\Xi_{s}dA_{\mathcal{S} ^{*}=\int_{0}^{t}\Xi_{s}D_{s}^{*}ds=\delta^ {+}(1_{[0,t]}\Xi)+\int_{0}^{t}D_{s}^{-}\Xi_{s}ds, \int_{0}^{t}\Xi_{s}d\Lambda_{s}=\int_{0}^{t}\Xi_{s}D_{s}^{*}D_{s}ds=\delta^{0} (1_{[0,t]}\Xi)+\delta^{-}(1_{[0,t]}D^{-}\Xi) \int_{0}^{t}(d\Lambda_{S})\Xi_{s}=\int_{0}^{t}D_{s}^{*}D_{s}\Xi_{S}ds= \delta^{0}(1_{[0,t]}\Xi)+\delta^{+}(1_{[0,t]}D^{+}\Xi) ,. ,. ,. 0. 0. ± However, D D_{t-tt}^{\Ξpm-t}- has no meaning directly. For example, we consider the annihilation process A t = a(1[0,t] ) and then A_{t}=a(1_{[0,t]}) − D t As = 1[0,s] (t). D_{t}^{-}A_{s}=1_{[0,s]}(t) .. But the annihilation process A A_{t}t can be defined defined as a(1 [0,t) ) and then we would have a(1_{[0,t)}) − − A = 1 (t). Therefore, D A cannot be defined D s t cannot be defined in [0,s) . Therefore, D_{t}^{-}A_{t} in aa unique unique way way [23]. [23]. From From t t D_{t}^{-}A_{s}=1_{[0,s)}(t) the above example, if we deal with quantum stochastic processes, then it is natural to ± ± consider two kinds of pointwisely defined defined annihilation derivative, D D_{t-}^t− {\pm} . Let D_{t+}^t+ {\pm} and D ∗ {Ξ define \{\Xi_{t}\t}_{}s\geqs≥0 0}\subset \mat⊂hcal{LL(G, }(\mathcal{G}, \matGhcal{G}^{)*}) be a quantum stochastic process. We define ± Ξt = lim Ds± Ξt , Dt+. ±. ±. D_{t^{-} ^{\pm-}+^{-t}= \lim_s↓t{s\downarrow t}D_{s-t}^{\pm-}-, D_{t- t}^{D\pm-}t−-=1 \dotΞt{ \im=ath} mlim D_s↑t{s-t}^{\pm-D}s\suparΞrowtt,-,. if the limits exist. ∗ Definition Definition 5.10 Let {Ξ \{\Xi_{t}\t}_{}t\geqt≥0 0}\subset \mat⊂hcal{LL(G, }(\mathcal{G}, \matGhcal{G}^{)*}) be a quantum stochastic process. − + (1) \mathbb{+ R}_{+} .. Then (1) Suppose Suppose that that δ\delta^{+(Ξ) }(_{-}^{-}- ) exists, exists, and and D exists and and it it is is integrable integrable on on R Then we we D_{t+-t}^{--}t+ Ξt exists define define − Dt+ Ξt dA∗t+ = δ + (Ξ) + Ξt dt.. (5.8) (5.8). \int_{\mathb {R}_{+} -t =\delta^{+} ( - )+\int_{\mathb {R}_{+} D_{+^{-t} ^{- }dt ^{-} R+. R+. 15.

(17) 44 − + (2) (Ξ)) exists, \mathbb{+ R}_{+} .. Then D_{t-}^{-}\Xi_{t} (2) Suppose Suppose that that δ\delta^{+}(\Xi exists, and and D exists and and it it is is integrable integrable on on R Then we we t− Ξt exists define define − Dt− Ξt dt.. (5.9) Ξt dA∗t− = δ + (Ξ) + (5.9). \int_{\mathb {R}_{+} \Xi_{t}dA_{t-}^{*}=\delta^{+}(\Xi)+\int_{\mathb {R}_{+} D_{t-}^{-}\Xi_{t}dt R+. R+. − − + (Ξ)) exists, (3) \mathbb{+R}_{+.}. D_{t-}^{-}\Xi_{t} (3) Suppose Suppose that that δ\delta^{+}(\Xi exists, and and D exist as as integrable integrable functions functions on on R t+ Ξt , D t− Ξt exist D_{t+}^{-}\Xi_{t}, define Then we define ∗ + − − Ξt ◦ dAt = δ (Ξ) + α1 <α> Dt+ Ξt dt + α2 Dt− Ξt dt (5.10) (5.10). < \alpha>\int_{\mathb {R}_{+} \Xi_{t}\circ dA_{t}^{*}=\delta^{+}(\Xi)+ \alpha_{1}\int_{\mathb {R}_{+} D_{t+}^{-}\Xi_{t}dt+\alpha_{2}\int_{\mathb {R}_{+ } D_{t-}^{-}\Xi_{t}dt R+. R+. R+. 2 <\alpha> for α = pha_{ (α1}, 1\al,pha_{α22})\i)n \mat∈hbb{R α >-creation ‐creation integral. \alpha=(\al R}^{2} , which is called the < 2 Theorem 5.11 Let p, ∈ R be p, qq\in \mathbb{R} be given and and let Ξ be aa quantum quantum \Xi\in L^{∈2}(\mathLbb{R}_{+(R }, \math+cal{L,}(\matL(G hcal{G}_{pp}, \mat, Ghcalq{G})) _{q}) be stochastic process. − − 1 (1) Ξ_{t}t exists for some > 0.. Then (1) Suppose Suppose that that D exists and and D some rr>0 Then \in L^{L 1}(\math(R bb{R}_{+}+, \mat,hcalL(G {L}(\mathcalp{G,}_{pG }, \matq−r hcal{G}_{)) q-r}) for ·+ Ξ· ∈ D_{t+}^{D_{+}^{-}\Xi. t+}\Xi ∗ exists as an operator in L(G , G ). the integral \inRt_{\+mathbb{ΞR}_{t+}dA exists as an operator . p q−r \mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{q-r}) \Xi_{t}dA_t+ {t+}^{*} − − 2 (2) for some > 0.. Then D_{-}^{-}\Xi. D_{t-}^{-}\Xi_{t} (2) Suppose Suppose that that D exists and and D some rr>0 Then \in L^{L 2}(\math(R bb{R}_{+}+, \mat,hcalL(G {L}(\mathcalp{G,}_{pG }, \matq−r hcal{G}_{)) q-r}) for ·− Ξ· ∈ t− Ξt exists ∗ exists as an operator in L(G , G ). the integral \inRt_{\+mathbb{ΞR}_t{+}dA exists as an operator . p q−r \mat h cal { L } ( \ mat h cal { G } _ { p } , \mat h cal { G } _ { q r } ) t− \Xi_{t}dA_{t-}^{*}. 2 Proof. PROOF. (1) Since Ξ Hitsuda‐ \Xi\in ∈ L^{2}(\matLhbb{R(R }_{+}, \mat+hcal,{L}(L(G \mathcal{G}_{pp}, ,\matGhcalq{G})), _{q}) , by Theorem 5.4, the quantum Hitsuda+ \Xi exists as an admissible operator in L(G (Ξ)) of Ξ Skorohod creation integral δ\delta^{+}(\Xi \mathcal{L}(\matphcal,{G}_G {p}, \matq−s hcal{G}_{q-)s}) for − − 1 Ξ_{t}t exists and D any ss>0 > 0.. Also, since, by assumption, D \in L^{1L }(\mathbb{(R R}_{+}+, \mat,hcalL(G {L}(\mathcalp{G},_{pG }, \matq−r hcal{G}_{)) q-r}) ·+ Ξ· ∈ D_{t+}^{D_{+}^{-}\Xi. t+}\Xi for some rr>0 > 0,, for any φ\phi\in∈\mathcalG{G} we obtain that − − Dt+ Ξt φ dt ≤ Dt+ Ξt p;q−r dt φ p , . \Vert\int_{\mathb {R}_{+} D_{t+}^{-}\Xi_{t}\phi dt\Vert_{q-r} \leq(\int_{\mathb {R}_{+} \Vert D_{t+^{-t} ^{- }-\Vert_{p,q-r}dt)\Vert\phi\Vert_ {p}, R+. q−r. R+. . − which implies that the integral \iRnt_{+\matD \mathcal{L}(\matphcal,{G}_G {p}, \matq−r hcal{G}_{q-). r}) . t+ h bb{ R }_{+} ΞD_{t+t-t}^dt {- }-dt exists as an admissible operator in L(G ∗ Finally, the integral \inRt_{\+mathbb{ΞR}_{t+}dA exists as an operator in L(G , G ). . p q−r \mat h cal { L } ( \ mat h cal { G } _ { p } , \mat h cal { G } _ { q r } ) \Xi_{t}dA_t+ {t+}^{*} \square (2) (2) The The proof proof is is similar similar to to the the proof proof of of (1). (1).. By similar arguments used in Definition Definition 5.10, we can define define the quantum stochastic integrals integrals of of types types given given as as in in (5.7) (5.7) of of which which the the study study will will be be appear appear in in some some other other papers. papers. Acknowledgements This work was supported by Basic Science Research Program through through the the NRF NRF funded funded by by the the MEST MEST (NRF-2016R1D1A1B01008782). (NRF‐2016R1D1A1B01008782).. References \emptyset ksendal, N. [1] Ub\emptyset e : White [1] K. K. Aase, Aase, B. B. Øksendal, N. Privault Privault and and J. J. Ubøe: White noise noise generalizations generalizations of of the Clark–Haussmann–Ocone finance, FiClark−Haussmann−Ocone theorem with application application to mathematical finance, Fi‐ nance nance Stochast. Stochast. 4 4 (2000), (2000), 465–496. 465‐496.. [2] [2] S. S. Attal Attal and and J. J. M. M. Lindsay: Lindsay: Quantum Quantum stochastic stochastic calculus calculus with with maximal maximal operator operator domains, Ann. Probab. 32 (2004), 488–529. domains, Ann. Probab. 32 (2004), 488‐529. 16.

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