Solution Properties of a 3D Stochastic Euler Fluid Equation

We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.


Introduction
The present paper shows that two important analytical properties of deterministic Euler fluid dynamics in three dimensions possess close counterparts in the stochastic Euler fluid model introduced in Holm (2015). The first of these analytical properties is the local-in-time existence and uniqueness of deterministic Euler fluid flows. The second property is a criterion for blow-up in finite time due to Beale et al. (1984). For a historical review of these two fundamental analytical properties for deterministic Euler fluid dynamics, see e.g. Gibbon (2008). We believe this fidelity of the stochastic model of Holm (2015) investigated here with the analytical properties of the deterministic case bodes well for the potential use of this model in, for example, uncertainty quantification of either observed or numerically simulated fluid flows. The need and inspiration for such a model can be illustrated, for example, by examining data from satellite observations collected in the National Oceanic and Atmospheric   (2017)). Upon looking carefully at the individual Lagrangian paths in this figure, one sees that each of them evolves as a mean drift flow, composed with an erratic flow comprising rapid fluctuations around the mean (Colour figure online) Administration (NOAA) "Global Drifter Program", a compilation of which is shown in Fig. 1. Figure 1 (courtesy of Lilly 2017) displays the global array of surface drifter displacement trajectories from the National Oceanic and Atmospheric Administration's "Global Drifter Program" (www.aoml.noaa.gov/phod/dac). In total, more than 10,000 drifters have been deployed since 1979, representing nearly 30 million data points of positions along the Lagrangian paths of the drifters at 6-h intervals. This large spatiotemporal data set is a major source of information regarding ocean circulation, which in turn is an important component of the global climate system. For a recent discussion, see for example Sykulski et al. (2016). This data set of spatiotemporal observations from satellites of the spatial paths of objects drifting near the surface of the ocean provides inspiration for further development of data-driven stochastic models of fluid dynamics of the type discussed in the present paper.
Inspired by this drifter data, the present paper investigates the existence, uniqueness and singularity properties of a recently derived stochastic model of the Euler fluid equations for incompressible flow (Holm 2015) that is consistent with this data. For this purpose, we combine methods from geometric mechanics, functional analysis and stochastic analysis. In the model under investigation, one assumes that the Lagrangian particle paths in the fluid motion x t = η t (X ) with initial position X ∈ R 3 each follow a Stratonovich stochastic process given by dη t (X ) = u(η t (X ), t)dt + i ξ i (η t (X )) • dB i t .
(1.1) This approach immediately introduces the issue of spatial correlations. In particular, an important feature of the data in Fig. 1 is that the ocean currents show up as persistent spatial correlations, easily recognized visually as spatial regions in which the colours representing individual paths tend to concentrate. To capture this feature, we transform Lagrangian trajectory description (1.1) into the spatial representation of the Eulerian transport velocity given by the Stratonovich stochastic vector field, (1.2) In Eqs. (1.1) and (1.2), the B i t with i ∈ N are scalar independent Brownian motions, and the ξ i (x) represent the spatial correlations which may be obtained as eigenvectors of the two-point velocity-velocity correlation matrix C i j (x, y), i, j = 1, 2, . . . , N , as an integral operator. Namely, j C i j (x, y)ξ j (y)dy = λξ i (x) . (1.3) These correlation eigenvectors exhibit a spectrum of spatial scales for the trajectories of the drifters, indicating the variety of spatiotemporal scales in the evolution of the ocean currents which transport the drifters. This feature of the data is worthy of further study.
In what follows, we will assume that the velocity correlation eigenvectors ξ i (x) with i = 1, . . . , N have been determined by reliable data assimilation procedures, so we may take them to be prescribed, divergence-free, three-dimensional vector functions.
For explicit examples of the process of determining the ξ i (x) eigenvectors at coarse resolution from finely resolved numerical simulations, see Cotter et al. (2018a, b).
For an extension of this method to include non-stationary correlation statistics, see Gay-Balmaz and Holm (2018). A rigorous analysis of the stochastic process η t in (1.1) is under way by the authors. Following from classical results (e.g. Kunita 1984Kunita , 1990, we show in a forthcoming paper that η t is a temporally stochastic curve on the manifold of smooth invertible maps with smooth inverses (i.e. diffeomorphisms). Thus, although the time dependence of η t in (1.1) is not differentiable, its spatial dependence is smooth. The stochastic process dη t (X ) in (1.1) is also the pullback by the diffeomorphism η t of the stochastic vector field dy t (x) in (1.2). That is, η * t dy t (x) = dη t (X ) (see e.g. Holm (2015) for details). Conversely, the stochastic vector field in (1.2) is the Eulerian representation in fixed spatial coordinates x of the stochastic process in (1.1) for the Lagrangian fluid parcel paths, labelled by their Lagrangian coordinates X .
The expression for the Lagrangian trajectories in Eq. (1.1) is clearly in accord with the observed behaviour of the Lagrangian trajectories displayed in Fig. 1. Moreover,expression (1.2) for the corresponding Eulerian transport velocity has been derived recently in Cotter et al. (2017) by using multi-time homogenization methods for Lagrangian trajectories corresponding to solutions of the deterministic Euler equations, in the asymptotic limit of timescale separation between the mean and fluctuating flow. In particular, the fluctuating dynamics in the second term in (1.2) has been shown in Cotter et al. (2017) to affect the mean flow. Thus, beyond being potentially useful as a means of uncertainty quantification, the decomposition in (1.2) represents a bona fide decomposition of the Eulerian fluid velocity into mean plus fluctuating components.
The approach of incorporating uncertainties in incompressible fluid motion via stochastic Lagrangian fluid trajectories as in Eq. (1.1) has several precedents, including Brzézniak et al. (1991), Mikulevicius and Rozovskii (2004) and Mémin (2014). However, the Eulerian fluid representation in (1.2) will lead us next to a stochastic partial differential equation (SPDE) for the Eulerian drift velocity u driven by cylindrical noise represented by the Stratonovich term in (1.2) which differs from the Eulerian equations treated in these precedents. For detailed discussions of SPDE with cylindrical noise, see Prato and Zabczyk (2015), Pardoux (2007), Prévôt and Röckner (2007), Schaumlöffel (1988).

Stochastic Euler Fluid Equations
As shown in Holm (2015) via Hamilton's principle and re-derived via Newton's law in "Appendix A" of the present paper, the stochastic Euler fluid equations we shall study in this paper may be represented in Kelvin circulation theorem form, as in which the closed loop c(t) follows the Lagrangian stochastic process in (1.1), which means it moves with stochastic Eulerian fluid velocity dy t in (1.2). In Kelvin's circulation theorem (1.4), the mass density is denoted as ρ, and F j denotes the jth component of the force exerted on the flow. In the present work, the mass density ρ will be assumed to be constant. Notice that the covariant vector with components v j (x, t) in the integrand of (1.4) is not the transport velocity in (1.2). Instead, v j (x, t) is the jth component of the momentum per unit mass. In what follows, the force per unit mass ρ −1 F j = − ρ −1 ∂ j p will be taken to be proportional to the pressure gradient. For this force, the Kelvin loop integral in (1.4) for the stochastic Euler fluid case will be preserved in time for any material loop whose motion is governed by Stratonovich stochastic process (1.1). That is, Eq. (1.4) implies, for every rectifiable loop c ⊂ R 3 , the momentum per unit mass v t has the property that for all t ∈ [0, T ], pathwise Kelvin theorem (1.5) is reminiscent of the Constantin-Iyer Kelvin theorem in Constantin and Iyer (2008) which has the beautifully simple implication that smooth Navier-Stokes solutions u t are characterized by the following statistical Kelvin theorem which holds for all loops ⊂ R 3 , where A t is the back-to-labels map for a stochastic flow of a certain forward Itô equation and E denotes expectation for that flow. Unlike pathwise Kelvin theorem (1.5) which holds for solutions of the stochastic Euler fluid equations, Constantin-Iyer Kelvin theorem in (1.6) is completely deterministic, since the fluid velocity u t is a solution of the Navier-Stokes equations. For more discussion of Kelvin circulation theorems for stochastic Euler fluid equations, see Drivas and Holm (2018).
In the case of the stochastic Euler fluid treated here in Euclidean coordinates, applying the Stokes theorem to the Kelvin loop integral in (1.4) yields the equation for ω = curl v proposed in Holm (2015), as where the loop integral on the right-hand side of (1.4) vanishes for pressure forces with constant mass density.
Main Results This paper shows that two well-known analytical properties of the deterministic 3D Euler fluid equations are preserved under the stochastic modification in (1.7) we study here. First, 3D stochastic Euler fluid vorticity Eq. (1.7) is locally wellposed in the sense that it possesses local-in-time existence and uniqueness of solutions, for initial vorticity in the space W 2,2 (R 3 ) (Ebin and Marsden 1970). See Lichtenstein (1925) as mentioned in Frisch and Villone (2014) for a historical precedent for local existence and uniqueness for the Euler fluid equations. Second, vorticity Eq. (1.7) also possesses a Beale-Kato-Majda (BKM) criterion for blow-up which is identical to the one proved for the deterministic Euler fluid equations in Beale et al. (1984).
Our result corresponding to the celebrated Beale-Kato-Majda characterization of blow-up (Beale et al. 1984) is stated in the following.

Plan of the Paper
• Section 2 discusses our assumptions and summarizes the main results of the paper.
-Section 2.1 formulates our objectives and sets the notation. -Section 2.2 discusses the cylindrical noise properties of (1.1) and provides basic bounds on the Lie derivatives needed in proving the main analytical results. -Section 2.3 provides additional definitions needed in the context of explaining the main results of the paper.
• Section 3 provides proofs of the main results -Sections 3.1 and 3.3 prove the uniqueness properties needed for establishing Theorem 1. -Section 3.5 introduces a cut-off function which is instrumental in the proof of the BKM theorem for the stochastic Euler equations given in Section 3.4.
• Section 4 summarizes the proofs of several key technical results which are summoned in establishing Theorems 1 and 2.
-Section 4.1 discusses fractional Sobolev regularity in time.
-Section 4.2 provides the a priori bounds needed to prove estimate (3.19).
-Section 4.3 proves the bounds needed to complete the proof that estimate (3.19) is uniform in time. -Section 4.4 establishes the key estimates for the bounds involving Lie derivatives that are needed in the proofs.
• "Appendix A" provides a new derivation of the stochastic Euler equations introduced in Holm (2015) from the viewpoint of Newton's second law and derives the corresponding Kelvin circulation theorem. The deterministic (resp. stochastic) equations of motion are derived using the pullback of Newton's second law by the deterministic (resp. stochastic) diffeomorphism describing the Lagrange-to-Euler map. The Kelvin circulation theorems for both cases are then derived from their corresponding Newtonian 2nd Laws. The importance of the distinction between transport velocity and transported momentum is emphasized in "Appendix A" for both the deterministic and stochastic Newton's Laws and Kelvin's circulation theorems.

Formulating Objectives and Setting Notation
Our aim from now on will be to prove local-in-time existence and uniqueness of regular solutions of the stochastic Euler vorticity equation which was proposed in Holm (2015). Here, L v ω (resp. L ξ k ω) denotes the Lie derivative with respect to the vector fields v (resp. ξ k ) as in (A.27) applied to the vorticity vector field. In particular, A natural question is whether we should sum only over a finite number of terms or, on the contrary, it is important to have an infinite sum, and not only for generality. An important remark is that a finite number of eigenvectors arises in the relevant case associated with a "data-driven" model based on what is resolvable in either numerics of observations, and it would simplify some technical issues [we do not have to assume (2.11)]. However, an infinite sum could be of interest in regularization-by-noise investigations: see an example in Delarue et al. (2014) (easier than 3D Euler equations) where a singularity is prevented by an infinite-dimensional noise. However, it is also true that in some cases a finite-dimensional noise is sufficient also for regularization by noise (see examples in Flandoli et al. (2010Flandoli et al. ( , 2011Flandoli et al. ( , 2014).
As mentioned in Remark 32, for the case of the Euler fluid equations treated here in Cartesian R 3 coordinates, the two velocities denoted u and v in the previous section may be taken to be identical vectors for the case at hand in R 3 . Consequently, for the remainder of the present work, in a slight abuse of notation, we simply let v denote the both fluid velocity and the momentum per unit mass. Then, ω = curl v is the vorticity, and ξ k comprise N divergence-free prescribed vector fields, subject to the assumptions stated below. The processes B k with k ∈ N are scalar independent Brownian motions. The result we present next will extend the known analogous result for deterministic Euler equations to the stochastic case.
To simplify some of the arguments, we will work on a torus T 3 = R 3 /Z 3 . However, the results should also hold in the full space, R 3 .
Stochastic Euler vorticity Eq. (2.1) above is stated in Stratonovich form. The corresponding Itô form is where we write for the double Lie bracket of the divergence-free vector field ξ k with the vorticity vector field ω. Indeed, let us recall that Stratonovich integral is equal to Itô integral plus one half of the corresponding cross-variation process 1 : By the linearity and the space independence of B k , L ξ k ω, where B h are independent, the cross-variation process ω, B k t is given by and therefore, in differential form, Among different possible strategies to study Eq. (2.3), some of them based on stochastic flows, we present here the extension to the stochastic case of a classical PDE proof (see, for instance, Kato and Lai 1984;Lions 1996;Majda and Bertozzi 2002).
The proof is based on a priori estimates in high-order Sobolev spaces. The deterministic classical result proves well-posedness in the space ω (t) ∈ W 3/2+ ,2 T 3 ; R 3 , for some > 0, when ω 0 belongs to the same space. Here we simplify (due to a number of new very non-trivial facts outlined in Sect. 4.2) and work in the space ω (t) ∈ W 2,2 T 3 ; R 3 . Consequently, we may consider ω (t) (to avoid fractional derivatives) and investigate existence and uniqueness in the class of regularity We consider the basis of L 2 T 3 ; C of functions e 2πiξ ·x ; ξ ∈ Z 3 , and for every f ∈ L 2 T 3 ; C , we introduce the Fourier coefficients If v ∈ L 2 T 3 ; R 3 is a vector field with components v i , i = 1, 2, 3, we write v (ξ ) = T 3 e 2πiξ ·x v (x) dx and we may easily check using the components that have Since functions which are partial derivatives of other functions, on the torus, must have zero average, we shall always restrict ourselves to functions f ∈ L 2 T 3 ; C such that T 3 f (x) dx = 0. In this case, f (0) = 0 and the term with ξ = 0 does not appear in the sums above. We introduce, for every s ≥ 0, the fractional Sobolev space W s,2 T 3 ; C of all f ∈ L 2 T 3 ; C such that |ξ | 2s f (ξ ) 2 < ∞.
As stated above, we are assuming zero average functions; hence, we have excluded ξ = 0. We denote by W s,2 σ T 3 , R 3 the space of all zero mean divergence-free (divergence in the sense of distribution) vector fields v ∈ L 2 T 3 ; R 3 such that all components v i , i = 1, 2, 3, belong to W s,2 T 3 ; C . For a vector field v ∈ W s,2 For f ∈ W s,2 T 3 ; C , we denote by (− ) s/2 f the function of L 2 T 3 ; C with Fourier coefficients |ξ | s f (ξ ). Similarly, we write − −1 f for the function having Fourier coefficients |ξ | −2 f (ξ ). We use the same notations for vector fields, meaning that the operations are made componentwise.
The Biot-Savart operator is the reconstruction of a zero mean divergence-free vector field u from a divergence-free vector field ω such that curl u = ω. On the torus, it is given by u = − curl −1 ω. In Fourier components, it is given by u (ξ ) = |ξ | −2 ξ × ω (ξ ). We have the following well-known result: for all s ≥ 0 (2.4) Indeed, using the definition given above of u 2 W s+1,2 σ , the formula which relates u (ξ ) to ω (ξ ) and the rule |a × b| ≤ |a| |b|, we get |ξ | 2s+2 |ξ | −2 | ω (ξ )| 2 and the latter is precisely equal to ω W s,2 σ , by the definition above. We shall denote the dual operator of the Lie derivative L α of a vector field as L * α , defined by the identity for all smooth vector fields α, β, γ . When div α = 0, the dual Lie operator is given in vector components by (2.5)

Assumptions on { k } and Basic Bounds on Lie Derivatives
We assume that the vector fields ξ k : T 3 → R 3 are of class C 4 and satisfy (2.8) These properties will be used below, both to give a meaning to the stochastic terms in the equation and to prove certain bounds. In addition, a recurrent energy-type scheme in our proofs requires comparisons of quadratic variations and Stratonovich corrections. Making these comparisons leads to sums of the form L 2 In dealing with them, we have observed the validity of two striking bounds, which a priori may look surprising. They are: k . For these estimates to hold, the regularity of f must be, respectively, W 2,2 T 3 ; R 3 and W 4,2 T 3 ; R 3 . The proofs of estimates (2.9) and (2.10) are given in Sect. 4.4. 2 Concerning inequality (2.9), it is clear that the second-order terms in L 2 ξ k f , f and L ξ k f , L ξ k f will cancel. However, the cancellations among the first-order terms are not so obvious. Remarkably, though, these terms do cancel each other, so that only the zero-order terms remain. Similar remarks apply to the other inequality.
In addition, we must assume (2.11) Because the constants C (i) k are rather complicated, we will not write them explicitly here. In the relevant case of a finite number of ξ k 's, there is obviously no need of this assumption. In the case of infinitely many terms, see a sufficient condition in Remark 28 of Sect. 4.4.

Statement of the Main Results
Let B k k∈N be a sequence of independent Brownian motions on a filtered probability space ( , F, F t , P). We do not use the most common notation for the probability space, since ω is the traditional notation for the vorticity. Thus, the elementary events will be denoted by θ ∈ . Let {ξ k } k∈N be a sequence of vector fields, satisfying the assumptions of Sect. 2.2. Consider Eq. (2.3) on [0, ∞).

Definition 3 (Local solution)
A local solution in W 2,2 σ of the stochastic 3D Euler Eq. (2.3 ) is given by a pair (τ, ω) consisting of a stopping time τ : → [0, ∞) and a process ω : , is adapted to (F t ), and Eq. (2.3) holds in the usual integral sense; more precisely, for any bounded stopping timeτ ≤ τ holds as an identity in L 2 T 3 ; R 3 .
Remark 5 Due to assumptions (2.6) and (2.7) and the regularity of ω, the two terms related to the noise in Eq. (2.3) are well defined, as elements of L 2 T 3 ; R 3 .

Remark 6 Recall that, for every
which explains why the term [v, ω] is in L 2 T 3 ; R 3 . (Recall that Definition 3 instructs us to interpret Eq. (2.3) as an identity in L 2 T 3 ; R 3 .)
In this paper, we will also prove a corresponding result to the celebrated Beale-Kato-Majda criterion for blow-up of vorticity solutions of the deterministic Euler fluid equations.

Remark 10
As in the deterministic case, Theorem 9 can be used as a criterion for testing whether a given numerical simulation has shown finite-time blow-up. Following Gibbon (2008), the classical Beale-Kato-Majda theorem implies that algebraic singularities of the type ω ∞ ≥ (t * − t) − p must have p ≥ 1. In our paper, we have shown that a corresponding BKM result also applies for the stochastic Euler fluid equations; hence, the same criterion applies here. In Constantin et al. (1996), the L ∞ condition in the BKM theorem was reduced to L p , for finite p, at the price of imposing constraints on the direction of vorticity. We hope to obtain a similar L ∞ result for the stochastic 3D-Euler equation in future work.
In Sects. 3.1 and 3.2, we prove uniqueness. The rest of the paper will be devoted to proving local existence of the solution and Theorem 9.

Local Uniqueness of the Solution of the Stochastic 3D Euler Equation
In the following proposition, we prove that any two local solutions of the stochastic 3D Euler Eq. (2.3) that are defined up to the same stopping time must coincide. The proof hinges on bound (2.9) and assumption (2.11).
Proposition 11 Let τ be a stopping time and ω (1) , ω (2) It follows We rewrite and use the following inequalities: Here and below, we repeatedly use the Sobolev embedding theorems for all s ≥ 0; see (2.4). For instance, the sequences of inequalities used above in the case of the terms V L 4 and ∇v (2) L ∞ were We omit similar detailed explanations sometimes below, when they are of the same kind.
Using also (2.11), we get where Y is defined as The inequality (recall 0 = 0) t∧τ 2 L 2 = 0 and thus, for every t, Recalling the continuity of trajectories, this implies The proof of the proposition is complete.

Existence of a Maximal Solution
Given R > 0, consider the modified Euler equations where C is a constant chosen so that Proof Obviously, because for t ∈ [0, τ R ] we have ∇v ∞ ≤ C ω W 2,2 ≤ R and thus κ R (ω R ) = 1; namely, the equations are the same.
The following proposition is the cornerstone of the existence and uniqueness of a maximal solution of stochastic 3D Euler Eq. (2.3) We postpone the proof of Proposition 13 to the later sections. For now, let us show how it implies the existence of a maximal solution.

Uniqueness of the Maximal Solution
Let us start by justifying the uniqueness of solution truncated Euler Eq. (3.2). The proof is similar with that of Proposition 11 so we only sketch it here. Let ω (1) 2 ). We preserve the same notation as in the proof of Proposition 11, i.e. denote by = ω (1) R . We also assume that the truncation function f R is Lipschitz and we will denote by K R the quantity and observe that, 4 To simplify notation, we will omit the dependence on R in the following. We are looking to prove uniqueness using the W 2,2 -topology; therefore, we need to estimate the sum (2) and thus Then, It follows that, on this set, there exists a constant c R such that (recall that 0 ≤ κ ≤ 1) and, similar to the proof of Proposition 11, we deduced that W 2,2 ≥ R C and 3.3 holds true, as seen by observing that there exists a constant c R such that (1) , .
Next we have From the Lemma 25, we have Moreover, by similar arguments, Similar estimates hold true for L V ω (2) , and L v (1) , . Next, as above, on the set τ (2) ≤ τ (1) observe there exists a constant c R such that Similarly, on the set τ (2) ≤ τ (1) , Summarizing, we deduce that It is then sufficient to repeat the argument of the proof of Proposition 11 to control 2 L 2 + 2 L 2 and obtain the uniqueness of the truncated Euler equation. The computation required here requires more regularity in space than what we have for our solutions (we have to compute, although only transiently, L 2 ξ k ). In order to make the computation rigorous, one has to regularize solutions by mollifiers or Yosida approximations and do the computations on the regularizations. In this process, commutators will appear and one has to check at the end that they converge to zero. The details are tedious, but straightforward and we do not write all of them here.
Proof From the local uniqueness result proved above, we deduce that ω =ω on [0, min (τ, τ max )). By an argument similar to the one in Theorem 14, we cannot have τ max < τ on any non-trivial set. Hence, τ < τ max . But then from the maximality property of (τ,ω), it follows that necessarily τ = τ max and therefore ω =ω on [0, τ max ).

Lemma 16
There is a constant C such that 5 Proof By comparing Beale et al. (1984), the following inequality holds true The result is then obtained from (3.4), the obvious inequality 1 + log + a ≤ C log (a + e) for C sufficiently large (say C ≥ 2) and the fact that ||ω|| 2 ≤ C ||ω|| ∞ on a torus.
Step 2 τ 2 ≤ τ 1 . P-a.s. We prove that for any n, k > 0 we have In particular, sup s∈[0,τ 2 n ∧k] ||ω t || 2,2 is a finite random variable P-almost surely, that is P sup ||ω t || 2,2 < ∞ = 1. 5 We thank James-Michael Leahy for pointing out an error in an earlier version of the proof of this result.
which gives us (3.7). The proof is now complete.

Remark 18
The original Beale-Kato-Majda result refers to a control of the explosion time of ||v|| 3,2 in terms of ||ω|| ∞ . Our result refers to a control of the explosion time for ||ω|| 2,2 in terms of ||ω|| ∞ . However, due to (3.4), we can restate our result in terms of ||v|| 3,2 as well.

Global Existence of the Truncated Solution
Consider the following regularized equation with cut-off, with ν, R > 0, where ω ν R = curl v ν R , div v ν R = 0. On the solutions of this problem, we want to perform computations involving terms like L 2 v ω (t), so we need ω (t) ∈ W 4,2 T 3 ; R 3 . This is why we introduce the strong regularization ν 5 ω ν R ; the precise power 5 can be understood from the technical computations of Step 1. While not optimal, it a simple choice that allows us to avoid more heavy arguments.
This regularized problem has the following property. We understand Eq. (3.12) either in the mild semigroup sense (see below the proof) or in a weak sense over test functions, which are equivalent due to the high regularity of solutions. However, 5 ω ν R cannot be interpreted in a classical sense, since the solutions, although very regular, will not be in W 10,2 T 3 ; R 3 . The other terms of Eq. (3.12) can be interpreted in a classical sense.

Proof
Step 1 (preparation) In the following, we assume to have fixed T > 0 and that all constants are generically denoted by C > 0 any constant, with the understanding that it may depend on T .
Let D (A) = W 10,2 σ T 3 ; R 3 and A : D (A) ⊂ L 2 σ T 3 , R 3 → L 2 σ T 3 , R 3 be the operator Aω = ν 5 ω; L 2 σ T 3 , R 3 denotes here the closure of D (A) in L 2 T 3 , R 3 (the trace of the periodic boundary condition at the level of L 2 spaces can be characterized; see Temam 1977). The operator A is self-adjoint and negative definite. Let e t A be the semigroup in L 2 σ T 3 , R 3 generated by A. The fractional powers (I − A) α are well defined, for every α > 0, and are bi-continuous bijections between W β,2 σ T 3 ; R 3 and W β−10α,2 σ T 3 ; R 3 , for every β ≥ 10α, in particular In the sequel, we write f , g = T 3 f (x) · g (x) dx. We work on the torus, which simplifies some definitions and properties; thus, we write (1 − ) s/2 f for the function having Fourier transform 1 + |ξ | 2 s/2 f (ξ ) ( f (ξ ) being the Fourier transform of f ); similarly, we write −1 f for the function having Fourier transform |ξ | −1 f (ξ ).
The fractional powers commute with e t A and have the property (from the general theory of analytic semigroups, see Pazy 1983) that for every α > 0 and T > 0 for all t ∈ (0, T ] and f ∈ L 2 σ T 3 ; R 3 . From these properties, it follows that, for p = 2, 4 In particular, the map f → It is here that we use the power 5 of , otherwise a smaller power would suffice. Step 2 (preparation, cont.) The function ω → κ R (ω) L v ω from W 2,2 T 3 ; R 3 to L 2 T 3 ; R 3 is Lipschitz continuous, and it has linear grows (the constants in both properties depend on R). Let us check the Lipschitz continuity; the linear growth is an easy consequence, applying Lipschitz continuity with respect to a given element ω 0 .
It is sufficient to check Lipschitz continuity in any ball B (0, r ), centred at the origin of radius r , in W 2,2 T 3 ; R 3 . Indeed, when it is true, one can argue as follows. Take ω (i) , i = 1, 2, in W 2,2 T 3 ; R 3 . If they belong to B (0, R + 2), we have Lipschitz continuity. The case that both are outside B (0, R + 2) is trivial, because the cut-off function vanishes. If one is inside B (0, R + 2) and the other outside, consider the two cases: if the one inside is outside B (0, R + 1), it is trivial again, because the cut-off function vanishes for both functions. If the one inside, say L 2 ≤ C R (same computations done below) and ω (1) − ω (2) W 2,2 ≥ c R , for two constants c R , C R > 0, hence Therefore, let us prove that the function ω → κ R (ω) L v ω from W 2,2 T 3 ; R 3 to L 2 T 3 ; R 3 is Lipschitz continuous on B (0, r ) ⊂ W 2,2 T 3 ; R 3 . Given ω (i) ∈ B (0, r ), i = 1, 2, let us use the decomposition Then, by the Sobolev embedding theorem and (2.4). Finally because ∇v (1) 2 ∞ ≤ C ω (2) 2 W 2,2 as above, and then, we use the Lipschitz continuity of the function ω → κ R (ω).
Step 3 (local solution by fixed point). Given ω 0 ∈ L 2 ; W 2,2 σ T 3 , R 3 , F 0measurable, consider the mild equation The map , applied to an element ω ∈ Y T , gives us an element ω of the same space. Indeed: (i) e t A is bounded in W 2,2 T 3 ; R 3 (for instance because it commutes with we apply property (3.14) and get that The proof that is Lipschitz continuous in Y T is based on the same facts, in particular the Lipschitz continuity proved in Step 2. Then, using the smallness of the constants for small T in properties (3.13) and (3.14) of Step 1, one gets that is a contraction in Y T , for sufficiently small T > 0.
Step 4 (a priori estimate and global solution). The length of the time interval of the local solution proved in Step 2 depends only on the L 2 ; W 2,2 σ T 3 , R 3 norm of ω 0 . If we prove that, given T > 0 and the initial condition ω 0 , there is a constant C > 0 such that a solution ω defined on [0, T ] has sup t∈[0,T ] E ω (t) 2 W 2,2 ≤ C, then we can repeatedly apply the local result of Step 2 and cover any time interval.
Step 5 (Regularity) Let ω be the solution constructed in the previous steps; it is the sum of the four terms given by the mild formulation ω = ω. By the property e t A ω 0 ∈ D (A), namely Ae t A ω 0 ∈ L 2 σ T 3 , R 3 , for all t > 0 and ω 0 ∈ L 2 σ T 3 , R 3 (see [Pazy], property (5.7) in Theorem 5.2 of Chapter 2, due to the fact that e t A is an analytic semigroup), we may take δ > 0 and have

Remark 21
The Prohorov theorem states that, for a tight family of probability measures, one can extract a sequence μ ν n n∈N which weakly converges to some probability measure, for all bounded continuous functions ϕ : X → R. We repeatedly use these facts below.
In order to prove Proposition 14, we want to prove that the family of solutions ω ν R ν>0 (R is given) provided by Lemma 19 is compact is a suitable sense and that a converging subsequence extracted from this family converges to a solution of Eq. (3.17). Since ω ν R ν>0 are random processes, the classical method we follow is to prove compactness of their laws {μ ν } ν>0 . For this purpose, we have to prove that {μ ν } ν>0 is tight and we have to apply Prohorov theorem, as recalled above. The metric space where we prove tightness of the laws will be the space E given by (3.15). 6

Lemma 22
Let T > 0, R > 0 and ω 0 ∈ W 2,2 σ T 3 , R 3 be given. Assume that the family of laws of ω ν R ν>0 is tight in the space for some β > 3 2 and satisfies, for some constant C R > 0, for every ν > 0. Then, the existence claim of Proposition 13 holds true, and thus, Theorem 8 is proved.

Proof
Step 1 (Gyongy-Krylov approach). We base our proof on classical ingredients, but also on the following fact proved in Gyongy and Krylov (1996) Step 2 (Preparation by the Skorokhod theorem). Let us enlarge the previous pair by the noise and consider the following triple: the sequence ω We have to prove that the marginal μ E×E of μ on E × E is supported on the diagonal. By the Skorokhod representation theorem, there exists a probability space , F , P Step 3 (Property of being supported on the diagonal). The passage to the limit in Eq. (3.16) when there is strong convergence ( P-a.s.) in L 2 0, T ; L 2 T 3 , R 3 is relatively classical (see Flandoli and Gatarek (1995)). We sketch the main points in Step 4. One deduces in the weak sense explained in Remark 7. Since ω i R have paths in C [0, T ] ; W 2,2 T 3 , R 3 (see Step 4), the derivatives can be applied on ω i R by integration by parts and we get the equation in the strong sense. Now we apply the pathwise uniqueness of solutions for Eq. (3.2) in W 2,2 as deduced in Sect. 3.3 to deduce ω 1 R = ω 2 R . This means that the law of ω 1 R , ω 2 R is supported on the diagonal of E × E. Since this law is equal to μ E×E , we have that μ E×E is supported on the diagonal of E × E.
Step 4 (Convergence) In this step, we give a few details about the passage to the limit, as j → ∞, from Eqs. (3.16) to (3.17). We do not give the details about the linear terms, except for a comment about the term ν The difficult term is the nonlinear one, also because of the cut-off term κ R ω i R (s) . We want to prove that, given φ ∈ C ∞ T 3 , R 3 , with probability one. From the Skorokhod preparation in Step 2, we know that ω i, j R → ω i R as j → ∞ in the strong topology of E, P-a.s., for i = 1, 2. In the sequel, we fix the random parameter and the value of i = 1, 2. Since W β,2 σ T 3 , R 3 is continuously embedded into C T 3 , R 3 (recall that β > 3/2), it follows that ω i, j R → ω i R in the uniform topology over [0, T ] × T 3 . By the continuity of Biot-Savart map from W β,2 σ , and because these functions are bounded by 1, we can take the limit in (3.18). Therefore, it remains to prove that, P-a.s., κ R ω i, j R (s) converges to κ R ω i R (s) for a.e. s ∈ [0, T ], or at least in probability w.r.t. time. This is true because strong convergence in L 2 (0, T ) in time implies convergence in probability w.r.t. time, and we have strong convergence in converges to κ R ω i R in probability w.r.t. time. Finally, from the integral identity satisfied by the limit processω i , one can deduce thatω i ∈ C [0, T ] ; W 2,2 T 3 , R 3 following the argument in Kim (2009).
Based on this lemma, we need to prove suitable bounds on ω ν R ν>0 .
Proof We shall use the following variant of Aubin-Lions lemma, which can be found in Simon (1987). Recall that, given an Hilbert space W , a norm on W α,4 (0, T ; W ) is the fourth root of Assume that V , H , W are separable Hilbert spaces with continuous dense embedding V ⊂ H ⊂ W such that there exists θ ∈ (0, 1) and M > 0 such that Simon (1987), Corollary 9). We apply it to the spaces where β ∈ 3 2 , 2 . The constraint β < 2 is imposed because we want to use the compactness of the embedding W 2,2 T 3 , R 3 ⊂ W β,2 T 3 , R 3 . The constraint β > 3 2 is imposed because we want to use the embedding W β,2 T 3 , R 3 ⊂ C T 3 , R 3 .
Let {Q ν } be the family of laws of ω ν R , supported on by the assumption of the theorem. We want to prove that {Q ν } is tight in E. The sets K R 1 ,R 2 ,R 3 defined as and this is smaller than /3 when R 1 is large enough. Similarly, we get when R 3 is large enough. Finally, Hence, also this quantity is smaller than 3 when R 2 is large enough. We deduce Q ν K c R 1 ,R 2 ,R 3 ≤ and complete the proof.
The difficult part of the estimates above is bound (3.19). Thus, let us postpone it and first show bound (3.20).

Fractional Sobolev Regularity in Time
In this section, we show that bound (3.20), with N = 1, follows from (an easier version of) bound (3.19).

Proof
Step 1 (Preparation) In the sequel, we take t ≥ s and denote by C > 0 any constant. From Eq. (3.12), we have Recall that f W −3,2 ≤ C f L 2 , which follows by duality from f L 2 ≤ C f W 3,2 . Hence, again denoting any of the constants in the calculation below as C > 0, we have The only term where W −3,2 is necessary is the term 5 ω ν R (r ) 4 W −3,2 ; we keep it also in the first term, but this is not essential. Now let us estimate each term.
Step 2 (Estimates of the deterministic terms) We have Moreover, also Summarizing Therefore, For the next term, we have by assumption (2.6), and therefore, W 2,2 dr ≤ C as above.
Step 3 (Estimate of the stochastic term) One has, by the Burkholder-Davis-Gundy inequality by assumption (2.7), by the assumption of this lemma.
Step 4 (Conclusion) From the previous steps, we have

Some a priori Estimates
In order to complete the proof of Theorem 8, we still need to prove estimate (3.19). To be more explicit, since now a long and difficult computation starts, what we have to prove is that, given R > 0, called for every ν ∈ (0, 1) by ω ν R the solution of equation for every ν ∈ (0, 1). In order to simplify notations, we shall simply write not forgetting that all bounds have to be uniform in ν ∈ (0, 1).

Difficulty Compared to the Deterministic Case
In the deterministic case, d dt T 3 | ω (t, x)| 2 dx is equal to the sum of several terms. Using Sobolev embedding theorems (3.1), one can estimate all terms as except for the term with higher-order derivatives However, this term vanishes, being equal to In the stochastic case, though, we have many more terms, coming from two sources: (i) the term 1 2 k L 2 ξ k ω dt, which is a second-order differential operator in ω, hence much more demanding than the deterministic term L v ω; (ii) the Itô correction term in Itô formula for d T 3 | ω (t, x)| 2 dx.
A quick inspection in these additional terms immediately reveals that the highest order terms compensate (one from (i) and the other from (ii)) and cancel each other. These terms are of "order 6" in the sense that, globally speaking, they involve 6 derivatives of ω. The new outstanding problem is that there remains a large amount of terms of "order 5", hence not bounded by C T 3 | ω (t, x)| 2 dx (which is of "order 4"). After a few computations, one is naïvely convinced that these terms are too numerous to compensate and cancel one another.
But this is not true. A careful algebraic manipulation of differential operators, as well as their commutators and adjoint operators, finally shows that all terms of "order 5" do cancel each other. At the end, we are able to estimate remaining terms again by C T 3 | ω (t, x)| 2 dx (now in expectation) and obtain the a priori estimates we seek. Preparatory Remarks By again using the regularity result of Lemma 19, we may write the identity and we may apply a suitable Itô formula in the Hilbert space L 2 T 3 (see Krylov and Rozovskii 1979) to obtain Being hence, the Itô correction above is given by (we have to integrate in dx the previous identity) Let us list the main considerations about identity (4.1).
(1) The term ν T 3 2 ω 2 dx will not be used in the estimates, since they have to be independent of ν; we only use the fact that this term has the right sign.
(2) The term can be estimated by C T 3 | ω (t, x)| 2 dx exactly as in the deterministic theory. The computations are given in Sect. 4.2.
(3) The term ∞ k=1 T 3 L ξ k ω · ωdx dB k t is a local martingale. Rigorously, we shall introduce a sequence of stopping times, and then, taking expectation, this term will disappear. Then, the stopping times will be removed by a straightforward limit.
(4) The main difficulty comes from the term since it includes, as mentioned above in Sect. 4.2, various terms which are of "order 6" and of "order 5", where "order" means the global number of spatial derivatives. These terms cannot be estimated by C T 3 | ω (t, x)| 2 dx. As it turns out, the terms of "order 6" cancel each other: this is straightforward and expected. But a large number of intricate terms of "order 5" still remain, which, naïvely, may give the impression that the estimate cannot be closed. On the contrary, though, they also cancel each other: this is the content of Sect. 4.4, summarized in assumption (2.11).
Estimate of the Classical Term (4.2) The following lemma deals with the control of the classical term (4.2).
Lemma 25 Given u ∈ W 3,2 σ , ω ∈ W 2,2 σ (not necessarily related by curl u = ω), one has Proof Since the second inequality is derived from the first and the fact that ∇u W 2,2 ≤ C ω W 2,2 , we concentrate on the first. We use tools and ideas from the classical deterministic theory; see for instance (Beale et al. 1984;Kato and Lai 1984;Lions 1996;Majda and Bertozzi 2002). We have The term T 3 (v · ∇ ω) · ωdx is equal to zero, being equal to 1 2 T 3 v · ∇ | ω| 2 dx which is zero after integration by parts and using div v = 0. The terms ( ω · ∇v) · ωdx are immediately estimated by C ∇v L ∞ ω 2 W 2,2 . The term T 3 (ω · ∇ v) · ωdx is easily estimated by C ω L ∞ ω 2 W 2,2 . It remains to understand the other two terms. We have Hence, we only need to prove that for every α, β, γ = 1, 2, 3. Recall the following particular case of Gagliardo-Nirenberg interpolation inequality: which implies Moreover, due to the relation between v and ω, we also have ∇v W 2,2 ≤ C ω W 2,2 . Hence, and inequality (4.4) has been proved. The proof of the lemma is complete.

Estimates Uniform in Time
We introduce the following notations Consequently, following from the estimates of the previous section and assumption (2.11), we have (4.5) By the Burkholder-Davis-Gundy inequality (see e.g. Theorem 3.28, page 166 in Karatzas and Shreve 1991), we have that where [M] is the quadratic variation of the local martingale M and Lemma 26 Under the assumption (2.8), there is a constant C > 0 such that where R k ω contains several terms, each one with at most second derivatives of ω.
With a few more computations, it is possible to show that Hence, we use assumption (2.8).
From Lemma 26, we deduce and thus, finally from (4.5), (4.6) and (4.7) and Grönwall's inequality, we obtain independently of > 0. This proves bound (3.19) and completes the necessary a priori bounds, modulo the estimates of the next section.
Lemma 27 Inequality (2.9) holds for every vector field f of class W 2,2 .

Proof
Step 1 We have where S 2 and S 4 are certain zero-order operators (see below for a proof). We have However, since f , S 2 f = S 2 f , f for any f , f two square integrable vector fields (see below for a proof) Step 2 Now we prove that L * ξ k = − L ξ k + S 2 and that f , S 2 f = S 2 f , f for any two square integrable vector fields f , f . We also have by integration by parts and using ∇ · ξ k = 0 that Step 3. Finally, we prove that L ξ k S 2 = S 2 L ξ k − S 4 . We have ( Similarly, we find that

Remark 28
From this computation, one can easily deduce that where c is an independent constant (c = 48). Therefore, the first of assumptions (2.11) is fulfilled, provided The condition for the second assumption in (2.11) is similar.

Remark 29
A typical example arises when ξ k are multiples of a complete orthonormal system {e k } of L 2 , namely ξ k = λ k e k . In the case of the torus, if e k are associated with sine and cosine functions, they are equi-bounded. Moreover, if instead of indexing with k ∈ N, we use k ∈ Z 3 , typically |∇e k | ≤ C |k| and | e k | ≤ C |k| 2 . In such a case, the previous condition becomes which is a verifiable condition.
Lemma 30 Inequality (2.10) holds for every vector field f of class W 4,2 .
Proof Let us define S 1 to be the following operator S 1 f := L ξ k f − L ξ k f . By a direct computation, we find that Consequently, S 1 is a second-order operator, whose dominant part may be expressed as where B i is a first-order operator. Similarly, the computation shows that C f i − D i f is a first-order differential operation, whose dominant part may be expressed as the operator and D i is a zero-order operator. Let S 3 := S 1 L ξ k − L ξ k S 1 . Then, one computes We now note that both (AC−C A) and E i are second-order operators. Consequently, Hence, The last term satisfies Since the terms in both expressions (4.8) and (4.9) can be controlled by f 2 W 2,2 , it follows that indeed, there exists C for an assumed force density F i e j i (x) d 3 x in a coordinate system with basis vectors e j i (x). To accomplish this, we of course must compute the time derivative of the total momentum M i (t) in (A.15). The result for the time derivative dM i (t)/dt is the following, Upon defining u k := dx k dt (in a slight abuse of notation) and using Eqs. (A.9) and (A.11) this calculation now yields Perhaps not unexpectedly, one may also deduce the Lie derivative relation where, in the last step, we have applied the Lie derivative of continuity Eq. (A.12). We note that care must be taken in passing to Euclidean spatial coordinates, in that one must first expand the spatial derivatives of e j i (x), before setting e j i (x) = ∂ i x j = δ j i . One may keeping track of these basis vectors by introducing a 1-form basis. Upon using continuity Eq. (A.12), one may then write Newton's second law for fluids in Eq. (A.17) as a local 1-form expression, Remark 32 (Distinguishing between u and v) In formula (A.23), two quantities with the dimensions of velocity appear, denoted as u and v. The fluid velocity u is a contravariant vector field (with spatial component index up) which transports fluid properties, such as the mass density in continuity Eq. (A.12). In contrast, the velocity v is the transported momentum per unit mass, corresponding to a velocity 1-form v i dx i (the circulation integrand in Kelvin's theorem) and it is covariant (spatial component index down). In general, these two velocities are different, they have different physical meanings (velocity versus specific momentum) and they transform differently under the diffeos. Mathematically, they are dual to each other, in the sense that insertion (i.e. substitution) of the vector field u into the 1-form v yields a real number, u k v k , where we sum repeated indices over their range. Only in the case when the kinetic energy is given by the L 2 metric and the coordinate system is Cartesian with a Euclidean metric can the components of the two velocities u and v be set equal to each other, as vectors.
And, as luck would have it, this special case occurs for the Euler fluid equations in R 3 . Consequently, when we deal with the stochastic Euler fluid equations in R 3 in the later sections of the paper, our notation will simplify, because we will not need to distinguish between the two types of velocity u and v. That is, in the later sections of the paper, when stochastic Euler fluid equations are considered in R 3 , the components of the velocities u and v will be the denoted by the same R 3 vector, which we will choose to be v.
Deterministic Kelvin Circulation Theorem Formula (A.22) is the Reynolds Transport Theorem (RTT) for a momentum density. When set equal to an assumed force density, the RTT produces Newton's second law for fluids in Eq. (A.23). Further applying Eq. (A.23) to the time derivative of the Kelvin circulation integral I (t) = c(t) v j (x, t) dx j around a material loop c(t) moving with Eulerian velocity u(x, t), leads to Holm et al. (1998) Perhaps not surprisingly, the Lie derivative appears again, and the line-element stretching term in the deterministic time derivative of the Kelvin circulation integral in the third line of (A.24) corresponds to the transformation of the coordinate basis vectors in RTT formula (A.21). Moreover, the last line of (A.24) follows directly from the Newton's second law for fluids in Eq. (A.23).

The Deterministic Euler Fluid Motion Equations
The simplest case comprises the deterministic Euler fluid motion equations for incompressible, constant-density flow in Euclidean coordinates on R 3 , ∂ t u i (x, t) + u k ∂ k u i + u k ∂ i u k = −∂ i p , with ∂ j u j = 0 , (A.25) for which the two velocities are the same and the only force is the gradient of pressure, p.
Upon writing Euler motion Eq. (A.25) as a 1-form relation in vector notation, (A.26) one easily finds the dynamical equation for the vorticity, ω = curl u, by taking exterior differential of (A.26), since ω · dS = d(u · dx) and the differential d commutes with the Lie derivative L u . Namely, In terms of vector fields, this vorticity equation may be expressed equivalently as where [u, ω] is the commutator of vector fields.

A.2. Stochastic Reynolds Transport Theorem (SRTT) for Fluid Momentum
For the stochastic counterpart of the previous calculation we replace u =η t η −1 t written above in Eq. (A.6) with the Stratonovich stochastic vector field where the B i t with i ∈ N are scalar independent Brownian motions. This vector field corresponds to the Stratonovich stochastic process where η t is a temporally stochastic curve on the diffeomorphisms. This means that the time dependence of η t is rough, in that time derivatives do not exist. However, being a diffeo, its spatial dependence is still smooth. Consequently, upon following the corresponding steps for the deterministic case leading to Eq. (A.20), the Stratonovich stochastic version of the deterministic RTT in Eq. (A.20) becomes We compare (A.31) with the Lie derivative relation (cf. Eq. (A.22)) dv j (x, t) + dy k t ∂ x k v j + v k ∂ x j dy k t dx j = dy + L dy t ) v j (x, t) dx j . (A.32) Remark 33 As we have seen, the development of stochastic fluid dynamics models revolves around the choice of the forces appearing in Newton's second law (A.35) and Kelvin's circulation theorem (A.38). For examples in stochastic turbulence modelling using a variety of choices of these forces, see Mémin (2014), Resseguier (2017), whose approaches are the closest to the present work that we have been able to identify in the literature.

The Stochastic Euler Fluid Motion Equations in Three Dimensions
The simplest 3D case comprises the stochastic Euler fluid motion equations for incompressible, constant-density flow in Euclidean coordinates on R 3 which was introduced and studied in Holm (2015). These equations are given in (A.37) by (A.39) in which the stochastic transport velocity (dy t ) corresponds to the vector field in (A.29), the only force is the gradient of pressure, p, and the density ρ is taken to be constant. The transported momentum per unit mass with components v j , with j = 1, 2, 3, appears in the circulation integrand in (A.38) as v j dx j = v · dx. 3D stochastic Euler motion Eq. (A.39) may be written equivalently by using (A.37) as a 1-form relation where we recall that (d) denotes the stochastic evolution operator, while (d) denotes the spatial differential. We may derive the stochastic equation for the vorticity 2-form, defined as with dx j ∧ dx k = − dx k ∧ dx j , by taking the exterior differential (d) of (A.40) and then invoking the two properties that (i) the spatial differential d commutes with the Lie derivative L dy t of a differential form and (ii) d 2 = 0, to find In Cartesian coordinates, all of these quantities may treated as divergence-free vectors in R 3 , that is, ∇ · v = 0 = ∇ ·dy t . Consequently, Eq. (A.41) recovers the vector SPDE form of 3D stochastic Euler fluid vorticity Eq. (1.7), dω + (dy t · ∇)ω − (ω · ∇)dy t = 0 . (A.42) In terms of volume preserving vector fields in R 3 , this vorticity equation may be expressed equivalently as dω + dy t , ω = 0 , (A.43) where [dy t , ω] is the commutator of vector fields, dy t := dy t · ∇ and ω := ω · ∇. Equation (A.43) for the vector field ω implies where L dy t ω = [dy t , ω]. In vector components, this implies the pullback relation where ω A 0 (X ) is the Ath Cartesian component of the initial vorticity, as a function of the Lagrangian spatial coordinates X of the reference configuration at time t = 0, and η * t is the pullback by the stochastic process in (1.1). Equation (A.45) is the stochastic generalization of Cauchy's 1827 solution for the vorticity of the deterministic Euler vorticity equation, in terms of the Jacobian of the Lagrange-to-Euler map. See Frisch and Villone (2014) for a historical review of the role of Cauchy's relation in deterministic hydrodynamics.