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Stochastic heat equation on algebra of generalized functions. (English) Zbl 1278.60106

This paper treats an extension operator \(\Delta_G^{\mathrm{ext}}(K)\) of the Volterra-Gross Laplacian on the nuclear algebra \({\mathcal F}_{\theta}^*(N')\) of generalized functions on \(N'\). As is well-known now, surprisingly, without using the renormalization procedure, this extension \(\Delta_G^{\mathrm{ext}}(K)\) provides a continuous nuclear realization of the square white noise Lie algebra obtained by L. Accardi et al. [Commun.Math.Phys.228, No. 1, 123–150 (2002; Zbl 1006.60064)].
As a matter of fact, for an infinite-dimensional real separable Hilbert space \(H\) with inner product \(\langle \cdot, \cdot \rangle\) and norm \(| \cdot |_0\), and for an operator \(A\) on \(H\) such that \(A e_n = \lambda_n e_n\) \(\forall n\), where \(( e_n)\) is an orthonormal basis of \(H\) and \(\sum_{n=0}^{\infty} \lambda_n^{-2} < \infty\), since \(A^{-1}\) is of Hilbert-Schmidt type, we have the real standard nuclear Gel’fand triple \[ X := \projlim_{p \to \infty} X_p \subset H \subset \operatorname{ind lim}_{p \to \infty} X_{-p} =: X' \tag{1} \] with the Hilbert space \(X_p\) consisting of all \(\xi \in H\) with \(| \xi |_p = | A^p \xi |_0 < \infty\). Then \(N \subset {\mathcal H} \subset N'\) is a counter-part of the complexifications of \(X, H\) and \(X'\) in white noise analysis. Indeed, for a Young function \(\theta\), \({\mathcal F}_{\theta}(N')\) denotes the space of holomorphic functions on \(N'\) with exponential growth of order \(\theta\) and of minimal type, and \({\mathcal G}_{\theta}(N)\) denotes the space of holomorphic functions on \(N\) with exponential growth of order \(\theta\) and of arbitrary type. Furthermore, \({\mathcal F}_{\theta}(N')\) is the space of test functions on \(N'\) and \({\mathcal F}_{\theta}^*(N')\) is its topological dual space equipped with strong topology, called the space of generalized functions on \(N'\), (cf., e.g., R. Gannoun et al. [J.Funct.Anal.171, No. 1, 1–14, (2000; Zbl 0969.46018)]). Note that \[ \| f \|_{\theta, p, m} := \sup_{x \in N_p} | f(x) | e^{- \theta ( m | x |_p)}, \tag{2} \] \(\tau(K)\) is a symmetric kernel function, and \(\tau(K) \in ( N \otimes N)'\) with \(K \in {\mathcal L}(N, N')\), having \(\langle \tau(K), \xi \otimes \eta \rangle=\langle K \xi, \eta \rangle\) for \(\xi, \eta \in N\). \({\mathcal L}\) is the Laplace transform of a distribution \(\Phi \in {\mathcal F}_{\theta}^* ( N')\) defined by \(( {\mathcal L} \Phi )(\xi) \equiv \hat{\Phi}(\xi):= \langle\langle \Phi, e_{\xi} \rangle\rangle\) for \(\xi \in N\), with \(e_{\xi}(z)=e^{ \langle z, \xi \rangle}\), \(z \in N'\). In fact, \(\Delta_G^{\mathrm{ext}}(K)\) is a continuous linear operator from \({\mathcal F}_{\theta}^*(N')\) into itself, and for any generalized function \(\Phi \sim ( \Phi_n)_{n=0}^{\infty}\) in \({\mathcal F}_{\theta}^* (N')\), \[ \Delta_G^{\mathrm{ext}}(K) \Phi \sim \{ (n+2) (n+1) \tau(K) \hat{\otimes}_2 \Phi_{n+2} \}_{n \geq 0}. \tag{3} \] Moreover, there exist \(q >0\) and \(m' > 0\) such that, for any \(m' > m > 0\) and \(p > q\), the estimate \[ \| \Delta_G^{\mathrm{ext}}(K) \Phi \|_{\theta, -p, m} \leqslant \rho | \tau(K) |_p \cdot \| \Phi \|_{ \theta, -q, m'} \tag{4} \] holds for some constant \(\rho > 0\). Let \({\mathcal P}_{t K}\) \(=\) \(e^{ \frac{t}{2} \Delta_G^{\mathrm{ext}}(K)}\), \(t \in {\mathbb R}\), be a strongly continuous one-parameter group of continuous linear operators from \({\mathcal F}_{\theta}^*( N')\) into itself with infinitesimal generator \(\frac{1}{2} \Delta_G^{\mathrm{ext}}(K)\). Actually, \[ {\mathcal P}_{tK} \Phi \sim \left( \sum_{\ell =0}^{\infty} \frac{ (n + 2 \ell)! t^{\ell} }{ n! \ell! 2^{\ell} } \tau(K)^{\otimes \ell} \hat{\otimes}_{2 \ell} \Phi_{n + 2 \ell} \right)_{n \geq 0} \tag{5} \] holds for any \(\Phi \in {\mathcal F}_{\theta}^* ( N')\). Then \(U_t = {\mathcal P}_{t K} \Phi\) provides a unique solution in \({\mathcal F}_{\theta}^*( N')\) to the heat equation \[ \frac{\partial U}{\partial t} = \frac{1}{2} \Delta_G^{\mathrm{ext}}(K) U \qquad \text{with } U(0) = \Phi \in {\mathcal F}_{\theta}^*( N'). \tag{6} \] The authors study a probabilistic representation of the solution \(U_t\) to the heat equation (6). In what follows, \(K\) is assumed to be a symmetric, non-negative linear operator with finite trace. For the translation operator \(t_{- \eta}\) (for \(\eta \in N\)) on \({\mathcal G}_{\theta^*}(N)\), the translation operator \(T_{- \eta}\) on \({\mathcal F}_{\theta}^*(N')\) is defined by \(T_{- \eta} \Phi := ( {\mathcal L}^{-1} t_{- \eta} {\mathcal L} ) \Phi\), as a linear continuous operator from \({\mathcal F}_{\theta}^*(N')\) into itself. By virtue of the theory of stochastic integration in Hilbert space developed in (e.g., [G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press. (1992; Zbl 0761.60052)]), a stochastic integrals of \({\mathcal F}_{\theta}^*(N')\)-valued processes can be well defined mathematically. For a \(K\)-Wiener process \(W = ( W(t))\), \(t \in [0, T]\), on a filtered probability space \(( \Omega, {\mathcal F}, ({\mathcal F}_t)_{t \in [0, t]}, \operatorname{P})\), let \(Z(t)\), \(t \in [0, T]\), be a \({\mathcal F}_{\theta}^*(N')\)-valued continuous \({\mathcal F}_t\)-semimartingale of the form \[ Z(t) = Z(0) + \int_0^t \Phi(s) d W(s) + \int_0^t \Psi(s)\, ds \qquad \text{for } \Phi, \Psi \in {\mathcal F}_{\theta}^*(N'). \tag{7} \] The Itō formula (cf. Theorem 4.3) for generalized functions reads \(T_{- W(t)} \Phi\) is a \({\mathcal F}_{\theta}^*(N')\)-valued continuous \({\mathcal F}_t\)-semimartingale having the decomposition \[ T_{- W(t)} \Phi = T_{- W(0)} \Phi + \sum_{j=0}^{\infty} \int_0^t \partial_{e_j} ( T_{- W(s)} \Phi) d W(s) + \frac{1}{2} \int_0^t \Delta_G^{\mathrm{ext}}(K) ( T_{- W(s)} \Phi)\, ds, \tag{8} \] where \(\{ \partial_{e_j} ; j \in {\mathbb N} \}\) is a family of derivations on the nuclear algebra \({\mathcal F}_{\theta}^*(N')\). Here are the main results.
{Theorem A.} The solution of the Cauchy problem (6) is given by \[ U_t = \operatorname{E}_x ( T_{- W(t)} \Phi), \tag{9} \] where \(\operatorname{P}^x\) is a probability law of \(W(t)\) starting at \(W(0) = x \in X_p\) and \(\operatorname{E}_x\) is the expectation with respect to \(\operatorname{P}^x\).
{Theorem B.} Let \(K \in {\mathcal L}(N', N)\) and \(\Phi \in {\mathcal F}_{\theta}^*(N')\). Then \(G= G_K \Phi\) is a solution of the equation \[ \left( \lambda I - \frac{1}{2} \Delta_G^{\mathrm{ext}}(K) \right) G = \Phi. \tag{10} \]
The above theorem asserts that the solution of the generalized Poisson equation (10) associated with the extended-Gross Laplacian \(\Delta_G^{\mathrm{ext}}(K)\) is represented by the \(\lambda\)-potential related to the solution of (6), where a functional \(G_K \Phi\) : \({\mathcal F}_{\theta}(N')\to {\mathbb C}\) is defined by \[ \langle\langle G_K \Phi, \varphi \rangle\rangle := \int_0^{\infty} e^{- \lambda t} \langle\langle \operatorname{E}_x( T_{- W(t)} \Phi), \varphi \rangle\rangle\, dt. \tag{11} \] Note that \(G_K \Phi\) allows the representation: \[ \Delta_G^{\mathrm{ext}}(K) G_K \Phi = \int_0^{\infty} e^{- \lambda t} \Delta_G^{\mathrm{ext}}(K) \operatorname{E}_x [ T_{- W(t)} \Phi]\, dt. \tag{12} \]

MSC:

60H40 White noise theory
46F25 Distributions on infinite-dimensional spaces
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[1] DOI: 10.1142/S0219025706002329 · Zbl 1097.60065 · doi:10.1142/S0219025706002329
[2] Accardi L., Commun. Math. Phys. 228 pp 123150–
[3] DOI: 10.1142/S0219025709003513 · Zbl 1172.60019 · doi:10.1142/S0219025709003513
[4] L. Accardi and O. G. Smolyanov, Mathematical Approach to Fluctuations II (World Scientific, Singapore, 1995) pp. 31–47.
[5] Accardi L., Dokl. Akad. Nauk SSSR 350 pp 5–
[6] Barhoumi A., Soochow J. Math. 32 pp 113–
[7] Barhoumi A., Quantum Prob. White Noise Anal. 25 pp 267–
[8] Ben Chrouda M., Soochow J. Math. 28 pp 375–
[9] Chung D. M., Nagoya Math. J. 147 pp 1–
[10] DOI: 10.1142/S0219025799000072 · Zbl 0936.46033 · doi:10.1142/S0219025799000072
[11] DOI: 10.1017/CBO9780511666223 · doi:10.1017/CBO9780511666223
[12] Dôku I., Stoch. Anal. Infinite Dimens. Spaces pp 60–
[13] Erraoui M., Quantum Probab. White Noise Anal. 25 pp 230–
[14] Erraoui M., Random Oper. Stoch. Equation 1 pp 1–
[15] DOI: 10.1006/jfan.1999.3518 · Zbl 0969.46018 · doi:10.1006/jfan.1999.3518
[16] Gel’fand I. M., Generalized Functions 4 (1964)
[17] DOI: 10.1007/978-94-017-3680-0 · doi:10.1007/978-94-017-3680-0
[18] DOI: 10.1007/s10440-008-9273-8 · Zbl 1178.60050 · doi:10.1007/s10440-008-9273-8
[19] DOI: 10.1007/s10440-008-9401-5 · Zbl 1206.60066 · doi:10.1007/s10440-008-9401-5
[20] DOI: 10.1142/S0219025704001487 · Zbl 1079.60065 · doi:10.1142/S0219025704001487
[21] DOI: 10.1142/S0219025702000912 · Zbl 1050.60070 · doi:10.1142/S0219025702000912
[22] DOI: 10.1142/S0219025707002713 · Zbl 1118.60057 · doi:10.1142/S0219025707002713
[23] Kang S. J., Soochow J. Math. 20 pp 45–
[24] DOI: 10.1016/0022-1236(76)90029-X · Zbl 0375.60087 · doi:10.1016/0022-1236(76)90029-X
[25] DOI: 10.1016/0022-1236(90)90028-J · Zbl 0749.46029 · doi:10.1016/0022-1236(90)90028-J
[26] DOI: 10.1142/S0219025702000882 · Zbl 1050.60071 · doi:10.1142/S0219025702000882
[27] Kuo H.-H., White Noise Distrubution Theory (1996)
[28] DOI: 10.1090/S0002-9947-1981-0632538-9 · doi:10.1090/S0002-9947-1981-0632538-9
[29] Lévy P., Lecons d’Analyse Fonctionnelle (1922)
[30] DOI: 10.1515/9783110845563 · Zbl 0503.60054 · doi:10.1515/9783110845563
[31] Obata N., Lecture Notes in Math 1577, in: White Noise Calculus and Fock Space (1994) · Zbl 0814.60058 · doi:10.1007/BFb0073952
[32] Obata N., RIMS Kokyuroku 1278 pp 130–
[33] DOI: 10.1090/S0002-9947-1974-0350231-3 · doi:10.1090/S0002-9947-1974-0350231-3
[34] DOI: 10.1007/BF02829609 · Zbl 1052.35065 · doi:10.1007/BF02829609
[35] Saitô K., Nagoya Math. J. 108 pp 67–
[36] DOI: 10.1142/S0219025799000291 · Zbl 1043.35526 · doi:10.1142/S0219025799000291
[37] Volterra V., Atti della Reale Accademia dei Lincei 5 pp 3939–
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