# Operator differential-algebraic equations with noise arising in fluid dynamics

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## Abstract

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

## Keywords

Operator DAE Noise disturbances Chaos expansion Itô-Skorokhod integral Stochastic convolution Regularization## Mathematics Subject Classification

65J10 60H40 60H30 35R60## 1 Introduction

The governing equations of an incompressible flow of a Newtonian fluid are described by the Navier–Stokes equations [43]. Therein, one searches for the evolution of a velocity field *u* and the pressure *p* to given initial data, a volume force, and boundary conditions. For results on the existence of a (unique) solution, we refer to [20], [42, Ch. 25], and [43, Ch.III].

In this paper, we consider the linear case but allow a more general constraint, namely that the divergence of the velocity does not vanish. Note that this changes the analysis and numerics since the state-of-the-art methods are often tailored for the particular case of a vanishing divergence. An application with non-vanishing divergence is given by the optimal control problem constrained by the Navier–Stokes equations where the cost functional includes the pressure [23].

The Navier–Stokes equations, as well as the corresponding linearized equations, can be formulated as differential-algebraic equations (DAEs) in an abstract setting [3, 4]. These so-called *operator DAEs* correspond to the weak formulation in the framework of partial differential equations (PDEs). As generalization of finite-dimensional DAEs, see [19, 25, 26] for an introduction, also here considered constrained PDEs suffer from instabilities and ill-posedness. This is the reason why the stable approximation of the pressure (which is nothing else than a Lagrange multiplier to enforce the incompressibility) is a great challenge.

One solution strategy is to perform a regularization which corresponds to an *index reduction* in the finite-dimensional setting. With this, the issue of instabilities with respect to perturbations is removed. In the case of fluid dynamics, this has been shown in [4].

*u*(0). In order to preserve the mean dynamics, we deal with stochastic perturbations of zero mean. This implies that the expected value of the stochastic solution equals the solution of the corresponding deterministic operator DAE. For the “noise” processes we consider either a general Gaussian white noise process or perturbations which can be expressed in the form of a stochastic convolution.

Within this paper, we consider the Gaussian white noise space \((\Omega , \mathcal F, \mu )\) with the Gaussian probability measure \(\mu \) to be the underlying probability space. Instead, the same analysis can be provided also on Poissonian white noise space \((\Omega , \mathcal F, \nu )\), with the Poissonian probability measure \(\nu \), on fractional Gaussian white noise space \((\Omega , \mathcal F, \mu _{H})\), or on fractional Poissonian white noise space \((\Omega , \mathcal F, \nu _H)\), for \(H\in (0,1)\). This follows from the existence of unitary mappings between Gaussian and Poissonian white noise spaces, and between Gaussian and fractional Gaussian white noise spaces [27].

With the application of the polynomial chaos expansion method, also known as the *propagator method*, the problem of solving the initial stochastic equations is reduced to the problem of solving an infinite triangular systems of deterministic operator DAEs, which can be solved recursively. Summing up all coefficients of the expansion and proving convergence in an appropriate space of stochastic processes, one obtains the stochastic solution of the initial problem.

The chaos expansion methodology is a very useful technique for solving many types of stochastic differential equations, linear and nonlinear, see e.g. [6, 18, 29, 30, 32, 33, 34, 40, 46]. The main statistical properties of the solution, its mean, variance, and higher moments, can be calculated from the formulas involving only the coefficients of the chaos expansion representation [16, 36].

The proposed method allows to apply regularization techniques from the theory of deterministic operator DAEs to the related stochastic system. Applications arise in fluid dynamics, but are not only restricted to this case. The same procedure can be used to regularize other classes of equations that fulfill our setting. A specific example with the operators of the Malliavin calculus is described in Sect. 5. For this reason, in the present paper, we develop a general abstract setting based on white noise analysis and chaos expansions. Numerical experiments with truncated chaos expansions, i.e., stochastic Galerkin methods, are not included in this paper. However, once we regularize each system, it becomes numerically well-posed [3] and then the stochastic equation is well-posed as well.

The paper is organized as follows. In Sect. 2 we introduce the concept of (deterministic) operator DAEs with special emphasis on applications in fluid dynamics. Considering perturbation results for such systems, we detect the necessity of a regularization in order to allow stochastic perturbations. The stochastic setting for the chaos expansion is then given in Sect. 3. Furthermore, we discuss stochastic noise terms in the differential as well as in the constraint equation and the systems which result from the chaos expansions, Theorems 6, 8 and 9. The extension to more general cases is then subject of Sect. 4. Therein, we consider more general operators and stochastic convolution terms. We also provide proofs of the convergence of the obtained solutions in appropriate spaces of generalized stochastic processes, Theorem 11. In Sect. 5 we consider shortly a specific example of DEAs that involve stochastic operators arising in Malliavin calculus. The proof of existence of a unique solution in a space of generalized stochastic processes is given in Theorem 13. Finally, we discuss extensions of our results to specific types of nonlinear equations.

## 2 Operator DAEs

In this section we introduce the concept of operator DAEs, analyze the influence of perturbations, provide regularization of operator DAEs, and state stability results.

### 2.1 Abstract setting

First we consider operator DAEs (also called PDAEs) which equal constrained PDEs in the weak setting or DAEs in an abstract framework [3, 15]. Thus, we work with generalized derivatives in time and space. In particular, we consider semi-explicit operator DAEs for which the constraint equation is explicitly stated.

*T*with values in \(\mathcal {V} \), see [14, Ch. 7.1] for an introduction. The corresponding norm of \(L^2(T;\mathcal {V} )\), which we denote by \(\Vert \cdot \Vert _{L^2(\mathcal {V} )}\), is given by

Furthermore, we need the operators and right-hand sides of (1 to satisfy the following assumptions.

### Assumption 1

- 1.The right-hand sides of (1 satisfy$$\begin{aligned} F \in L^2(T;\mathcal {V} ^*) \quad \text {and} \quad G \in H^1(T;\mathcal {Q} ^*) \hookrightarrow C(T; \mathcal {Q} ^*). \end{aligned}$$
- 2.
The constraint operator \(B:\mathcal {V} \rightarrow \mathcal {Q} ^*\) is linear and there exists a right-inverse which is denoted by \(B^-\).

- 3.
Operator \(K:\mathcal {V} \rightarrow \mathcal {V} ^*\) is linear, positive on the kernel of

*B*, and continuous.

Note that the involved operators \(B:\mathcal {V} \rightarrow \mathcal {Q} ^*\) and \(K:\mathcal {V} \rightarrow \mathcal {V} ^*\) can be extended to Nemytskii mappings of the form \(B:L^2(T;\mathcal {V} ) \rightarrow L^2(T;\mathcal {Q} ^*)\) and \(K:L^2(T;\mathcal {V} ) \rightarrow L^2(T;\mathcal {V} ^*)\), see [41, Ch. 1.3]. From here onwards, we restrict ourselves to the linear case.

### Remark 1

*B*is not applicable to \(u^0\). In this case, the condition has the form

*B*in \(\mathcal {H} \) [4, 15]. If \(u^0 \in \mathcal {V} \) is given, then we get the same decomposition but with \(u^0_B \in {{\mathrm{Ker}}}B\).

In the following, we write \(a\lesssim b\) meaning that there exists a positive constant *c* such that \(a\le cb\). We show that the solution is bounded in terms of the initial data, the right-hand sides, and their derivative, cf. [3, Sect. 6.1.3].

### Theorem 1

### Proof

*B*and \(\mathcal {V}^\text {c}\) is any complementary space. This gives a unique decomposition \(u=u_1+u_2\) where \(u_1\), \(u_2\) take values in \(\mathcal {V}_{B}\), \(\mathcal {V}^\text {c}\), respectively. Thus, we have \(Bu = B u_2 = G\) and therefore \(u_2 = B^- G\). The assumption on

*G*implies \(u_2 \in H^1(T;\mathcal {V} )\) and

### Remark 2

Throughout the paper, we concentrate on results for the variable *u* which corresponds to the velocity in terms of fluid flow applications. Similar results for the Lagrange multiplier \(\lambda \) (respectively the pressure) are valid but require stronger regularity assumptions on *F* and \(u^0\). For a detailed stability analysis of the Lagrange multiplier, we refer to [50, Ch. 3.1.2]. Note that Assumption 1 is not sufficient to prove \(\lambda \in L^2(T;\mathcal {Q} )\).

Since this paper focuses on fluid flows, we show that the linear Stokes equations fit into the given framework. Note that also the Navier–Stokes equations may be considered in the given setting if we allow the operator *K* in (1 to be nonlinear. However, we exclude the nonlinear case in this paper.

### Example 1

*D*, cf. [43]. We consider homogeneous Dirichlet boundary conditions and set

*K*which equals the weak form of the Laplace operator, i.e.,

*u*describes the velocity of the fluid whereas \(\lambda \) measures the pressure. The operator equations (1 then equal the weak formulation of the Stokes equations

### Example 2

(Linearized Navier–Stokes equations) With a simple modification of the operator *K*, the framework given in Example 1 includes any linearization of the Navier–Stokes equations such as the Oseen equations.

*u*describes the ’disturbance velocity’, i.e., the variation around \(u_\infty \).

Although we focus here on applications in fluid dynamics, we emphasize that the given framework is not restricted to this class. Further examples are given by PDEs with boundary control [11] (with *B* being the trace operator) as well as applications in elastodynamics which leads to second-order systems of similar structure [2].

### 2.2 Influence of perturbations

DAEs are known for its high sensitivity to perturbations. The reason for this is that derivatives of the right-hand sides appear in the solution. In particular, this implies that a certain smoothness of the right-hand sides is necessary for the existence of solutions. Furthermore, the numerical approximation is much harder than for ODEs, since small perturbations - such as round-off errors or errors within iterative methods - may have a large influence [38].

The resulting level of difficulty in the numerical approximation of DAEs is measured by the so-called *index*. There exist several index concepts [35] and we use here the *differentiation index*, see [10, Def. 2.2.2] for a precise definition. A comparable index concept for operator DAEs which may be used to classify systems of the form (1 does not exist. Thus, in order to obtain information about stability issues it is advisable to analyse the influence of perturbations. Furthermore, a spatial discretization of system (1 by finite elements (under some basic assumptions) leads to a DAE of index 2. Note that the understanding of the index is not crucial for the further reading of this paper. However, we comment on the index from time to time for additional insight.

*u*and \(\hat{u}\) projected to the kernel of the constraint operator

*B*. Accordingly, we denote the projected initial error by \(e_{1,0}\). In [5] it is shown that with the given assumptions on the operators

*K*and

*B*of Assumption 1, we have

### 2.3 Regularization of operator DAEs

In this subsection, we introduce an operator DAE which is equivalent to (1, but where the solution of the perturbed system does not depend on derivatives of the perturbations. Furthermore, a semi-discretization in space of the regularized system directly leads to a DAE of index 1 and thus, is better suited for numerical integration [25].

In the case of the Stokes equations, the right-hand side *G* vanishes since we search for divergence-free velocities. In this case, the constrained system is often reduced to the kernel of the constraint operator *B* which leads to an operator ODE, i.e., a time-dependent PDE. However, with the stochastic noise term in the constraint, we cannot ignore the inhomogeneity anymore. In addition, the inclusion of *G* enlarges the class of possible applications. Thus, we propose to apply a regularization of the operator DAE.

*hidden constraint*, to the system. In order to balance the number of equations and variables, we add a so-called

*dummy variable*\(v_2\) to the system. The assumptions are as before, but we split the space \(\mathcal {V} \) into \(\mathcal {V} =\mathcal {V}_{B}\oplus \mathcal {V}^\text {c}\) were

*B*is invertible, i.e., there exists a right-inverse of

*B*, namely \(B^-:\mathcal {Q} ^* \rightarrow \mathcal {V}^\text {c}\) with \(B B^- q=q\) for all \(q\in \mathcal {Q} ^*\). In the example of the Stokes equations, cf. Example 1, \(\mathcal {V}_{B}\) is the space of divergence-free functions which build a proper subspace of \(\mathcal {V} \) and \(\mathcal {V}^\text {c}\) equals its orthogonal complement in \(\mathcal {V} \). We then search for a solution \((u_1, u_2, v_2, \lambda )\) where \(u_1\) takes values in \(\mathcal {V}_{B}\) and \(u_2\), \(v_2\) in the complement \(\mathcal {V}^\text {c}\). The extended (but equivalent) system then reads

### Theorem 2

### Proof

*t*and using the initial condition, we obtain the stated assertion. \(\square \)

Note that, in contrast to the original formulation, estimate (5) does not depend on derivatives of the perturbations. This is crucial when we consider stochastic perturbations.

## 3 Inclusion of stochastic perturbations

From the modeling point of view, noise may enter the physical system either as temporal fluctuations of internal degrees of freedom or as random variations of some external control parameters; internal randomness often reflects itself in additive noise terms, while external fluctuations gives rise to multiplicative noise terms. Moreover, the additive noises may appear in various forms, ranging from the space time white noise to colored noises generated by some infinite dimensional Brownian motion with a prescribed covariance operator [13].

### 3.1 Preliminaries

In this section, we recall some basic facts and notions of the white noise theory, random variables, stochastic processes, and operators. Then we apply the chaos expansion method in order to solve stated problems.

#### 3.1.1 White noise space

*Schwartz spaces*of rapidly decreasing test functions \(S(\mathbb R)\) and tempered distributions \(S ' (\mathbb R)\), respectively, and \(\mathcal B\) the Borel sigma algebra generated by the weak topology on \(S ' (\mathbb R)\). By the Bochner-Minlos theorem, there exists a unique measure \(\mu \) on \(( S ' (\mathbb R), \mathcal B)\) such that for each \(\phi \in S(\mathbb R)\) the relation

*Fourier-Hermite polynomials*defined by use of the Hermite functions and the Hermite polynomials. We denote by \(\{h_{n}(x)\}_{n\in \mathbb {N}_0}\) the family of

*Hermite polynomials*and \(\{\xi _{n}(x)\}_{n\in \mathbb {N}}\) the family of

*Hermite functions*, where

#### 3.1.2 Spaces of random variables

Let \(\mathcal {I}=(\mathbb {N}_{0}^{\mathbb {N}})_{c}\) be the set of sequences of non-negative integers which have only finitely many nonzero components \(\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _m,0,0,\ldots )\), \(\alpha _i\in \mathbb {N}_0\), \(i=1, 2,\dots , m\), \(m\in \mathbb {N}\). The *k*-th unit vector \(\varepsilon ^{(k)}=(0,\ldots ,0,1,0,\ldots ), \, k\in \mathbb {N}\), is the sequence of zeros with the entry 1 as the *k*-th component and \(\mathbf{0}\) is the multi-index with only zero components. The length of a multi-index \(\alpha \in \mathcal {I}\) is defined as \(|\alpha |=\sum _{k=1}^\infty \alpha _k\). We say \(\alpha \ge \beta \) if \(\alpha _k \ge \beta _k\) for all \(k\in \mathbb N\) and thus \(\alpha -\beta = (\alpha _1-\beta _1, \alpha _2-\beta _2,\ldots )\). For \(\alpha < \beta \) the difference \(\alpha -\beta \) is not defined. Particularly, for \(\alpha _k >0\) we have \(\alpha -\varepsilon ^{(k)} = (\alpha _1,\ldots , \alpha _{k-1}, \alpha _k - 1, \alpha _{k+1},\ldots , \alpha _m, 0,\ldots )\), \(k\in \mathbb N\). We denote \((2\mathbb {N})^\alpha =\prod _{k=1}^\infty (2k)^{\alpha _k}\).

### Theorem 3

([49]) It holds that \(\sum \nolimits _{\alpha \in \mathcal {I}}(2\mathbb {N})^{-p\alpha }<\infty \) if and only if \(p>1\).

The proof can be foud in the paper of Zhang [49], also in [22, Prop. 2.3.3].

*Fourier-Hermite*orthogonal polynomial basis of \(L^2(\Omega )\) such that \(\Vert H_\alpha \Vert ^2_{L^2(\Omega )} = \mathbb E (H_\alpha )^2 = \alpha !\). In particular, \(H_{\mathbf{0}} (\omega ) = H_{(0,0,\ldots )} (\omega )=1\), and for the

*k*-th unit vector \(H_{\varepsilon ^{(k)}}(\omega )=h_1(\langle \omega ,\xi _{k}\rangle ) = \langle \omega ,\xi _{k}\rangle \), \(k\in \mathbb {N}\).

### Theorem 4

*exponential growth spaces*of stochastic test functions \(\exp (S)_{\rho }\) and stochastic generalized functions \(\exp (S)_{-\rho }\) [21, 22, 27, 39, 40]. In this paper, we consider the largest Kondratiev space of stochastic distributions, i.e., \(\rho =1\). For definition of the Kondratiev spaces we follow [22].

*Kondratiev test random variables*\((S)_1\) can be constructed as the projective limit of the family of spaces

*Kondratiev generalized random variables*\((S)_{-1}\) can be constructed as the inductive limit of the family of spaces

The problem of pointwise multiplications of generalized stochastic functions in the white noise analysis is overcome by introducing the *Wick product*, which represents the stochastic convolution. The fundamental theorem of stochastic calculus states the relation of the Wick multiplication to the Itô-Skorokhod integration [22].

*L*and

*S*be random variables given in their chaos expansion representations \(L= \sum _{\alpha \in \mathcal I} \ell _\alpha H_\alpha \) and \(S= \sum _{\alpha \in \mathcal I} s_\alpha \, H_\alpha \), \(\ell _\alpha , s_\alpha \in \mathbb R\) for all \(\alpha \in \mathcal I\). Then, the

*Wick product*\(L\lozenge S\) is defined by

### Example 3

Kondratiev spaces \((S)_1\) and \((S)_{-1}\) are closed under the Wick multiplication. For the proof we refer to [22, Lemma 2.4.4].

#### 3.1.3 Stochastic processes

Classical stochastic process can be defined as a family of functions \(v:T \times \Omega \rightarrow \mathbb R\) such that for each fixed \(t \in T\), \(v(t,\cdot )\) is an \(\mathbb R\)-valued random variable and for each fixed \(\omega \in \Omega \), \(v(\cdot , \omega )\) is an \(\mathbb R\)-valued deterministic function, called *trajectory*. Here, following [39], we generalize the definition of a classical stochastic process and define generalized stochastic processes. By replacing the space of trajectories with some space of deterministic generalized functions, or by replacing the space of random variables with some space of generalized random variables, different types of generalized stochastic processes can be obtained. In this manner, we obtain processes generalized with respect to the *t* argument, the \(\omega \) argument, or even with respect to both arguments [22, 39].

A very general concept of generalized stochastic processes, based on chaos expansions was introduced in [39] and further developed in [27, 28]. In [22] generalized stochastic processes are defined as measurable mappings \(T \rightarrow (S)_{-1}\). Thus, they are defined pointwise with respect to the parameter \(t \in T\) and generalized with respect to \(\omega \in \Omega \). We define such processes by their chaos expansion representations in terms of an orthogonal polynomial basis.

Let \(\tilde{X}\) be a Banach space endowed with the norm \(\Vert \cdot \Vert _{\tilde{X}} \) and let \(\tilde{X}'\) denote its dual space. If, for example, \(\tilde{X}\) is a space of functions on \(\mathbb R\) such as \(\tilde{X}=C^k(T)\) or \(\tilde{X}=L^2(\mathbb R)\), we obtain stochastic processes. The definition of processes where \(\tilde{X}\) is not a normed space, but a nuclear space topologized by a family of seminorms, e.g. \(\tilde{X}=S(\mathbb R)\) is given in [39].

*u*have the formal expansion \(u=\sum _{\alpha \in \mathcal I} u_\alpha \otimes H_\alpha \), where \(f_\alpha \in X\) and \(\alpha \in \mathcal I\). We define the spaces

*X*denotes an arbitrary Banach space (both possibilities \(X=\tilde{X}\) and \(X=\tilde{X}'\) are allowed).

### Definition 1

*Generalized stochastic processes*and

*test stochastic processes*in Kondratiev sense are elements of the spaces respectively

In this case the symbol \(\otimes \) denotes the projective tensor product of two spaces, i.e., \(\tilde{X}'\otimes (S)_{-1}\) is the completion of the tensor product with respect to the \(\pi \)-topology.

### Remark 3

From the nuclearity of the Kondratiev space \( (S)_{1}\) it follows that \((\tilde{X}\otimes (S)_{1})'\cong \tilde{X}'\otimes (S)_{-1}\). Moreover, \(\tilde{X}'\otimes (S)_{-1}\) is isomorphic to the space of linear bounded mappings \(\tilde{X}\rightarrow (S)_{-1}\), and it is also isomporphic to the space of linear bounded mappings \((S)_{1}\rightarrow \tilde{X}'\). More details can be found in [28, 32, 39].

*u*which belong to \(X\otimes (S)_{-1}\) and are given by the chaos expansion form

*p*corresponds to the

*level of singularity*of the process

*u*. Note that the deterministic part of

*u*in (9) is the coefficient \(u_{\mathbf {0}}\), which represents the generalized expectation of

*u*. In the applications of fluid flows, the space

*X*equals one of the Sobolev-Bochner spaces \(L^2(T;\mathcal {V} )\) or \(L^2(T;\mathcal {Q} )\).

### Example 4

*F*is the zero-th coefficient in the expansion representation (10), i.e., it is given by \(\sum _{k\in \mathbb {N}} a_{\mathbf {0}, k} \otimes \xi _k = b_{\mathbf {0}}\).

Space of processes with finite second moments and square integrable trajectories \(X\otimes L^2(\mathbb R)\otimes (L)^2\). It is isomporphic to \(X\otimes L^2(\mathbb R\times \Omega )\) and if *X* is a separable Hilbert space, then it is also isomorphic to \(L^2(\mathbb R \times \Omega , X)\).

### Example 5

*T*denotes a time interval. From the nuclearity of \((S)_{1}\) and the arguments provided in Remark 3 it follows that \(C^k(T;(S)_{-1})=C^k (T) \otimes (S)_{-1}\), i.e., differentiation of a stochastic process can be carried out componentwise in the chaos expansion, cf. [28, 32]. This means that a stochastic process \(u(t,\omega )\) is

*k*times continuously differentiable if and only if all of its coefficients \(u_\alpha \), \(\alpha \in \mathcal I\) are in \(C^k (T)\). The same holds for Banach space valued stochastic processes, i.e., for elements of \(C^k(T;X) \otimes (S)_{-1}\), where

*X*is an arbitrary Banach space. These processes can be regarded as elements of the tensor product space

In this way, by representing stochastic processes in their polynomial chaos expansion form, we are able to separate the deterministic component from the randomness of the process.

### Example 6

*Brownian motion*\(B_t(\omega ) := \langle \omega , \chi _{[0, t]} \rangle \), \(\omega \in S'(\mathbb R)\), \(t\ge 0\) is defined by passing though the limit in \(L^2(\mathbb R)\), where \(\chi _{[0,t]}\) is the characteristic function on [0,

*t*]. The chaos expansion representation has the form

*t*, \(B_t\) is an element of \(L^2(\Omega )\). Brownian motion is a Gaussian process with zero expectation and the covariance function \(E(B_t(\omega ) B_s(\omega )) = \min \{t,s \}\). Furthermore, almost all trajectories are continuous, but nowhere differentiable functions.

*Singular white noise*is defined by the formal chaos expansion

*X*of the form

*F*and

*G*be stochastic processes given in their chaos expansion forms \(F= \sum _{\alpha \in \mathcal I} f_\alpha \, \otimes \, H_\alpha \) and \(G= \sum _{\alpha \in \mathcal I} g_\alpha \,\otimes \, H_\alpha \), \(f_\alpha , g_\alpha \in X\) for all \(\alpha \in \mathcal I\). Assuming that \(f_\alpha \, g_\beta \in X\), for all \(\alpha , \beta \in \mathcal I\), the

*Wick product*\(F\lozenge G\) is defined by

### Theorem 5

([28]) Consider \(F\in X\otimes (S)_{-1, -p_1}\) and \(G\in X\otimes (S)_{-1, -p_2}\) for some \(p_1, p_2\in \mathbb N_0\). Then the Wick product \(F\lozenge G\) is a well-defined element in the space \(X\otimes (S)_{-1, -q}\) for \(q\ge p_1+p_2 +2\).

#### 3.1.4 Coordinatewise operators

*coordinatewise operator*if it is composed of a family of operators \(\{A_\alpha \}_{\alpha \in \mathcal I}\), \(A_\alpha : X \rightarrow X\), \(\alpha \in \mathcal I\), such that for a process \(u= \sum _{\alpha \in \mathcal I} \, u_\alpha \, \otimes \, H_\alpha \in X\otimes (S)_{-1}\), \(u_\alpha \in X\), \(\alpha \in \mathcal I\) it holds that

*simple coordinatewise operator*.

### 3.2 Chaos expansion approach

*K*and

*B*are linear and that they satisfy Assumption 1. For the right-hand side of the differential equation (6a), namely stochastic process \(\mathcal {F} \), and the constraint (6b), namely stochastic process \(\mathcal {G} \), we assume that they are given in the chaos expansion forms

*p*it holds that

### Remark 4

Since the family of spaces \((S)_{-1, -p} \) is monotone, i.e., it holds that \((S)_{-1, - p_1} \subset (S)_{-1, - p} \) for \(p_1 < p\), we may assume in (17) that all the convergence conditions hold for the same level of singularity *p*. Clearly, for two different \(p_1\) and \(p_2\) we can take *p* to be \(p=\max \{p_1, p_2\}\) and thus, obtain that generalized stochastic processes satisfies (17) in the biggest space \((S)_{-1,-p}\). In that sense, we use in the sequel always the same level of singularity *p*.

*u*and \(\lambda \) of stochastic operator DAEs (6 and (7, which are

*stochastic processes*belonging to \(L^2(\mathcal {V} )\otimes (S)_{-1}\) and \(L^2(\mathcal {Q} )\otimes (S)_{-1}\), respectively. Their chaos expansions are given by

*u*and \(\lambda \). Furthermore, we are going to prove bounds on the solutions, provided that the stated assumptions on the given processes \(\mathcal F\) and \(\mathcal G\), the initial condition, and the noise terms are fulfilled.

*B*in \(\mathcal {H} \) and \(B^-\) denoting the right-inverse of the operator

*B*, cf. Remark 1.

### 3.3 Noise in the differential equation

### Example 7

We summarize the needed requirements in the following assumption.

### Assumption 2

- 1.
Operators \(\mathcal {K} \) and \(\mathcal {B} \) are simple coordinatewise operators with corresponding deterministic operators \(K:\mathcal {V} \rightarrow \mathcal {V} ^*\) and \(B:\mathcal {V} \rightarrow \mathcal {Q} ^*\), which satisfy the assumptions stated in Assumption 1.

- 2.
The stochastic processes \(\mathcal {F} \) and \(\mathcal {G} \) are given in their chaos expansion forms (16) such that the conditions in (17) hold.

- 3.
The process \(G_t\) is a Gaussian noise term represented in the form (12), with \(m_k \in L^2(T;\mathcal {V} ^*)\), \(k\in \mathbb N\), such that (13) holds.

- 4.The stochastic process \(u^0\) has the chaos expansion form \(u^0 = \sum _{\alpha \in \mathcal I} \, u_\alpha ^0 \, H_\alpha \) such that for some \(p\in \mathbb N_0\) it holds that$$\begin{aligned} \sum _{\alpha \in \mathcal I} \left\| u^0_{\alpha } \right\| _\mathcal {H} ^2 (2\mathbb N)^{-p\alpha } < \infty . \end{aligned}$$(21)

### Remark 5

### Theorem 6

Let Assumption 2 be satisfied. Then, for any consistent initial data there exists a unique solution \(u\in L^2(T;\mathcal {V} )\otimes (S)_{-1}\) of the stochastic DAE (20).

### Proof

- 1.for \(|\alpha | =0\), i.e., for \(\alpha = \mathbf {0} = (0,0,\ldots )\), we have to solveNote that system (23) is a deterministic problem of the form (1, where$$\begin{aligned} \dot{u}_{\mathbf {0} }(t) + {K} u_{\mathbf {0} }(t) + {B}^* \lambda _{\mathbf {0} } (t) = f_{\mathbf {0}}(t), {B} u_{\mathbf {0} }(t) = g_{\mathbf {0}}(t), \quad u_{\mathbf {0}} = u^0_{B,\mathbf {0}} + B^- g_{\mathbf {0}}(0). \end{aligned}$$(23)
*F*and*G*from (1 are equal to \(f_{\mathbf {0}}\) and \(g_{\mathbf {0}}\), respectively. Moreover, the system (23) can be obtained by taking the expectation of the system (20). The assumptions on the operators and right-hand sides \(f_\mathbf {0}\in L^2(T;\mathcal {V} ^*)\), \(g_\mathbf {0}\in H^1(T;\mathcal {Q} ^*)\) imply the existence of a solution \(u_{\mathbf {0} }\), \(\lambda _{\mathbf {0}}\). - 2.for \(|\alpha | =1\), i.e., for \(\alpha = \varepsilon ^{(k)}\), \(k\in \mathbb N\), we obtain the systemwith initial condition \(u_{\varepsilon ^{(k)} } (0) = u^0_{B, \varepsilon ^{(k)}} + B^- g_{\varepsilon ^{(k)}} (0)\). For each \(k\in \mathbb N\) system (24) is a deterministic initial value problem of the form (1, with the choice \(F =f_{\varepsilon ^{(k)}} + m_k\) and \(G =g_{\varepsilon ^{(k)}}\).$$\begin{aligned} \dot{u}_{\varepsilon ^{(k)} }(t) + {K} u_{\varepsilon ^{(k)} }(t) + {B}^* \lambda _{\varepsilon ^{(k)}} (t)= f_{\varepsilon ^{(k)}} (t) + m_{k}(t), {B} u_{\varepsilon ^{(k)}}(t) = g_{\varepsilon ^{(k)}}(t) \end{aligned}$$(24)
- 3.for \(|\alpha |>1\), we finally solveAgain, system (25) is a deterministic operator DAE, which can be solved in the same manner as the system (23).$$\begin{aligned} \dot{u}_{\alpha }(t) + {K} u_{\alpha }(t) + {B}^* \lambda _{\alpha } (t) = f_\alpha (t), {B} u_{\alpha }(t)= g_{\alpha }(t), \quad u_{\alpha }(0) = u_{B,\alpha }^0 + B^- g_{\alpha }(0). \end{aligned}$$(25)

### Remark 6

If the process \(\mathcal F\) in (20) is a deterministic function, then it can be represented by \(\mathcal F = f_{\mathbf 0}\), since the remaining coefficients satisfy \(f_\alpha =0\) for all \(|\alpha |>0\). Therefore, systems (24) and (25) further simplify.

As mentioned in Remark 2, a similar result can be formulated for the Lagrange multiplier if we assume stronger regularity assumptions. For completeness we state the following result for the Lagrange multiplier but leave out the proof.

### Theorem 7

Let Assumption 2 be satisfied. Assume additionally \(f_\alpha \in L^2(T;\mathcal {H} ^*)\) and \(u^0_{B,\alpha } \in \mathcal {V} \) and let the operator *K* be symmetric. Then, for any consistent initial data there exists a unique Lagrange multiplier \(\lambda \in L^2(T;\mathcal {Q} )\otimes (S)_{-1}\) of the stochastic operator DAE (20).

### Theorem 8

### Proof

*u*belongs to \(L^2(T;\mathcal {V} )\otimes (S)_{-1}\). \(\square \)

### Remark 7

Let the assumptions of Theorem 8 hold. If we assume additionally that \(f_\alpha \in L^2(T;\mathcal {H} ^*)\), \(u^0_{B,\alpha } \in \mathcal {V} \) and the operator *K* is symmetric, then there exists a unique Lagrange multiplier \(\lambda \in L^2(T;\mathcal {Q} )\otimes (S)_{-1}\) of the stochastic operator DAE (20).

### 3.4 Noise in the constraint equation

- 1.for \(|\alpha | =0\), i.e., for \(\alpha = \mathbf {0} = (0,0,\ldots )\), we obtain$$\begin{aligned} \dot{u}_{\mathbf {0} }(t) + {K} u_{\mathbf {0} }(t) + {B}^* \lambda _{\mathbf {0} } (t)= f_{\mathbf {0}}(t), {B} u_{\mathbf {0} }(t) = g_{\mathbf {0}}(t), \quad u_{\mathbf {0}} = u^0_{\mathbf {0}}. \end{aligned}$$(32)
- 2.for \(|\alpha | =1\), i.e., for \(\alpha = \varepsilon ^{(k)}\), \(k\in \mathbb N\), we have$$\begin{aligned} \dot{u}_{\varepsilon ^{(k)} }(t) + {K} u_{\varepsilon ^{(k)} }(t) + {B}^* \lambda _{\varepsilon ^{(k)} } (t)= & {} f_{\varepsilon ^{(k)}}(t) + m_k (t),\nonumber \\ {B} u_{\varepsilon ^{(k)}}(t)= & {} g_{\varepsilon ^{(k)}}(t) + m_k^{(1)}(t), \nonumber \\ u_{\varepsilon ^{(k)} } (0)= & {} u^0_{\varepsilon ^{(k)}}. \end{aligned}$$(33)
- 3.for \(|\alpha |>1\), we solve$$\begin{aligned} \dot{u}_{\alpha }(t) + {K} u_{\alpha }(t) + {B}^* \lambda _{\alpha } (t)= & {} f_\alpha (t), \nonumber \\ {B} u_{\alpha }(t)= & {} g_{\alpha }(t),\nonumber \\ u_{\alpha }(0) = u_{\alpha }^0. \end{aligned}$$(34)

### 3.5 Regularization

### Theorem 9

### Proof

### 3.6 Convergence of the truncated expansion

*P*, i.e., up to a certain order

*P*, can be computed. Thus, the infinite sum has to be truncated such that a given tolerance is achieved. Clearly, denoting by \(\tilde{u}\) the approximated (truncated) solution and \(u_r\) the truncation error, i.e.,

Similar results for specific equations can be found, e.g., in [1, 7, 9]. A general truncation method is stated in [24]. The same ideas can be applied to our equations once we have performed the regularization to the deterministic system (such that operator DAE is well-posed in each level), the convergence of the truncated expansion is, in general, guaranteed by the stability result of Theorem 6. The main steps of the numerical approach are sketched in Algorithm 3.1.

## 4 More general cases

This section is devoted to the discussion of two generalizations. First, we consider general coordinatewise operators instead of simple coordinatewise operators as in the previous section. Thus, following the definition from Sect. 3.1.4, we allow the operators \(\mathcal {K} \) and \(\mathcal {B} \) to be composed out from families of deterministic operators \(\{K_\alpha \}_{\alpha \in \mathcal I}\) and \(\{B_\alpha \}_{\alpha \in \mathcal I}\), respectively, which may not be the same for all multi-indices. Second, we replace the Gaussian noise term by a stochastic integral term. The mean dynamics will remain unchanged, while the perturbation in the differential equation will be given in the form of a stochastic convolution.

Throughout this section we keep the following assumptions.

### Assumption 3

- 1.
The operator \(\mathcal K\) is a coordinatewise operator that corresponds to a family \(\{K_\alpha \}_{\alpha \in \mathcal I}\) of deterministic operators \(K_\alpha :\mathcal {V} \rightarrow \mathcal {V} ^*\), \(\alpha \in \mathcal I\). The operators \(K_\alpha \), \(\alpha \in \mathcal I\), are linear, continuous, and positive on the kernel of

*B*. - 2.
The constraint operator \(\mathcal B\) is a coordinatewise operator that corresponds to a family \(\{B_\alpha \}_{\alpha \in \mathcal I}\) of deterministic operators \(B_\alpha :\mathcal {V} \rightarrow \mathcal {Q} ^*\), \(\alpha \in \mathcal I\). The opertors \(B_\alpha \) are linear and for every \(\alpha \in \mathcal I\) there exists a right-inverse which is denoted by \(B_\alpha ^-\).

- 3.
The operators \(K_\alpha \) and \(B_\alpha \) are uniformly bounded.

- 4.
The stochastic processes \(\mathcal {F} \) and \(\mathcal {G} \) are given in their chaos expansion forms (16) such the conditions (17) hold.

### 4.1 Coordinatewise operators

- 1.for \(|\alpha | =0\), i.e., for \(\alpha = \mathbf {0}\),$$\begin{aligned} \dot{u}_{\mathbf {0} }(t) + {K}_\mathbf {0} u_{\mathbf {0} }(t) + {B}_\mathbf {0}^* \lambda _{\mathbf {0} } (t)&= f_{\mathbf {0}}(t),\\ {B}_\mathbf {0} u_{\mathbf {0} }(t)&= g_{\mathbf {0}}(t), u_{\mathbf {0}} = u^0_{\mathbf {0}}. \end{aligned}$$
- 2.for \(|\alpha | =1\), i.e., for \(\alpha = \varepsilon ^{(k)}\), \(k\in \mathbb N\),with \(u_{\varepsilon ^{(k)} } (0) = u^0_{\varepsilon ^{(k)}}.\)$$\begin{aligned} \dot{u}_{\varepsilon ^{(k)}}(t) + {K}_{\varepsilon ^{(k)}} u_{\varepsilon ^{(k)} }(t) + {B}^*_{\varepsilon ^{(k)}} \lambda _{\varepsilon ^{(k)} } (t)&= f_{\varepsilon ^{(k)}}(t) + m_k^{(1)}(t),\\ {B}_{\varepsilon ^{(k)}} u_{\varepsilon ^{(k)}}(t)&= g_{\varepsilon ^{(k)}}(t) + m_k^{(2)}(t), \end{aligned}$$
- 3.for the remaining \(|\alpha |>1\),$$\begin{aligned} \dot{u}_{\alpha }(t) + {K}_\alpha u_{\alpha }(t) + {B}_\alpha ^* \lambda _{\alpha } (t)&= f_\alpha (t), \\ {B}_\alpha u_{\alpha }(t)&= g_{\alpha }(t), u_{\alpha }(0) = u_{\alpha }^0. \end{aligned}$$

### 4.2 Stochastic convolution

*Itô-Skorokhod stochastic integral*. The Skorokhod integral is a generalization of the Itô integral for processes which are not necessarily adapted. The fundamental theorem of stochastic calculus connects the Itô-Skorokhod integral with the Wick product by

The operator \(\delta \) is the adjoint operator of the Malliavin derivative \(\mathbb D\). Their composition is known as the *Ornstein-Uhlenbeck operator* \(\mathcal R\) which is a self-adjoint operator. These operators are the main operators of an infinite dimensional stochastic calculus of variations called the *Malliavin calculus* [37]. We consider these operators in Sect. 5.

*v*the Itô integral and the Skorokhod integral coincide, i.e., \(I(v) = \delta (v)\). Because of this fact, we refer to the stochastic integral as the Itô-Skorokhod integral. Applying the definition of the Wick product (14) to the chaos expansion representation (9) of a process

*v*and the representation (11) of a singular white noise in the definition (45) of \(\delta (v)\), we obtain a chaos expansion representation of the Skorokhod integral. Clearly, for \(v=\sum _{\alpha \in \mathcal I} v_\alpha (t) H_\alpha \) we have

*v*, see also [22, 28].

### Definition 2

*v*is of the form (46) and we write \(v\in {{\mathrm{Dom}}}(\delta )\).

### Theorem 10

### Proof

*v*satisfy condition (47). Then we have

A detailed analysis of the domain and the range of operators of the Malliavin calculus in spaces of stochastic distributions can be found in [28, 29, 31].

First, we solve the stochastic operator DAE (44) with the stochastic perturbations given in terms of a stochastic convolution and without disturbance in the constraint equation. In order to prove the convergence of obtained solution in the Kondratiev space of generalized processes it is necessary to assume uniform boundness of the family of operators \(C_{\alpha }\), \(\alpha \in \mathcal I\). Then, we consider the stochastic operator DAE (44) with perturbation in the constraint equation that is given by a Gaussian noise term.

### Theorem 11

### Proof

- \(1^\circ \) for \(|\alpha | =0\), i.e., for \(\alpha = \mathbf {0}\),$$\begin{aligned} \dot{u}_{\mathbf {0} }(t) + {K}_{\mathbf {0} } u_{\mathbf {0} }(t) +{B}_{\mathbf {0} }^* \lambda _{\mathbf {0} } (t)= & {} f_\mathbf {0}(t),\nonumber \\ {B}_{\mathbf {0} } u_{\mathbf {0} }(t)= & {} g_\mathbf {0}(t). \end{aligned}$$(50)
- \(2^\circ \) for \(|\alpha | =1\), i.e., for \(\alpha = \varepsilon ^{(k)}\), \(k\in \mathbb N\),$$\begin{aligned} \begin{array}{rrl} \dot{u}_{\varepsilon ^{(k)} }(t) + {K}_{\varepsilon ^{(k)} }\, u_{\varepsilon ^{(k)} }(t) +{B}_{\varepsilon ^{(k)} }^* \, \lambda _{\varepsilon ^{(k)} } (t) &{} = &{} f_{\varepsilon ^{(k)}} + (C_{\mathbf {0}} \, u_{\mathbf {0}})_k, \\ {B}_{\varepsilon ^{(k)} } \, u_{\varepsilon ^{(k)}}(t) &{} = &{} g_{\varepsilon ^{(k)}}(t) . \end{array} \end{aligned}$$(51)
- \(3^\circ \) for \(|\alpha |>1\),$$\begin{aligned} \dot{u}_{\alpha }(t) + {K}_{\alpha } \, u_{\alpha }(t) + {B}_{\alpha }^* \, \lambda _{\alpha } (t)= & {} f_\alpha (t) + \sum \limits _{k\in \mathbb N} \, \left( C_{\alpha - \varepsilon ^{(k)} } \, u_{\alpha -\varepsilon ^{(k)}}\right) _k, \nonumber \\ {B}_{\alpha } \, u_{\alpha }(t)= & {} g_{\alpha }(t) . \end{aligned}$$(52)

*k*th component of the action of the operator \(C_{\mathbf {0}}\) on the solution \(u_{\mathbf {0}}\) obtained in the previous step, i.e., on the solution of the system (50). Similarly, the term \((C_{\alpha -\varepsilon ^{(k)} } \, u_{\alpha -\varepsilon ^{(k)}})_k\) from (52) represents the

*k*th coefficient obtained by the action of the operator \(C_{\alpha - \varepsilon ^{(k)} } \) on \(u_{\alpha - \varepsilon ^{(k)} }\) calculated in the previous steps. We use the convention that \(C_{\alpha - \varepsilon ^{(k)} }\) exists only for those \(\alpha \in \mathcal I\) for which \(\alpha _k \ge 1\). Therefore, the sum \( \sum _{k\in \mathbb N} (C_{\alpha - \varepsilon ^{(k)} } \, u_{\alpha -\varepsilon ^{(k)}})_k\) has as many summands as the multi-index \(\alpha \) has non-zero components. For example, for \(\alpha =(2, 0, 1, 0, 0,\ldots )\) with two non-zero components \(\alpha _1=2\) and \(\alpha _3=1\), the sum has two terms \((C_{(1, 0, 1, 0, 0,\ldots )} u_{(1, 0, 1, 0, 0,\ldots )})_1\) and \((C_{(2, 0, 0, 0, 0,\ldots )} u_{(2, 0, 0, 0, 0,\ldots )})_3\).

We point out that, in contrast to the previous cases, the unknown coefficients are obtained by recursion. Thus, in order to calculate \(u_\alpha \), we need the solutions \(u_\beta \) for \(\beta <\alpha \) from the previous steps. Also this case can be found in applications, see for example [24, 29, 31, 33].

*u*in \(L^2(T;\mathcal {V} )\otimes (S)_{-1}\). Particularly, we have to show that for some \(p\in \mathbb N\) it holds

*p*large enough so that \(1 - Md >0\). With this, we have proven that the solution

*u*of (49 the norm \(\Vert u\Vert ^2_{{L^2(\mathcal {V} )\otimes (S)_{-1}}}\) is finite and thus, complete the proof of theorem norm. \(\square \)

Let us now consider briefly the stochastic operator DAE (44). This problem corresponds to the stochastic operator DAE (49 with additional disturbance in the constraint equation. Similar to Theorem 9, the regularization is needed and will be provided only for the coefficients \(u_\alpha \), when \(|\alpha |=1\). Thus one can obtain the error estimate of the solution of the initial problem (49 and the perturbed one (44), i.e. of the solutions of their corresponding problems in extended forms. Here we state the theorem, but omit the proof.

### Theorem 12

With this result, we close this section and consider a further generalization, namely the fully stochastic case.

## 5 An example involving operators of Malliavin calculus

*u*and

*F*belonging to appropriate spaces [37].

*X*. Taking in (1 the operators \(\mathcal B = \mathbb D\) and thus \(\mathcal B^* = \delta \), we can consider the stochastic operator DAE of the form

*v*and

*y*.

The results concerning the generalized Malliavin calculus and the equations involving these operators can be found in [28, 29, 31, 32]. The chaos expansion method combined with the regularization techniques presented in the previous sections can be applied also in this case. Here we present the direct chaos expansion approach and prove the convergence of the obtained solution.

- 1.The
*Malliavin derivative*, namely \(\mathbb D\), as a stochastic gradient in the direction of white noise, is a linear and continuous mapping \(\mathbb {D}:{{\mathrm{Dom}}}(\mathbb D) \subseteq X\otimes (S)_{-1} \rightarrow X\otimes S'(\mathbb R) \otimes (S)_{-1}\) given byfor \(u=\sum _{\alpha \in \mathcal I}u_\alpha \otimes H_\alpha \), \(u_\alpha \in X\), \(\alpha \in \mathcal I\). We say that a process$$\begin{aligned} \mathbb {D} u=\sum _{\alpha \in \mathcal {I}}\sum _{k\in \mathbb {N}}\, \alpha _k \, u_\alpha \, \otimes \, \xi _k\, \otimes {H}_{\alpha -\varepsilon _k}, \end{aligned}$$(55)*u*is differentiable in Malliavin sence, i.e., it belongs to the domain \({{\mathrm{Dom}}}(\mathbb D)\) if and only if for some \(p\in \mathbb N_0\) it holds thatThe operator \(\mathbb D\) reduces the order of the Wiener chaos space and it holds that the kernel \({{\mathrm{Ker}}}(\mathbb D)\) consists of constant random variables, i.e., random variables having the chaos expansion in the Wiener chaos space of order zero. In terms of quantum theory, this operator corresponds to the annihilation operator.$$\begin{aligned} \sum _{\alpha \in \mathcal I} \, |\alpha |^2 \, \Vert u_\alpha \Vert ^2_X \, (2\mathbb N)^{-p\alpha } < \infty . \end{aligned}$$ - 2.The
*Itô-Skorokhod integral*, namely \(\delta \), is a linear and continuous mapping \(\delta :X\otimes S'(\mathbb R) \otimes (S)_{-1}\rightarrow X\otimes (S)_{-1}\) given byNote that the domain \({{\mathrm{Dom}}}(\delta ) = X\otimes S'(\mathbb R) \otimes (S)_{-1}\). The operator \(\delta \) is the adjoint operator of the Malliavin derivative. It increases the order of the Wiener chaos space and in terms of quantum theory \(\delta \) corresponds to the creation operator.$$\begin{aligned} \delta (F )=\sum _{\alpha \in \mathcal {I}}\sum _{k\in \mathbb {N}} f_{\alpha }\otimes v_{\alpha ,k} \otimes H_{\alpha +\varepsilon _k}, \,\,\text{ for } F=\sum _{\alpha \in \mathcal I} f_\alpha \otimes \left( \sum _{k\in \mathbb {N}}v_{\alpha , k}\, \xi _k\right) \otimes H_\alpha . \end{aligned}$$ - 3.The
*Ornstein-Uhlenbeck operator*, namely \(\mathcal R\), as the composition \(\delta \circ \mathbb D\), is the stochastic analogue of the Laplacian. It is a linear and continuous mapping \(\mathcal R:X\otimes (S)_{-1}\rightarrow X\otimes (S)_{-1}\) given byClearly, \(\mathcal R\) is a coordinatewise operator and its domain \({{\mathrm{Dom}}}(\mathcal R)\) coincides with the domain \({{\mathrm{Dom}}}(\mathbb D)\). In terms of quantum theory, the operator \(\mathcal R\) corresponds to the number operator. It is a self-adjoint operator with eigenvectors equal to the basis elements \(H_\alpha \), \(\alpha \in \mathcal I\), i.e., \(\mathcal R(H_\alpha )=|\alpha |H_\alpha \), \(\alpha \in \mathcal I\). Therefore, Gaussian processes from the Wiener chaos space of order one with zero expectation are the only fixed points for the Ornstein-Uhlenbeck operator [28, 31].$$\begin{aligned} \mathcal R(u) = \sum _{\alpha \in \mathcal I}|\alpha |u_\alpha \otimes H_\alpha \quad \text{ for } u=\sum _{\alpha \in \mathcal I}u_\alpha \otimes H_\alpha . \end{aligned}$$

*u*in the space of stochastic processes \(X\otimes (S)_{-1}\). Then by subtracting the obtained solution

*u*in the first equation of (54) we solve an integral equation and obtain the explicit form of \(\lambda \) in the space of generalized \(S'(\mathbb R)\)-stochastic processes.

### Theorem 13

### Proof

### Theorem 14

### Proof

Let \(u_1\) and \(u_2\) be the solutions of the system (59). From the linearity of the operator \(\mathbb D\) and the linearity of \(\mathbb E\) it follows \(\mathbb D u = \mathbb D (u_1+u_2) = \mathbb D u_1 + \mathbb D u_2 = y\) and \( \mathbb E u = \mathbb E (u_1+u_2) = \mathbb E u_1 + \mathbb E u_2 = u^0\). Thus the superposition of \(u_1\) and \(u_2\) solves (58).

Let now *u* be the solution of (58). By Theorem 13 it has chaos expansion representation form (56). The kernel of \(\mathbb D\), i.e., \({{\mathrm{Ker}}}(\mathbb D) \) is equal to \(\mathcal H_0\) and therefore *u* can be expressed in the form \(u=u_1 + u_2\), where \(u_1\in {{\mathrm{Ker}}}(\mathbb D)\) and \(u_2\in {{\mathrm{Im}}}(\mathbb D)\). Thus, by (56) we conclude that \(\mathbb D u_1 =0\) and \(\mathbb E u_1 = u^0\), while \(\mathbb D u_2 = y\) and \(\mathbb E u_2 =0\). \(\square \)

### 5.1 Extension to nonlinear equations

*u*and

*v*,

*i*th order of the Malliavin derivative operator, one can construct approximations of finite stochastic order. Particularly, the nonlinear advection term in the Navier–Stokes equations can be approximated by

*Q*denotes the highest stochastic order in the Wick-Malliavin expansion. The zero-order approximation \((u\cdot \nabla ) u \simeq (u \lozenge \nabla ) u \) is known as the Wick approximation, while \((u\cdot \nabla ) u \simeq (u \lozenge \nabla ) u + (\mathbb D u \lozenge \nabla ) \, \mathbb D u \) is the first-order Wick-Malliavin approximation [44]. As the Malliavin derivate has an explicit chaos expansion representation form (55), the formula (60) allows us to express the nonlinear advection term in terms of chaos expansions. Therefore, the ideas presented in this paper for the linear semi-explicit stochastic operator DAEs can be extended to Navier–Stokes equations and in general to equations with nonlinearities of the type (60). Moreover, the multiplication formula

## 6 Conclusion

We have analyzed the influence of stochastic perturbations to linear operator DAEs of semi-explicit structure. With the application of the polynomial chaos expansion, we could reduce the problem to a system of deterministic operator DAEs. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. With this, we have proven the existence and uniqueness of a solution of the stochastic operator DAE in a weighted space of generalized stochastic processes. Examples analyzed in this paper are the Stokes equations and the linearized Navier–Stokes equations. Moreover, the results of this paper can be extended to a certain type of nonlinear equations including Navier–Stokes.

## Notes

### Acknowledgments

Open access funding provided by University of Innsbruck and Medical University of Innsbruck. R. Altmann was supported by the ERC Advanced Grant “Modeling, Simulation and Control of Multi-Physics Systems” MODSIMCONMP. H. Mena was supported by the project *Solution of large-scale Lyapunov Differential Equations* (P 27926) founded by the Austrian Science Foundation. Moreover, the authors would like to thank the referees for their valuable suggestions which helped to improve this paper.

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