LINEAR APPROXIMATION OF NONLINEAR SCHR¨ODINGER EQUATIONS DRIVEN BY CYLINDRICAL WIENER PROCESSES

. In this paper the existence and uniqueness of the solution for a stochastic nonlinear Schr¨odinger equation, which is perturbed by a cylindrical Wiener process is investigated. The existence of the variational solution and of the generalized weak solution are proved by using sequences of successive approximations, which are the solutions of certain linear problems.


1.
Introduction. Linear or nonlinear Schrödinger equations describe how the quantum state of a physical system changes in time. The Schrödinger equation takes several different forms, depending on the physical situation. Using stochastic processes in Schrödinger equations one can model spontaneous emissions or thermic fluctuations or general random disturbances. Many authors investigated stochastic equations of Schrödinger type. Here we investigate a stochastic evolution equation of Schrödinger type over a triplet of complex rigged Hilbert spaces (V, H, V * ): dX(t) = −iAX(t)dt + f (t, X(t))dt + g(t, X(t))dW (t), where A is a certain linear and continuous operator from V to V * , the nonlinear terms f and g are Lipschitz continuous. Usually, in the applications A is the Laplacian operator, while typical Lipschitz continuous nonlinearities are the exponential law and the saturating law f (t, z) = λ|z| 2 1 + λ|z| 2 z , where λ > 0 and z ∈ C.
The saturating law describes the variation of the dielectric constant of gas vapors through which a laser beam propagates [2]. The exponential nonlinearity serves as a useful model in homogenous, unmagnetized plasmas and laser-produced plasmas [5].
Stochastic partial differential equations can be interpreted as stochastic evolution equations and the solutions are defined in a generalized sense (see, e.g., [8], [16]). For example, the mild solution of a stochastic Schrödinger type equation is discussed in [3], [4], [7], see also the literature therein. The variational solution and generalized weak solution of (1) are considered in [10], where the solution process is approximated by the Galerkin method, but no error estimates were given.
Often it is possible to approximate the solutions of partial differential equations by linearization, see, e.g., [13]. This idea was also considered in the stochastic case (see, e.g., [12], [15]). Such approximations are useful for further investigations of optimal control problems. The linearization method discussed in detail in this paper enables us to give error estimates, i.e. to derive expressions for the bound of the difference between the solution of the considered equation and its approximations.
In our paper we approximate the variational solution and generalized weak solution of (1) by the sequence of solutions of the linearized problem: for n = 0, 1, . . . we consider dX n+1 (t) = −iAX n+1 (t)dt + f (t, X n (t))dt + g(t, X n (t))dW (t), X 0 (t) = x 0 . (2) This paper has the following structure: In Section 2 we introduce a list of assumptions and give the definitions for the variational solution and generalized weak solution and present the method to approximate them. In Section 3 we investigate an auxiliary linear problem, which helps us to construct the sequence of successive approximations (2). This sequence is proved to converge to the variational solution in Section 4 and error estimates are derived. Section 5 is devoted to the approximation of the generalized weak solution. In Section 6 we give concrete examples for Schrödinger type problems for which the solution can be approximated by our method.

2.
Assumptions and formulation of the problem. Let (Ω, F, (F t ) t≥0 , P ) be a filtered complete probability space. Consider (V, (·, ·) V ) and (H, (·, ·)) be complex Hilbert spaces, such that (V, H, V * ) forms a triplet of rigged Hilbert spaces. Let K be a separable real Hilbert space and W (t) t≥0 to be a K-valued cylindrical Wiener process adapted to the filtration (F t ) t≥0 .
We indicate a list of assumptions and for each specific solution type and properties of it, we mention which assumptions we use: [A] A : V → V * has the following properties: • A is linear and continuous • there exists constants α 1 ∈ R and α 2 > 0, such that for all v ∈ V it holds • Let (h n ) n ⊂ H be the eigenvectors of the operator A, for which we assume that Ah n ∈ H for all n ∈ N * and (h n ) n is a complete orthonormal system in H. We denote by (λ n ) n the sequence of the corresponding eigenvalues of A and suppose that λ n > 0 for all n ∈ N * . [ for all t ∈ [0, T ], u, v ∈ H and for a.e. ω ∈ Ω; (2) for all t ∈ [0, T ], u ∈ V and a.e. ω ∈ Ω we have f (t, u) ∈ V and there exists (1) there exists a constant c g > 0 such that it holds T ], u, v ∈ H and for a.e. ω ∈ Ω; (2) for all t ∈ [0, T ], u ∈ V and a.e. ω ∈ Ω we have g(t, u) ∈ L 2 (K, V ) and there exists k g > 0 such that Remark 1. In [14] p. 110 is mentioned: Let A be a symmetric operator. Then A considered as an unbounded operator in H with the domain D(A) = {h ∈ V : Ah ∈ H} is self-adjoint. If the injection map of V into H is compact, then the operator has a basis of eigenfunctions (h j ) j such that Ah j = λ j h j , h j ∈ V, λ j ≤ λ j+1 , λ j → ∞, (h j , h k ) = δ jk , and (h j ) j is a complete orthonormal system in H.
2) We denote by C HV the embedding constant of V → H. , which can be approximated by a sequence of successive approximations (X n ) n in the following sense is the solution of the following linearized problem for all t ∈ [0, T ], v ∈ V , n ≥ 0, a.e. ω ∈ Ω, and X 0 = x 0 . where X n ∈ L 2 (Ω; C([0, T ]; H)) is the solution of the following linearized problem Av ∈ H}, n ≥ 0, a.e. ω ∈ Ω and X 0 = x 0 . Note, that the methods to prove the existence of the variational solution of (3) and of the generalized weak solution of (5) are different than those given in [10]. In our paper we obtain the solution as the limit of a sequence of successive approximations, which enables us to give error estimates, i.e. to derive expressions for the bound of the difference between the solution of the considered equation and its approximations. where the series above converges in L 2 (Ω; H), (w n (t)) t≥0 n is a sequence of mutually independent real-valued Wiener processes and (e n ) n is an orthonormal basis in K. One can prove that For more details, see [8].
2) Note, that any variational solution is also a generalized weak solution.
3) The operator iA with D(A) = {v ∈ V : Av ∈ H} generates a C 0 group (U t ) t∈R of unitary operators in H. This fact follows from Stone's Theorem (see Theorem 3.1.4 in [1]). We can introduce the mild solution by for all t ∈ [0, T ] and for a.e. ω ∈ Ω. The equivalence of generalized weak and mild solutions is well known for the case of C 0 semigroups, see for example Theorem 1.1 of Chapter 2 in [11]. Consequently, the generalized weak solution definition (5) is equivalent to the mild solution definition (7).
3. An auxiliary linear problem. Consider a stochastic process ξ ∈ L 2 (Ω; C([0, T ]; H)) ∩ L 2 (Ω × [0, T ]; V ) and the following stochastic linear equation for all t ∈ [0, T ], v ∈ V and a.e. ω ∈ Ω. Since we will use the Galerkin method to prove the existence of the solution of (8), for each n ≥ 1 we introduce the finite dimensional spaces H n := sp{h 1 , h 2 , . . . , h n } (equipped with the norm induced from H) and K n := sp{e 1 , e 2 , . . . , e n } (equipped with the norm induced from K). We define π n : H → H n the orthogonal projection and we denote x 0n = π n x 0 and W n (t) = n j=1 e j w j (t) ∈ K n .

WILFRIED GRECKSCH AND HANNELORE LISEI
Consider the finite dimensional Galerkin approximations corresponding to (8): Proof. Consider the complex valued processes (β j,n (t)) t∈[0,T ] , j ∈ {1, . . . , n}, to be the unique solution of the following finite dimensional system (which is linear in which obviously verifies (AY ξ n , h j ) = λ j (Y ξ n , h j ) = λ j β j,n and is the solution of (9). In what follows we derive some estimates for Y ξ n . Using the finite dimensional Itô formula we have and taking expectation, we get By using the assumptions on f we have and by the assumptions on g it holds as well as by the Burkholder-Davis-Gundy inequality (see [19, p. 44 Then, where C 0 > 0 is a constant independent of n, but depending on c f , c g , c HV , k f , k g , T. This implies that Y ξ n ∈ L 2 (Ω; C([0, T ]; H n )). Now we derive estimates in the V norm. Multiplying both sides of the equality (11) with λ j and then summing up for j = 1 to n, using [A], [f- (2)], [g- (2)] and taking expectation we get

Moreover, by [A] and [I-(1)] it follows
By (12) and by Gronwall's lemma it follows that there exists a constant The following estimate holds Proof. The almost sure uniqueness of the solution of (8), follows by assuming that there exist two variational solutions Y andŶ of (8) and computing for all t ∈ [0, T ] and a.e. ω ∈ Ω. By the properties of A we obtain Y (t) =Ŷ (t) for all t ∈ [0, T ] and a.e. ω ∈ Ω. By Theorem 3.1 it follows that (Y ξ n ) n≥1 is bounded in the space L 2 (Ω×[0, T ]; V ). Then by [21, p. 258, Proposition 21.23 (i)] there exist a subsequence of (Y ξ n ) n≥1 , for which we use the same notation, and Y ∈ L 2 (Ω × [0, T ]; V ) such that Taking limit in (9) and using that There exists an F t -adapted H-valued process which is equal to Y (t) for a.e. (ω, t) ∈ Ω × [0, T ] and equal to the right-hand side of the above equality for all t ∈ [0, T ] and a.e. ω ∈ Ω. We also denote this process by (Y (t)) t∈[0,T ] . Hence, for each t ∈ [0, T ], each j ≥ 1 and a.e. ω ∈ Ω it holds The process (Y (t)) t∈[0,T ] has in H almost surely continuous trajectories (see [19, p. 73, Theorem 2]). In fact, Y ξ := Y is the unique solution of (8). We know that a subsequence of (Y ξ n ) n converges to Y ξ strongly in L 2 (Ω × [0, T ]; H) and weakly in L 2 (Ω × [0, T ]; V ). In fact, the whole sequence has these properties because of [20,p. 480,Proposition 10.13 (1) and (2)] and since (8) possesses a unique solution. By using Theorem 3.1 and the above mentioned weak convergence, the following estimate holds 4. Approximation of the variational solution. We consider successively ξ := X n and denote X n+1 := Y Xn for each n ≥ 0 with X 0 = x 0 , where x 0 ∈ L 2 (Ω; V ) is the initial condition. Then by (8)   (g(s, X(s))e j − g(s,X(s))e j , X(s) −X(s))dw j (s) + t 0 g(s, X(s)) − g(s,X(s)) 2 L2(K,H) ds.
Then by D'Alembert's criterion for convergent series, it follows that the series We successively apply the inequality (13) from Theorem 3.2, where we assume without loss of generality that C 1 > 1, to get Hence (X n ) n is a bounded sequence in L 2 (Ω×[0, T ]; V ) and there exist a subsequence (X n k ) k of (X n ) n and X ∈ L 2 (Ω × [0, T ]; V ) such that By (17) we must have
In the following theorem we give some error estimates. We denote by where C 2 was the constant determined during the proof of Theorem 4.1.  2)] are satisfied and n ∈ N such that n > C − 1, then the following error estimates hold: 2) E sup Proof. 1) By using the notation a n = C n n! from Theorem 4.1 (where C = C 2 T ) we compute for each k > 1 For n sufficiently large such that C < n + 1 we have n+k−1 j=n a j ≤ a n k j=1 C n + 1 .

LINEAR APPROXIMATION OF NONLINEAR SCHRÖDINGER EQUATIONS 3107
Hence, by taking k → ∞ in the inequality (16) which was derived in the proof of Theorem 4.1 we obtain 2) Similar to (15) we derive for j > 1 Then, a j and by k → ∞ we obtain (15) we have

3) As in
Taking k → ∞ and using the strong convergence result from Theorem 4.1 we get Remark 3. Observe that the smallest n ∈ N such that n > C − 1 is n = [C]. Then a possible bound for the convergent series mentioned in the above theorem is where [C] is the integer part of the positive constant C.

5.
Approximation of the generalized weak solution. where X n is the solution of (6). Moreover, for n ∈ N such that n > C − 1 (with C given in (20)), the following error estimates hold:
By the Burkholder-Davis-Gundy inequality and estimates similar to those computed in the proof of Theorem 4.1 we get where C 2 is the same constant as in (14). But the convergence result from (23) implies Successively applying for ξ = X n and η = X n−1 inequality (24) we obtain Exactly as in the proof of Theorem 4.1 we get that (X n ) n is a Cauchy sequence in the Banach space L 2 (Ω; C([0, T ]; H)). Therefore, there exists X ∈ L 2 (Ω; C([0, T ]; H)) such that We pass to the limit in (6) as n → ∞, then it follows that for each t ∈ [0, T ], v ∈ D(A) and a.e. ω ∈ Ω. The error estimates are derived similarly to those given in Theorem 4.2.

Stochastic equations of Schrödinger type.
In all examples (K, (·, ·) K ) is a separable real Hilbert space and (e n ) n is an orthonormal basis in K, and (w n (t) t≥0 ) n is a sequence of mutually independent real-valued Wiener processes.
If we assume that (27) holds, then g : [0, T ] → L 2 (K, H) and it follows that there exists a generalized weak solution for the following problem with initial condition 3112 WILFRIED GRECKSCH AND HANNELORE LISEI for all t ∈ [0, T ], v ∈ V ∩ H 2 ([0, 1]; C) and a.e. ω ∈ Ω. Each of the problems (30) and (31) describes a particle in [0, 1] driven by a potential ψ and an additive infinite dimensional noise. If ψ(x) ≡ 0, then we get the Schrödinger equation for a randomly disturbed free particle.
For x 0 ∈ H the following problem has generalized weak solution for all t ∈ [0, T ], v ∈ H 2 (R d ; C) and a.e. ω ∈ Ω. It is well known that in the deterministic case for ψ(x) = 1 2 |x| 2 problem (32) defines the linear harmonic oscillator (see [9]). In our case we have additionally the influence of random noise.