Incompressible Euler equations with stochastic forcing: a geometric approach

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic Euler--Arnold equations.


Introduction and main result
Since formulated in 1757 [Eul57], Euler's equations for the motion of an inviscous incompressible fluid have had a profound role in science; in geophysics, in meteorology, in aerospace engineering, in astrophysics, and, of course, in mathematics where advanced techniques for existence and uniqueness in 2D and 3D provide important mathematical tools and new theoretical insights.
To accommodate external influence for which a precise model is missing, it is natural to consider stochastic versions of the Euler equations. The main result in this paper is a new framework for local existence and uniqueness of stochastic nonlinear partial differential equations (PDEs) of hydrodynamic type 1 evolving on compact manifolds. A cornerstone in the analysis of deterministic hydrodynamic PDEs is the infinitedimensional geometric theory developed in 1970 by Ebin and Marsden [EM70]. Our framework constitutes an extension of this theory to stochastic PDEs.
We introduce noise as stochastic forcing. That is, the noise is an external fluctuating force acting on the fluid particles. The force is assumed to have Gaussian distribution, uncorrelated in time but correlated in space: the latter condition means roughly that, at any time, nearby fluid particles should experience nearly the same force. As a first case, we consider here additive noise in the forcing, i.e. noise not depending on the solution itself. We leave the more general case of multiplicative noise for future investigation.
Early works on Euler equations with stochastic forcing are mostly in the twodimensional (2D) case [Bes99,BF99,BP01]. The three-dimensional (3D) case has been treated in [MV00,Kim09,GHV14]. In particular, in [GHV14] local existence and uniqueness among smooth solutions in 2D and 3D domains is proved for a wide class of noises. Let us also mention that, complementary to stochastic forcing, transport noise offers a different way to introduce noise in the Euler equations: here one considers the vorticity formulation and noise is added in the Poisson equation relating the stream function and the vector field, see e.g. [CFH19,Hol15,BFM16]. Other types of noises are also possible, like non-Gaussian noises (for example Lévy noises) or not time-uncorrelated noises (for example rough paths).
In the aforementioned work, the analysis is based on PDE techniques combined with stochastic analysis. Our approach is different; it is based on the infinite-dimensional geometric technique first devised by Arnold [Arn66], who discovered that solutions to the deterministic Euler equations can be interpreted as geodesic curves on the infinitedimensional configuration manifold of volume preserving diffeomorphisms equipped with a right-invariant Riemannian metric. Ebin and Marsden [EM70] thereafter used Arnold's geometric viewpoint to obtain local well-posedness of the deterministic equations, including smooth dependence on intial conditions. As the streamlined presentation in [Ebi15] shows, their strategy is to prove that the Lagrangian formulation of the Euler equations, as a second order system on the tangent bundle of a Hilbert manifold of diffeomorphisms, is a smooth, infinite-dimensional ordinary differential equation (ODE). Once this is achieved, standard Picard iterations yield the local wellposedness (since the finite-dimensional ODE analysis extends to Banach spaces). Let us highlight here that this approach is not just a formal way to view PDEs as equations in infinite dimension: this ODE on the Hilbert manifold is driven by a smooth velocity field and not by an unbounded operator.
Extending the Ebin and Marsden framework to a stochastic setting requires us to deal with stochastic differential equations (SDEs) on infinite-dimensional manifolds. A theory for SDEs, in the case of Hilbert manifolds, appears in [Elw82]. It is based on Stratonovich integration, a stochastic integral invariant under change of charts, and can be extended to Banach manifolds, cf. [BE00]. Another approach to stochastic integration on manifolds is developed in [BD89], which is more based on Itô integration and needs a strong Riemannian structure on the (infinite-dimensional) manifold. Applications of SDEs on infinite-dimensional manifolds to mathematical physics and stochastic PDEs have been considered, for example in the case of SDEs on loop manifolds (e.g. [BE00]), manifold-valued stochastic PDEs (e.g. [Fun92,BO13,Hus15]), SDEs on infinite-dimensional Lie groups (e.g. [AD00]). The lifting to an infinite-dimensional manifold of diffeomorphisms has been used in [Elw78,BE96] to show the existence of stochastic flows for finite-dimensional SDEs. However, none of these obtain a stochastic extension of the Ebin and Marsden result as we accomplish in this paper.
Closer to our framework is instead an interesting approach to deterministic, viscous PDEs like Navier-Stokes equations. Roughly speaking, the idea is that the Laplacian at the Eulerian level corresponds to noise at the Lagrangian level, provided an average (over the noise) is taken. An example is given by Gliklikh in [Gli11,Chapter 16] (based on previous works like [BG02]): in particular, the Lagrangian SDEs [Gli11,§16.24] and [Gli11,§16.25] are similar to our Lagrangian SDE (2) (more precisely, to (16)). However, in those equation the noise is taken on the velocity (not on the force) and mean derivatives are used for the velocity equation, so that the Eulerian counterpart becomes the deterministic Navier-Stokes equations (and not stochastic Euler equations). Also, Gliklikh uses an Itô-like formulation (which needs a connection at the infinite-dimensional level, something avoided here). This line of research has been developed by Cruzeiro and coauthors [CS09,ACF18]. In particular, the recent paper [Cru18] also considers stochastic Euler equations (with transport noise), where the Eulerian-to-Lagrangian link is shown without going into the infinite-dimensional analysis. Another important contribution to the geometric viewpoint on stochastic Euler equations is given in [CM08,CFM07]: these papers take Arnold's viewpoint and, in 2D, construct a solution to Euler equations via Girsanov transform (a transformation of the SDE that removes the drift), though no direct analysis of the infinite-dimensional Lagrangian SDE is given.
In summary, the notion of using the Ebin and Marsden approach to stochastic PDEs has been considered by many authors, but to fully develop such an analysis has remained an open problem. Here we solve this problem in a first setting: stochastic Euler equations with additive noise. As a byproduct, we get local existence and uniqueness for stochastic Euler equations on any compact manifold (of any finite dimension d ≥ 2). To the best of our knowledge this has not been obtained before.
We stress that our purpose is not to improve on previous existence results 2 . Rather, we anticipate the framework presented here to be the first step in a stochastic analog of the Ebin and Marsden framework for deterministic non-linear PDEs, which for 50 years has been a valuable complement to more traditional PDE analysis, and which for various equations has led to deep insights: Fredholmness of solution maps [EMP06,MP10], global existence results [Mis02,MM13], vanishing geodesic distances [MM05,BHP20], limits between compressible and incompressible Euler equations [Ebi75], averaged Euler equations and sharp results on viscous limits [MRS00,MS03,CS07], stability of steady solutions [Arn65,FS01,Pre04], connections between fluid dynamics and optimal transport [KW09], improved numerical methods [MV20], etcetera. The key point is that a stochastic Ebin and Marsden framework enables stochastic calculus on manifolds as long as the techniques one use are oblivious to the manifold being modeled 2 The existence results in [GHV14], based on traditional SPDE techniques on R n , are sharper than what presented here in terms of regularity and in terms of classes of noise than what we obtain in this paper. It is possible to sharpen our results in terms of regularity and we believe we can treat more general types of noises. However, a refined analysis easily becomes technical and would diverge from the essence of this paper. Instead, we chose to present the most accessible results to illustrate our framework. We postpone refined analysis to future, more specialized publications. on a Banach (or Hilbert) space. One such technique is local existence of stochastic ordinary differential equations on Hilbert and Banach manifolds [Elw82,BE00], as we elevate in this paper. In analogous ways, our framework may also lead to progress on related stochastic topics. Some of these, not addressed in this paper, include: (i) A different, geometric approach to stationary measures for stochastic Euler and Navier-Stokes equations, developed e.g. in [KS12]. (ii) Stability analysis for stochastic Euler equations (with possibly multiplicative noise): the interplay between stability/completeness/explosion and geometry for a stochastic system has been investigated in the finite dimensional case for example in [Aze74,Li94]. (iii) Metastability (i.e. the notion where, for small noise and large time, the solution "jumps" from a stable steady solution to the deterministic equation to another one), see e.g. [BLZ14] for 2D Euler equation, one could try to study the transition paths between steady states from a geometric viewpoint. (iv) Interpretations of the stochastic Euler equations (with added damping and an potential) as a Hamiltonian Monte-Carlo method on an infinite-dimensional Riemannian manifold [GC11]. (v) Structure preserving numerical methods for stochastic Euler equations, inspired by the deterministic setting in [MV20]. We now continue the introduction by presenting the stochastic Euler model. Thereafter we state the main result. The fluid domain is a compact oriented Riemannian manifold K of dimension d, possibly with smooth boundary, and the equations we study are (formally) given by where u is a vector field on K of Sobolev regularity H s describing the fluid velocity, p is the pressure function, ∇ u denotes the co-variant derivative along u, and the vector valued noiseẆ corresponds to a fluctuating external force field; more precisely, W is a Wiener process with values in the space of Sobolev, divergence free vector fields. If K has a boundary, we require u and the noise fieldẆ to be tangential to it. The next step is to reformulate equation (1) using the Lagrangian variable Φ, witḣ Φ = u • Φ. The Lagrangian equation then takes the form (2) Here, we interpret the flow Φ as evolving on the infinite-dimensional manifold of volume preserving diffeomorphisms of Sobolev regularity H s . Using geometric and stochastic analysis on infinite-dimensional manifolds we prove the following result.
First main theorem Fix s > d/2 + 1 and suppose that the noise takes values in the space of H s+2 divergence-free (and tangential) vector fields. Then local existence and uniqueness hold for the Lagrangian formulation (2).
Relating this result to the original Eulerian formulation (1) is more complicated than in the deterministic case because of the stochastic terms. That is why the following theorem requires higher regularity.
Second main theorem Fix s > d/2 + 4 and suppose that the noise takes values in the space of H s+2 divergence-free (and tangential) vector fields. Then local existence and uniqueness for the stochastic Euler equation (1) hold.
We expect that the regularity required from noise terms and initial data is not optimal. Indeed, regularisation arguments (cf. Section 4) should lead to much lower regularity assumptions. The point of the paper is to develop, as neatly as possible, a stochastic version of the Ebin and Marsden results. For this reason we do not deal with optimal regularity questions here. Instead we plan show in a separate paper how to close the regularity gap between our two main theorems above, getting local well-posedness for the stochastic Euler equation (1) for s > d/2 + 1.
The paper is organized as follows: In Section 1, we give a self-contained presentation of Stratonovich SDEs on infinite-dimensional Hilbert manifolds. This is complemented by extensive background material on stochastic integration on infinite-dimensional spaces in Appendix A. In Section 2, we introduce the Hilbert manifold of Sobolev diffeomorphisms on K and analyze the regularity properties of the composition map. This is also complemented by background material on spaces of Sobolev maps in Appendix B. In Section 3, we state and prove the main results: we prove the local existence and uniqueness for the Lagrangian formulation and then we show, first formally and then rigorously, the link between the Lagrangian form and the Eulerian form, concluding the local existence and uniqueness for the latter. In Section 4, we discuss possible extensions and future developments; as another example of application of our methods, we show local well-posedness of averaged Euler equation (25).

Notation and Conventions
Let E, F be Banach spaces and U ⊆ E open. A map f : E ⊇ U → F is said to be of class C k if it is k times continuously Fréchet differentiable. We write Df for its derivative and say that a C k -map is of class C k,1 (or C k,1 loc ) if its kth derivative is (locally) Lipschitz continuous. Spaces of linear operators L(E, F ) are endow with the operator norm · L(E,F ) . If there is no possible confusion, we write · L for the operator norm.
We let K be a compact manifold (possibly with smooth boundary ∂K). Further, H, H ′ etc. will be separable Hilbert spaces and M a (possibly infinite-dimensional) metrisable and separable manifold modelled on H. Our main example will be M = Diff s µ (K), the group of volume preserving diffeomorphisms of class H s on K. For a differentiable map f : M → N we let T f be its tangent map. Generic charts will be denoted by κ : U → R d and by ψ : U → H. For a a vector field X we let X κ either be its representative in a chart κ, or (abusing notation) its principal part pr 2 •T κ•X •κ −1 .
Concerning the stochastic setting, we denote the Borel-σ-algebra of a topological space T by B(T ) and let 1 A be the indicator function of the set A. Moreover, we diverge from usual notation and denote Stratonovich integrals by f • dW . The usual "•" will be reserved for composition.

Stochastic differential equations on Hilbert manifolds
We assume familiarity with the basic objects of probability theory and stochastic processes such as Brownian motion. This section will review Stratonovich integration on Hilbert manifolds. We take the main results (with small modifications), from, and follow the approach in, [Elw82,BE00], taking also some facts from [DPZ14] and [BNVW08]. Moreover, Appendix A includes a self contained review of stochastic integration and stochastic differential equations on Hilbert spaces.
We will introduce SDEs on Hilbert manifolds and obtain the main local wellposedness result. This will enable us to establish the local well-posedness of Euler flows in Section 3. The Stratonovich integral is the right type of integral to use here: it is invariant under change of charts, because the Itô formula for Stratonovich integrals is analogue to the classical chain rule, without second order terms. This invariance is at the basis of the so-called Malliavin's "transfer principle": quoting [É90], 'geometric constructions involving manifold-valued curves can be extended to manifold-valued [stochastic] processes by replacing classical calculus with Stratonovich stochastic calculus'. On the contrary, the Itô integral instead would not be invariant under change of charts (cf. Appendix A). Let us now describe the basic setting used throughout this section.
1.1 Fix a probability space (Ω, A, P ) together with a filtration F = (F t ) t , t ∈ [0, +∞[, that is a non-decreasing (i.e. F s ⊆ F t for every s ≤ t) family of σ-algebras contained in A, indexed by t ∈ [0, +∞[. We assume that A is the σ-algebra F ∞ generated by all F t and that F is complete and right-continuous (these are technical and classical assumptions). Further, we let L p (Ω) be the space of real valued L p -functions on Ω. For X in L 1 (Ω), the symbol XdP denotes the expectation under P . For a random variable or process with values in a topological space, measurability and progressive measurability are understood with respect to the Borel σ-algebra on the topological space, unless differently specified.
1.2 Let H, E be separable Hilbert space. We choose an E-valued Q-Brownian motion W with respect to F (cf. A.2 for the definition of Brownian motion). Let now M be a metrizable separable differentiable (i.e. C ∞ ) manifold modelled on the Hilbert space H. 3 Let (ψ α : O α → V α ) be a countable atlas of C ∞ charts, with ψ α and ψ −1 α and their first 100 derivatives bounded (we have chosen 100 just as high enough number for what we need). Here "bounded" refers for the mappings to the natural metric on manifold and model space, and for the derivatives with respect to the natural operator norms induced by the Hilbert space norm and the norm on the tangent spaces induced by a strong Riemannian metric.
1.3 Let A and B be vector bundles over M whose fibres we denote by A x and B x . Recall from [Kli95, Lemma 1.2.12] the associated vector bundle of linear maps , ·)))), where ϕ A and ϕ B are bundle trivialisations over (U, ψ). Note that, since we are not considering maximal solutions here, it will suffice to work in a fixed bundle trivialisation. If one of the vector bundles is trivial with typical fibre E, we write (if A is trivial) L(E; B) to shorten the notation. As usual we denote the .
Let now f : M →H be a C k+1 map with values in a separable Hilbert spaceH and Df := pr 2 • T f : T M →H the principal part of the tangent map. Note that Df induces a C k -bundle morphism over the identity (π, Df ) : Differently from the case of Hilbert space, we do not have a notion of Itô differential that we can use to define a Stratonovich integral. However, we can give the notion of solution to a stochastic differential equation on M : Here the drift b : M → T M and the diffusion coefficient σ : M → L(E; T M ) are given sections assumed continuous and in C 1 resp., the initial datum ζ : Ω → M is a F 0 -measurable random variable.
Here the trace is understood as This definition extends to progressively measurable processes X, defined on the closed interval [0, τ ] for some P -a.s. finite stopping time τ with τ > 0 P -a.s., requiring that (4) holds for all t in [0, τ ]. Moreover, if X takes values P -a.s. in an open subset U of M , it is enough that b and σ are defined and in C 0 , in C 1 resp. on U . If M = H, thanks to Theorem A.21 in Appendix A, the above definition is equivalent to the definition of solution on Hilbert spaces.
Given a solution X to (3) and a C 1 map g : M → L(E,H), withH a separable Hilbert space, we define the Stratonovich integral . By this definition, the Stratonovich integral depends a priori on X, g and σ separately, though also here one can show that the definition depends only on g(X) (see Remark A.20 in Appendix A). The above definitions and properties are extended to Stratonovich differentials, that is the case of a more general drift B, namely where B is a T M -valued progressively measurable process, with P -a.s. continuous paths, requiring that, P -a.s., π(B t ) = X t for every t (we recall that π is the bundle projection on the base point). For example, a progressively measurable process X, defined on [0, τ ) for some accessible stopping time τ , satisfies (5) if it has P -a.s. continuous paths, it satisfies P -a.s. π(B t ) = X t for every t and, for every C 2 function h : M →H with values in a separable Hilbert spaceH, there holds, P -a.s., 1.5 Theorem (Itô formula for manifold-valued processes) Assume the above setting. Let X be a solution to (3) and let f : M → N be a C 2 function, where N is a metrizable separable differentiable manifold modelled on a (separable) Hilbert space H ′ . Then there holds, P -a.s., The meaning of (7) is understood rigorously as follows: For every C 2 function h : N → H with values in a separable Hilbert spaceH, there holds for all t ∈ [0, τ ), P -a.s., Proof. The result follows from the definition of Stratonovich differential, precisely formula (6), with h replaced by h • f , noting that D(h • f ) = Dh • T f .

Remark
If X takes values P -a.s. in an open subset U of M , then equation (6) holds P -a.s. also for every f ∈ C 2 (U,H), that is, X solves (5) also as U -valued process, where U inherits the manifold structure from M . We argue by localisation and take for each n, where dist is the distance induced by the metric on M . Now M is paracompact and modelled on a Hilbert space. As Hilbert spaces admit smooth bump functions, cf.
[ KM97,16.16. Corollary], the usual partition of unity argument shows that for every n there is a C 2 (even smooth) ψ n : M → R with ψ n = 1 on U n and ψ n = 0 on U c n+1 . Define τ n as the minimum of τ and the first exit time of X from U n . Then for any f : U →H, we apply (6) to ψ n · f (trivially extended to a C 2 function on M ), and get (4) for f before time τ n . Letting n go to ∞, we get (6) for f before τ .
As a consequence, we get the invariance of the equation under diffeomorphism: 1.7 Lemma Let U , V be open sets resp. on M , N , with N another metrizable separable Hilbert manifold (modelled possibly on another separable Hilbert space), let ϕ : Proof. The result follows from Itô formula (8), provided that there holds, for any C 2 function h : N →H (withH separable Hilbert space), But this follows from The proof is complete.
1.8 Remark As one sees from the proof, we can relax the assumption that ϕ is a C 2 diffeomorphism, requiring instead the following condition: ϕ : U → V is C 2 and there exist a continuous sectionb : V → T N and a C 1 sectionσ : We can also relax the assumptions on b andb, requiring that b andb are Borel sections and b(X) andb(Y ) coincide P -a.s. and are continuous in time P -a.s.
As a consequence, taking ϕ = ψ α (where (ψ α ) is a countable atlas of smooth charts), we get the expression of the SDE (3) in chart and the invariance of the solution under change of chart. In the following, for any α, we call In particular, if, for some α and β, (9) holds and X takes values in O α ∩ O β P -a.s., then (9) holds also with β in place of α.
Now we give the main local well-posedness result: 1.10 Theorem Assume that there exists an open set U in M and an index α, with U ⊆ O α , such that X 0 = ξ is in U P -a.s., b α is in C 0,1 on ψ α (Ū ) and σ α is in C 1,1 on ψ α (Ū ). Then existence and uniqueness up to the first exit time from U hold for (3), that is, for every T > 0: there exists a solution X on [0, τ U ∧ T ], where τ U is the exit time of X from U , of the SDE (3) and, ifX is another solution defined on [0,τ ], thenX = X on [0,τ ∧ τ U ∧ T ] P -a.s.; moreover τ U > 0 P -a.s..

Remark
In Theorem 1.10 the requirements on the coefficients are formulated with respect to a manifold chart. To define intrinsically a Lipschitz section (independent of the chart), an auxiliary structure, like a strong Riemannian metric, is needed, cf. [JL14] for the finite-dimensional case. Working in charts we avoid a lengthy discussion or the strengthening of the requirements on the coefficients by requiring e.g. more orders of differentiability.

Remark
The invariance under diffeomorphism Lemma 1.7, Remark 1.8 and Corollary 1.9 can be extended (with the same proof) to the case of a random drift, under the following assumption: given an accessible stopping time τ and an open set • it holds P -a.s.: for every t in [0, τ ), b(t, ω, ·) is a section on U , and • it holds P -a.s.: (t, x) → b(t, ω, x) is continuous.
Moreover, as in the flat case, the existence and uniqueness Theorem 1.10 can be extended to the case of a random drift, under the following assumption: given an accessible stopping time τ and an open set U as in Theorem 1.10, the drift b : [0, τ ) × Ω × U → T M is such that • for every x in ψ α (Ū ), b α (·, ·, x) is progressively measurable, • it holds P -a.s.: for every t in [0, τ ), b(t, ω, ·) is a section on U , • it holds P -a.s.: (t, x) → b α (t, ω, x) is continuous, and uniformly with respect to (t, ω).
The proof is analogous, applying Theorem A.22 together with Remark A.24.

Vector fields on Sobolev diffeomorphisms
In this section, we establish the geometric setting in which we will solve the stochastic differential equations. Thus we leave the stochastic considerations of the last section for the time being and consider the geometry of the Hilbert manifold of Sobolev diffeomorphisms preserving a volume form. See also [Gli11] for an introduction to the topic. Our aim will be to construct certain second order vector fields on this manifold. This construction drives our later investigation, as the vector fields are crucial ingredients in the formulation of the stochastic differential equations we aim to investigate.

2.1
All manifolds considered will be assumed to be smooth. Recall from [Lee13, Sec- We fix a compact d-dimensional Riemannian manifold (K, g K ) (possibly with C ∞ -boundary) 5 . Denote by µ the volume form associated to the metric g K . Further, (N, g N ) will be another Riemannian manifold (possibly also with boundary).
Spaces of mappings of Sobolev type between (subsets of) Euclidean space are well studied in the literature dealing with partial differential equations, see e.g. [Tri92,Pal68]. The corresponding notion for manifold valued mappings is also classical, but much less well known [IKT13, EM70, Pal68, Pal65].

Definition
where d is the dimension of the compact manifold K (possibly with boundary). Then a continuous map f : K → N between manifolds is locally of class H s around k ∈ K, if there exist a pair of charts (κ, λ) around k and f (k) such that λ • f • κ −1 makes sense and is a mapping of Sobolev class H s , whose distributional derivatives up to order s are in L 2 . Then we define the space of all H s -Sobolev maps. We recall in Appendix B that this space can be turned into a Hilbert manifold modelled on spaces H s (E) of Sobolev sections for certain vector bundles π E : E → K. For E = T M we write X s (K) := H s (T M ).
2.3 (Warning) On a manifold, a map is of class H s if it is everywhere locally of class H s . Note that the notion of locally being H s incorporates a boundedness concept on the derivatives which do not have intrinsic meaning on a manifold (without specifying a Riemannian structure). In particular, a map locally of class H s in some pair of charts might fail to be of class H s in another pair of charts, see [IKT13, 3.1].

From now on we choose an s > d 2 + 1 and define the group of volume preserving
For simplicity, we shall also assume without further notice that s ∈ N. Though most of the results will generalise also to fractional Sobolev spaces (cf. Section 4 for a discussion), the integer assumption allows us to conveniently cite most results needed in our approach. It is well known (compare [EM70] or Appendix B) that Diff s µ (K) is a Hilbert manifold modelled on the space of divergence free H s -vector fields X s µ (K). Now Diff s µ (K) is a topological group for which the right multiplication operator is smooth (a so called half Lie group [MN18]), that is here Φ → comp(Φ, φ) is smooth for every φ. Moreover, Diff s µ (K) is a metrizable manifold such that every component is separable. For our purpose separability of components is sufficient (albeit we asked the whole manifold to be separable in Section 1) since solutions of stochastic equations are continuous, whence they evolve in one component if the initial conditions live in one component.

2.5
If K is a manifold with smooth boundary, the model space of Diff s µ (K) consists of all divergence free H s -vector fields which are tangential to the boundary. While it is important to have the condition in the presence of boundary, it is of no consequence for the arguments we are about to develop. Hence for the rest of the article, we will suppress the boundary condition in our notation. Thus for K (with or without boundary) we shall simply write X s µ (K) and assume that elements in this space are tangential to the boundary. As is explained in Appendix B, the group Diff s µ (K) has the same properties we discussed in 2.4 in the boundary-less case.
To generate a second order equation which corresponds to the Euler equation, we construct a vertical vector fields from elements of the tangent space at the identity. Following [EM70, Section 11], we can combine this field with the metric spray to obtain the desired second order vector field (cf. [Lan99, IV, §3]). Let us recall some facts on Diff s µ (K) and its tangent bundle from Appendix B.
2.6 One can identify the tangent space at Φ ∈ Diff s µ (K) as follows: ) and we obtain a trivialisation of the tangent bundle via the homeomorphism 6 The inverse of J is given by the (continuous) composition map We stress here that the lack of differentiability of the maps J and comp is caused by the lack of differentiability of left composition, cf. B.9. However, the composition allows us to extend elements of the tangent space at the identity to vector fields on Diff s µ (K).

Definition
Let V be a divergence-free H s vector field on K (i.e. an element of X s µ (K)). Define a continuous vector fieldV ∈ C(Diff s µ (K), T Diff s µ (K)) on Diff s µ (K) as Furthermore, every such vector field yields a continuous map Due to B.9 the sectionV and thus also B V will be of class , called the vertical bundle. In local coordinates (see e.g. [Lan99, X, §4] for a detailed discussion), the vertical bundle is given by elements of the form ((x, v), (0, w)). Recall that the vertical lift is a smooth bundle isomorphism. Since B V from Definition 2.7 is fibre-preserving, we obtain a vector field By construction E V is continuous and a C k -vector field if V ∈ H s+k µ .
We will now represent E V in local charts. Fix a manifold chart ψ : such that the following properties are satisfied: . Thus for V ∈ X s+ℓ µ (K), the principal part of the representative is determined by the C ℓ -mapping The key result to obtain regularity will be a Lipschitz estimate for e ψ,ℓ and its derivatives. We will deduce these properties from the following technical lemma: makes sense and is a mapping of class C k,1 loc (i.e. of class C k with kth derivative being locally Lipschitz-continuous).
Proof. Let us note first that since F is a mapping of class C k,1 loc , the composition F •c of F with any C k,1 loc -curve c : R → U ×E is again a curve of class C k,1 loc . A mapping with this property is called Lip k map (cf. [KM97, Section 12] or [FK88] for a detailed discussion).
To see that F ∧ : U → L(F, G) is actually a C k,1 loc mapping we work with the directional derivatives. Recall from [Wal12, Proposition A.3.2] that (as F ∧ is C k−1 ) F ∧ will be a C k -map if we can show that the mapping is continuous. To see this, we exploit that due to the construction of F ∧ (x) = F (x, ·), the kth directional derivative satisfies where d k 1 denotes the kth iterated partial derivative with respect to the first component inductive argument easily shows that D k 1 F (x, ·) is linear for every fixed x. Using (12) we thus observe that D k F ∧ can be expressed as follows As a consequence, the computation shows that F ∧ will be of class We conclude that (D k F ) ∧ is indeed locally Lipschitz continuous as the operator norm can be estimated as: 2.10 Remark (a) It is essential that the map F from Lemma 2.9 is k-times differentiable with kth derivative being locally Lipschitz. Weakening the Lipschitz assumption to mere continuity, the statement of Lemma 2.9 becomes false as [KM97, 12.13.
Smolyanov's Example] shows. However, the converse statement between (open sets of) Banach spaces a Lip k -mapping is automatically of class C k,1 loc . Using this result, the proof of Lemma 2.9 could have been considerably shorter. The reason we did not use this is that the cited results hinge on the following statement: A map f : E ⊇ U → F from a normed space to a locally convex space is Lip 0 if and only if the mapping is locally Lipschitz [KM97,Lemma 12.7]. This is false if F is nonnormable as a counterexample due to H. Glöckner shows (cf. the errata of [KM97]). Studying the proof it was unclear to us whether the result holds for normed F (this was established in [FK88, Theorem 1.4.2] with essentially the same proof). Thus we chose to err on the side of caution and have avoided using these results.
We can now deduce from Lemma 2.9 the regularity of e ψ,ℓ . Due to linearity of the bundle trivialisation T ψ of T Diff s µ (K) the C ℓ -map . is linear in V . We can thus deduce from Lemma 2.9 the following proposition (see also [Elw82, Chapter VIII Section 1] and [BE96, Corollary 5.10] for a similar result): 2.11 Proposition The C ℓ−1,1 loc map e ψ,ℓ (see (11)) gives rise to a C ℓ−1,1 We finally note that the mapping e ψ,ℓ can not be expected to be of class C ℓ,1 loc . The reason for this is that e ψ,ℓ essentially is given by the composition map of the half-Lie group Diff s µ (K) and it is a folklore fact that composition with Sobolev vector fields of class H s+ℓ is only C ℓ but not C ℓ,1 loc . 7

Ebin-Marsden theory for the stochastic Euler equation
In this section we combine the stochastic and geometric considerations developed in the last sections to obtain existence and uniqueness results for a stochastic version of the Euler equation for an incompressible fluid on a manifold.
3.1 As in Section 2, we fix the following data: a compact (oriented) manifold K (possibly with boundary), a Riemannian metric with associated volume form µ and a pressure function p ∈ H s+1 (K, R). Now the classical Euler equation for an incompressible fluid occupying K is Moreover, we can consider the Euler equation with (deterministic) forces, where we add to the right hand side of (13) a forcing term f : R×K → T K, π•f (t, k) = k, ∀(t, k) of suitable regularity, i.e. f should be continuous and of class H s+2 with respect to K.
Following an idea by Arnold [Arn66], one can rewrite (13) as an ordinary differential equation on the Hilbert manifold Diff s µ (K). To do this, recall the following formal facts: First, if we consider Diff s µ (K) as manifold embedded into L 2 (K; K), then, for any Φ in Diff s } is orthogonal to ∇g • φ for any function g : K → R. Second, given a manifold M , a vector field B on T 2 M is the geodesic spray, that is the curve γ satisfying(γ,γ) = B(γ,γ) is a geodesic, if and only if the velocity component of B(V ) is V and the acceleration component is orthogonal to Note that, if u satisfies the Euler equation, and Π is the Leray projection on the divergence-free vector fields, then 3.2 To treat the Euler equation with forces as a second order equation on the infinitedimensional manifold Diff s µ (K), we augment f ∈ X s+2 µ (K) to a right invariant vector field which we then vertically lift to a second order vector field with values in T 2 Diff s µ (K). Thus we consider the following map . Then one modifies the geodesic spray B by defining By the chain rule, the second-order vector field B f satisfies Now the termB(η) has velocity component η and its acceleration component is orthogonal to T π(η) Diff s µ (K), henceB coincide with the geodesic spray B.
The equation is then called the Euler equation with forces in Lagrangian form; we will call the classical Euler equation (13) Euler equation in Eulerian form.
In their seminal paper [EM70], Ebin and Marsden established smoothness of the spray of the L 2 -metric, which is not automatic due to the metric being a weak Riemannian metric. Hence the Euler equation in Lagrangian form (14) is an ordinary differential equation (in infinite dimension) with smooth drift, opposed to many standard PDEs, and so it is solvable, at least locally, by standard Banach manifold techniques.
Here we consider the classical Euler equation (13) with an additive noise of the forṁ W (t, k), Gaussian, white in time and smooth in space: we take the stochastic Euler equation with (deterministic and stochastic) forces in Eulerian form where W is a Brownian motion with values in a suitable function space and f is as above a suitably regular forcing term. Hence the corresponding stochastic Euler equations in Lagrangian form reads formally with B the geodesic spray on Diff s µ (K). We have used Stratonovich form here according to the "transfer principle" (see [Bis81], [É90] and references therein), and ultimately because the Stratonovich chain rule has the same form of the classical chain rule, hence the formal computations in the deterministic case go through also in the stochastic case.
In the next subsection we show the first main result, that is existence and uniqueness for the stochastic Euler equation in Lagrangian form (16). In the subsequent subsection we prove rigorously the link between the Eulerian and the Lagrangian form and derive the second main result, that is existence and uniqueness for the stochastic Euler equation in Eulerian form (15).

Local well-posedness in Lagrangian formulation
The objective of this section is to combine Theorem 1.10 for SDE on Hilbert manifolds with Proposition 2.11 for regularity of right translated vector fields to obtain local wellposedness for the stochastic Euler equation in Lagrangian form (16). To deal with the diffusion term, we take a similar approach to [Elw82, Chapter VIII] and [BE96], which however are concerned with stochastic flows for finite-dimensional SDEs. Now we define rigorously the drift and diffusion in (16). We recall that K is a compact (oriented) manifold K (possibly with boundary), of dimension d, together with a Riemannian metric and its associated volume form µ. We take M = Diff s µ (K) with s > d/2 + 1. We assume to have a probability space (Ω, A, P ) and a filtration (F t ) t as in 1.1. (16) is the geodesic spray associated to the right invariant L 2 -metric on Diff s µ (K). 8 The smoothness of the drift is the main result by Ebin and Marsden: 3.3 ([EM70, Theorem 11.2]) If s > d/2 + 1 then the geodesic spray B on Diff s µ (K) corresponding to the deterministic Euler equations is C ∞ . Furthermore, let f : R → X s+1 µ (K) be continuous and consider B f := B + V f (with V f as in 3.2). The second order vector field B f is associated to the Euler equation with forces is continuous and for every fixed t, B f (t, ·) is of class C 1 . In particular, both B and B f are of class C ∞ (resp. B f (t, ·) is for t fixed of class C 0,1 loc ) in every chart.

Assumption
Given s ′ non-negative integer, the process W is a Q-Brownian motion (with respect to (F t ) t ) on X s ′ µ (K), for some symmetric, positive-semidefinite and trace-class operator Q on X s ′ µ (K).
For s ′ ≥ s + 2 (without loss of generality, s ′ ≥ s + 2), the diffusion coefficient in We assume that the initial datum η 0 satisfies π(η 0 ) = id. The equation (16) now makes sense as SDE on the manifold T Diff s µ (K), as in Definition 1.4 (as we will see in the proof of 3.5, the diffusion coefficient is C 1 in a neighbourhood of η 0 ).
We can now formulate our local well-posedness result: 3.5 Theorem Fix s > d/2 + 1 and suppose Assumption 3.4 with s ′ = s + 2, take η 0 in X s µ (K) with π(η 0 ) = id (the identity map). Then local strong well-posedness, in the sense of Theorem 1.10, holds for the Lagrangian formulation (16) of the stochastic Euler equations (with or without forces) on T Diff s µ (K).
Proof. We want to apply Theorem 1.10. We take a chart ψ as in 2.8; restricting the domain O of ψ, we can assume that O is an open bounded set and ψ is bounded with its 100 derivatives. Then B ψ is smooth, in particular C 0,1 (in the presence of a forcing term f as in 3.3 we consider instead B ψ f which is of class C 0,1 ), on ψ(O). Moreover Σ ψ = (e ψ,2 ) ∧ is C 1,1 on ψ(O) by Proposition 2.11. Hence the result follows from Theorem 1.10.

Link between Eulerian and Lagrangian formulations
In this subsection we come back to the stochastic Euler equation in Eulerian form (14): we prove the rigorous link between the Eulerian form and the Lagrangian form and we derive the well-posedness result for the Eulerian form. Here we will establish the results first only for the Euler equation without deterministic forcing term f . This keeps the formulae simpler. However, we stress that the same results hold also in the presence of a forcing term f (by taking a trivial modification of the proof), cf. Remark 3.12 below. Let Π denote the Leray projector on the divergence-free vector fields (cf. [Shk00, Section 5] and [EM70, Appendix A]). On the noise, we take Assumption 3.4 with s ′ ≥ s − 1. (15) is an X s µ (K)-valued F -progressively measurable process u = (u(t)) [0,τ ) , with τ accessible stopping time and τ > 0 P -a.s., with P -a.s. continuous paths in X s µ (K), such that

Definition A local (strong and smooth) solution to the stochastic Euler equation in Eulerian form
3.7 Remark For s > d/2 + 1, if t → u(t) is a continuous path with values in X s µ (K), then, by Sobolev embedding, t → u(t) is a continuous path with values in C 1 (K) (since K is compact, C 1 (K) is a Banach space, cf. e.g. [Pal68, p.24]). Hence t → ∇ u(t) u(t) and so t → Π[∇ u(t) u(t)] are continuous paths in H s−1 (K), in particular the integral of Π[∇ u(t) u(t)] makes sense. Hence the equality (17) holds in H s−1 .
The link between the Eulerian and the Lagrangian form is proved in the following: 3.8 Theorem The Eulerian form (15) and the Lagrangian form (16) of the stochastic Euler equation are equivalent, in the following sense: • Fix s > d/2 + 3 and assume that W satisfies 3.4 with s ′ = s + 2. If η is a solution on [0, τ ) to the Lagrangian form in T Diff s µ (K), with π(η 0 ) = id, then u(t) = η(t) • π(η(t)) −1 is a solution on [0, τ ) to the Eulerian form in X s µ .
• Conversely, fix s > d/2 + 4 and assume that W satisfies 3.4 with s ′ = s − 1. If u is a solution on [0, τ ) to the Eulerian form in X s µ and Φ is the unique flow solution on [0, τ ′ ) (for some accessible τ ′ ≤ τ ) to the (random) ODE Before we establish the theorem, we need establish differentiability and identities for certain tangent mappings first. These will be used in the proof.

Lemma
The following mappings are of class C 2 their tangent mappings are (up to canonical identification) given by: Here Y X : K → T 2 K is the vertical field locally (in a chart κ) conjugate to k → (k, X κ (k), 0, Y κ (k)).
Proof. Recall from Appendix B, (44) that we can identify the tangent manifold of the manifold of Sobolev mappings as T H s (K, K ′ ) = H s (K, T K ′ ) for all s ≥ dim K/2 + 1.
As Diff s−3 µ (K) ⊆ H s−3 (K, K) is a submanifold, we use the inclusion to identify the tangent bundle of Diff s−3 µ (K) as a subbundle of T H s−3 (K, K). By restricting the composition H s−1 (K, T K) × Diff s−3 µ (K) → H s−3 (K, T K) to a submanifold, we obtain G. Hence G is of class C 2 as a map to T Diff s−3 µ (K) (which we have identified as T H s−3 (K, T K)| Diff s−3 µ (K) ) by B.9 and B.10. It is well known that the tangent map of the composition is the sum of the tangent maps of left and right composition, B.9 and see e.g. [Mic80,Corollary 11.6]. We will use this to compute an explicit formula for To combine this with the formula for the left and right composition B.9, we need to identify T X s−1 µ (K) as a subspace of T H s−1 (K, T K) = H s−1 (K, T 2 K). By definition, any element X in X s−1 µ (K) satisfies π K • X = id K , for π K : T K → K the bundle projection. Hence the differential of the embedding X s−1 Then the identification together with the formulae for left and right composition B.9 yields (18).
We now turn to the mapping F : the composition map and inv : Diff s µ (K) → Diff s−2 µ (K) the inversion map. As all of these mappings are at least of class C 2 by B.9, we see that F is C 2 as a map to the closed subvectorspace X s−2 µ (K) ⊆ H s−2 (K, T K). Now we wish to leverage the formulae for the tangent mappings of composition and inversion in B.9 to obtain a formula for the tangent of F . To this end, we identify X s−1 µ (K) ⊆ H s (K, T K) and X s µ (K) × X s µ (K) = T X s µ (K) with mappings taking their image in the vertical part of the double tangent bundle. Identify T 2 Diff s µ (K) ⊆ H s (K, T 2 K) and apply the chain rule. Then the derivative of F viewed as an element in H s (K, T 2 K) becomes for any ζ in T 2 Diff s µ (K) ⊆ H s (K, T 2 K), (19) The symbol '−' in (19) means the operation in T V •Φ T Diff s−1 µ (K) and we observe that since V is a vector field, (19) indeed takes its values in the vertical part of T 2 K.
Proof of Theorem 3.8. We start with the second statement. We are given u solution to the stochastic Euler equations on [0, τ ). We start constructing the Euler flow Φ: we show that there exists a unique progressively measurable process Φ : [0, τ ′ ) × Ω → Diff s−1 µ (K), where τ ′ is a suitable positive accessible stopping time, such that, P -a.s., For this, we pick a chart ψ α of Diff s−1 µ (K), with domain O α , such that ψ α (O α ) is a bounded zero neighbourhood with ψ α (id) = 0. Then the mapping is C 1 . We observe that for every fixed u 0 ∈ X s µ (K) the norm Dξ op is bounded on the compact set C := {tu 0 | t ∈ [0, 1]} × {0}. Thus we find an open convex neighbourhood U of C such that the norm of Dξ is bounded, whence ξ is of class C 0,1 (with uniform Lipschitz constant) on U . Applying now the Wallace Lemma [Eng89, 3.2.10], we may shrink U to a neighbourhood U ′ × W of C such that, for every V in U ′ , ξ(V, ·) is C 0,1 on W and the Lipschitz constant is uniformly bounded in V ∈ U ′ . As a consequence, calling τ ′ the minimum between τ and the (accessible) exit time of u from U ′ , the random drift the random drift is, in the chart ψ α , P -a.s. Lipschitz in Φ, uniformly in t, and also continuous in (t, Φ), because ξ is continuous and the path t → u(t, ω) is continuous in X s µ (K). Therefore we can apply Theorem 1.10 and Remark 1.12 with this random drift and with zero diffusion coefficient: possibly taking a smaller τ ′ , we get existence and uniqueness on [0, τ ′ ) of a progressively measurable solution Φ to (20).
As a consequence, the process (u, Φ) satisfies the Stratonovich differential, on the manifold X s−1 is the inclusion map. Note that the drift can be extended to a Borel function on X s−1 µ (K) × Diff s−3 µ (K) (for example setting the drift equal to (0, u • Φ) for u not in X s µ (K)), which fits into Remark 1.8. In view of Itô formula, we need an expression for the derivative of the composition map G(V, Φ) = V • Φ from Lemma 3.9. To give an explicit formula, we evaluate (18) in k ∈ K and localise in a manifold chart κ for K around Φ(k). Writing k ⋆ = κ(Φ(k)), Apply the Itô formula to see that the process η(t) = u(t) • Φ(t) satisfies Now we evaluate the drift T G((V, −Π[∇ V V ]), V • Φ) in k, and localize via (21) in the chart κ: for every V ∈ X s µ (K), Φ ∈ Diff s−3 µ (K) and k ∈ K: As k was arbitrary, we obtain the desired identity of the drifts for V in X s µ (K) (which is enough, since u lives in X s µ (K) P -a.s.). For the diffusion coefficient, we argue similarly to obtain for every V ∈ X s−1 µ (K) and Φ ∈ Diff s−1 µ (K), for every U in X s ′ µ (K), for every k ∈ K, therefore also the diffusion coefficient T G((V, I), 0 • Φ) and Σ(V • Φ) coincide. Hence we apply Lemma 1.7 and Remark 1.8 to obtain the Eulerian form. The proof of the second statement is complete.
Let us now consider the first statement: We are given a solution η to the Eulerian form on [0, τ ), with π(η 0 ) = id, and use the auxiliary map F (V • Φ) := V • Φ• Φ −1 = V from Lemma 3.9. To turn (19) into an explicit formula for the derivative of F as a mapping to X s µ (K) × X s µ (K), we need to localise ζ ∈ H s (K, T 2 K) in a chart. Thus let κ be a chart around k and take the chart representation of ζ • Φ −1 as Z κ (κ(k)) = (κ(k), V κ , Z 1,κ , Z 2,κ ) (where we suppress the argument κ(k) for readability), where we used the identification . This allows us to obtain a local formula for DF as a mapping to X s µ (K) × X s µ (K) (the image viewed locally over O κ ): for arbitrary k in K. Since F is C 2 by Lemma 3.9, we can apply Itô formula to u = F (η) and get Now evaluate T F • B(V • Φ) in k ∈ K using (23). The principal part becomes for every V • Φ ∈ T Diff s µ (K) and k, where we have suppressed the argument κ(k) on the right hand side. Now (24) yields the desired identity for the drift. Likewise, for the diffusion coefficient, (23) reduces for every V • Φ, for every U in X s ′ µ (K), for every k ∈ K, to hence DF (Σ(V •Φ)) coincide with the Eulerian diffusion coefficient (V, I). We can now apply Lemma 1.7 and Remark 1.8 to obtain the Eulerian form. The equation holds a priori on X s−2 µ (K), but u and all the integrands are continuous paths in X s−2 µ (K), hence the equation holds on X s−1 µ (K) as well. The proof is complete.
3.10 Remark Theorem 3.8 above can be seen as a stochastic analogue of [EM70,Thm 14.4], where it is proven that the deterministic incompressible Euler equation with smooth forcing on X s µ (K) is equivalent to the deterministic Euler flow on Diff s µ (K) if s > d/2 + 2. Notice, however, that in our Theorem 3.8 we do not obtain equivalence: solutions to the stochastic Lagrangian form give solutions to the stochastic Eulerian form only for s > d/2 + 3, and the other direction requires even s > d/2 + 4. This comes mainly from the C 2 regularity necessary in the stochastic analysis to apply the Itô formula.
3.11 Corollary Fix s > d/2 + 4 and assume that W satisfies Assumption 3.4 with s ′ = s + 2. Then, for every u 0 in X s µ (K), local strong well-posedness, in the sense of Theorem 1.10, holds for the stochastic Euler equation in Eulerian form (15) among X s µ (K)-valued solutions.
Proof. The result follows from the well-posedness theorem 3.5 for the Lagrangian form and the link between the Eulerian and Lagrangian viewpoints Theorem 3.8.

Remark
The proof of Theorem 3.8 used certain algebraic identities of the spray B and its continuity. In the presence of an additional deterministic forcing term f , one has to replace the spray B by the second order vector field B f discussed in 3.2 and Theorem 3.5. Due to the work of Ebin and Marsden [EM70, Section 11], similar algebraic identifies as the ones used for B in the proof of Theorem 3.8 hold for B f . In addition, we have seen that for continuous f : R → X s+1 µ (K), also the second order vector field B f is locally C 0,1 . Thus, the results obtained in Theorem 3.8 and Corollary 3.11 hold (by the same proof) also for solutions of the stochastic Euler equation in the presence of an additional deterministic forcing term.

Extensions of the local well-posedness results
In this section we discuss various natural extensions of the results of the present paper. In order to get a simple introductory presentation, we chose in many cases not to give the most general version possible with the methods developed in the present work. These topics are now discussed.

Sobolev spaces of different regularity
Throughout the main text our assumption was that the Sobolev spaces H s (K, M ) and Diff s (K) are of integer order s and the derivatives are contained in L 2 . There are two natural ways to modify these requirements: 1. Pass to fractional order Sobolev spaces.
2. Replace derivatives in L 2 by derivatives in some L p for 1 < p < ∞.
Reviewing the arguments used, none of them are tied to integer order Sobolev spaces 9 whence they carry over verbatim to fractional order Sobolev spaces. The main issue why we refrained from using fractional order Sobolev spaces is due to the fact that the underlying manifold K is allowed to have a smooth boundary. Hence our discussion has to distinguish two cases: 4.1 Fractional order Sobolev spaces for K without boundary: If K has no boundary, it is no problem to define fractional order Sobolev spaces on manifolds via the approach outlined in [IKT13] (note that loc.cit. only defines integer order Sobolev spaces on compact manifolds, but the approach carries over to fractional order, as noted in [BV17, Section 5]). Indeed one then retains all the necessary tools (e.g. Sobolev embedding theorems, differentiability of the composition, etc.) to carry out the constructions leading to Theorems 3.5 and 3.8.
Hence we can state the following:

Proposition
If K is a compact smooth manifold without boundary. Then Theorems 3.5 and 3.8 remain valid for fractional order Sobolev index s which satisfy the assumptions of these theorems. In particular, both main theorems from the introduction remain valid for fractional order Sobolev index s.

4.3
Fractional order Sobolev spaces for K with smooth boundary: If K is allowed to have a smooth boundary, we still believe that the results stated in the boundary-less case carry over. However, to the best of our knowledge, the only source in the literature sketching the construction of a manifold of mappings for fractional Sobolev regularity on a manifold with smooth boundary is the sketch contained in [Mic19, Section 5]. We strongly believe that the cited results will allow a similar theory for equations on fractional order Sobolev spaces as in the boundary-less case. Working out the details is however beyond the scope of the present paper.
If we change the Sobolev spaces in requiring that the derivatives should be contained in L p (relaxing the requirement that p = 2) one notices first that the spaces of Sobolev mappings are no longer (modelled on) Hilbert spaces. Instead we are in the weaker Banach space setting. It is well known that the classical Ebin-Marsden analysis can be carried out in the L p setting, [BB74]. However, also the stochastic setting outlined in Section 1 has to be modified since it relied on Hilbert space techniques, see [BE00] for how to do this. Though the authors think that this is possible in principle, we remark here that it would require a significant amount of work. This is not only due to the stochastic setting used. For example, we have used cut-off functions which exist on Hilbert, but not necessarily on Banach spaces [KM97, Chapter III].

Maximal solutions and preservation of regularity
Our aim in the present paper was to establish a local existence and uniqueness theory for a stochastic version of the Euler equation. From the perspective of the infinitedimensional manifold of Sobolev diffeomorphisms, all solutions to the Euler equation start at the identity element. Hence, to treat local existence and uniqueness we could restrict to a single fixed chart around the identity. As a result, the solutions we construct will (in general) not be maximal as we stop the solution once it leaves the domain of the manifold chart. While this is sufficient for local existence and uniqueness considerations, it might be desirable to establish the existence of maximal (in time) solutions to the stochastic differential equations. With some additional effort this is possible, see [Elw82, Chapter VII], but again it requires some work (essentially because, even at fixed time, a solution X(ω) of an SDE can live in different chart domains for different ω). In order to keep the presentation self-contained, we did not include maximality here. Moreover, some attention is needed to relate maximality at the Lagrangian and Eulerian level (to our knowledge, this point is not investigated even in the deterministic setting in [EM70]). However we expect existence (and uniqueness) of a maximal solution to hold for both Lagrangian and Eulerian level.
Once maximality is established, one can study the problem of preservation of regularity. Indeed, it is well known that the deterministic Euler equation preserves regularity of initial conditions. Namely, one has the famous 4.4 ("no loss no gain" theorem [EM70, Theorem 12.1]) Let K be a compact manifold (possibly with boundary), s > dimK 2 + 1 and η the solution of a second order equation on Diff s µ (M ) given by a smooth right invariant second order vector field (in particular, by the geodesic spray). If (η(0), η ′ (0)) ∈ Diff s+k µ (K) × T η(0) Diff s+k µ (K) then η(t) is H s+k on the interior of K for all t in its domain of definition.
In particular, the interval of existence of the solution does neither increase nor decrease based on regularity of the initial data (that is, there cannot exist a solution which is H s+k up to some time, and only H s after). While we expect that a version of the theorem also holds in our setting, the argument in loc.cit. cannot be readily adapted to stochastic differential equations: the main reason is that this argument uses the existence of a flow solution to the ODE on the infinite-dimensional manifold T Diff µ (K), something which is not clear to hold in the stochastic case.

Optimality in the Sobolev index
In the deterministic setting, [EM70,Theorem 14.4] obtains equivalence of the Eulerian and Lagrangian forms at least for s > d/2+2, one can probably get even just s > d/2+1 with some attention in the proof. We expect that also in the stochastic setting this threshold can be reached and that Corollary 3.11 holds with the threshold s > d/2 + 1. This can be done using some technical tools, like embedding of split manifolds (possibly with boundary) and regularisation arguments. For example, in the passage from Eulerian to Lagrangian viewpoint, one may regularize the Eulerian solution and apply Itô formula, then send regularization parameter to 0, and so one should get an equation for position and velocity of each particle (that is, for (Φ(k), u(Φ(k))) for each fixed k). Finally, using the embedding of T Diff s µ as split manifold into the Hilbert space of H s mappings, one may show the equation for position and velocity (Φ, u(Φ)) as an element in T Diff s µ (K). However the use of these technical tools requires a longer and careful analysis, which is outside the scope of this paper, hence we will address this result in a subsequent paper.

Multiplicative noise
In this paper, we considered additive noise as the simplest example of noise. We expect that the framework can be extended to other kinds of noise, for example a multiplicative noise of Nemytskii type, namely g(x, u(t, x)) • dW (t, x) for g : T K → R sufficiently regular: in this case, the noise in the Lagrangian formulation should be of the form Σ(η) • dW = vl T Diff s µ (K) comp(g • dW, η), which has similar regularity properties to the additive noise vl T Diff s µ (K) comp(•dW, π(η)). We leave the precise analysis of Euler equations with this and other noises for future works.

Extending the mechanism to other Euler-Arnold equations
The Euler equation is the prototypical example of a PDE which can be rewritten as an ODE on an infinite-dimensional configuration space. However, the same is true for a large class of PDEs: the so-called Euler-Arnold equations. We refer to [KW09,  To demonstrate the flexibility of the approach we give here a new result for a stochastic version of the averaged Euler equations [Shk98,Shk00, MRS00,MS03]. This is a fluid model where nonlinear interactions for length-scales smaller than α > 0 are neglected. We consider a stochastically forced version of these equations Here, ∆ denotes the Laplace-de Rham operator and the new variable m is interpreted as momentum. For simplicity, let us assume that the underlying compact manifold K is without boundary. In the following, we take s > d/2 + 1 and assume Assumption 3.4 on the noise with s ′ ≥ s − 3. The precise definition of solution is the following one: (25) is an X s µ (K)-valued F -progressively measurable process u = (u(t)) [0,τ ) , with τ accessible stopping time and τ > 0 P -a.s., with P -a.s. continuous paths in X s µ (K), such that for all t ∈ [0, τ )

Definition A local (strong and smooth) solution to the stochastic averaged Euler equation in Eulerian form
As for the Euler equations, the equations (25) can be cast instead as a second order SDE in the Lagrangian variable Φ ∈ Diff s µ (K) viaΦ = u•Φ. Informally, in Lagrangian coordinates the equations take the form Rigorously, the Lagrangian formulation is the following SDE on T Diff s µ (K): where . Applying our stochastic Ebin and Marsden framework then yields the following results. Note that the regularity assumption on the noise is lower than what we get for the stochastic Euler equations. This is because the relation between the momentum m and the velocity field u has a regularizing effect (hence averaged Euler equations). We also obtain a corresponding result in Eulerian variables: 4.7 Theorem Fix s > d/2 + 4 and suppose Assumption 3.4 with s ′ ≥ s. Then local strong well-posedness (in the sense of Theorem 1.10) holds for the stochastic averaged Euler equation (25) on X s µ (K).
Proof of Theorem 4.6. From [Shk00] we know that the drift B α is a smooth spray on Diff s µ (K). Due to the regularization operator (id −α 2 ∆) −1 , the diffusion coefficient Σ α , localised in a chart around the origin, is a C 1,1 section, by Proposition 2.11. Hence the result follows from Theorem 1.10.
Proof of Theorem 4.7. The proof follows the same steps as in Section 3, with obvious changes.

A. Stochastic integration on Hilbert spaces
In this appendix we recall basic concepts and construction from stochastic integration on infinite-dimensional spaces and stochastic differential equations. Also here we take the main results and the approach from [Elw82,BE00], taking also some facts from [DPZ14] and [BNVW08]. We fix a probability space (Ω, A, P ) and a filtration (F t ) t as in 1.1. In the following, H and E are separable Hilbert spaces.
A.1 A stochastic process X : I × Ω → G on a time interval I = [a, b] or I = [a, +∞) is a collection indexed by time of random variables X t : Ω → G, t ∈ I, where (G, G) is a measurable space. Strictly speaking, for every t, X t is an equivalence class, but we can also consider a modification, or version,X of X: this is a map defined on [a, t] × Ω such that, for every t,X t is a representative of X; we often use the same symbol for X andX. A process X is called

Brownian motion and Itô integral
As a first example we recall the definition and the construction of a Brownian motion on a separable Hilbert space.
A.2 (Q-Wiener process) Let Q be a symmetric, positive semidefinite, trace-class operator on E. An E-valued Q-Wiener process, or Q-Brownian motion, with respect to F is an E-valued progressively measurable process W satisfying • W 0 = 0 P -a.s.; • for every s < t, W t − W s is a centred normal random variable with variance (t − s)Q (that is, for every v in E, E[e i v,Wt−Ws ] = e − v,Qv /2 ); • for every s < t, W t − W s is independent of F s ; • W has a.s. continuous paths (that is, P -a.s. t → W t is continuous).
By [DPZ14,Propositions 4.3,4.4], for any symmetric, positive semidefinite, trace-class operator Q on E, there exists a Q-Brownian motion W which is given by where (e k ) k is an orthonormal basis of eigenvectors of Q, with non-negative eigenvalues (λ k ) k .
A.3 Example (Trace-class Brownian motion) Let (W k ) k∈N be independent real Brownian motions and σ k be elements of E such that Then the series W t = k σ k W k t converges, in L 2 (Ω) for every t fixed and also Pa.s., uniformly in t, and it defines an E-valued Brownian motion, with covariance matrix Q = k σ k σ * k , that is, Qa, b = k σ k , a σ k , b . This result is proved in [DPZ14, Propositions 4.3, 4.4 and Theorem 4.5], in the case of σ k orthogonal basis of eigenvectors of Q, but the proof can be extended in the same line to the case of possibly not orthogonal σ k , as here.

A.4 Example (Cylindrical Brownian motion) To define a cylindrical Brownian motion
W on a separable Hilbert space H, we set formally W = k W k f k , where (f k ) k is a orthonormal basis ofH and W k are independent real Brownian motions, defined on some filtered probability space (Ω, A, (F t ) t , P ) with complete and right-continuous filtration. Note that the above sum does not converge inH.
Let now E be a separable Hilbert space, such that there exists a dense embedding i :H → E which is Hilbert-Schmidt, that is Then W := i(W ) is an E-valued Brownian motion, in the sense that k f k W k converges in E as in Example A.3. See [DPZ14, Section 4.1.2] for more facts on cylindrical Brownian motion.
For stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), one has to deal with objects of the form b a ξ r dW r , where W is a Brownian motion. The problem is that W is not differentiable in time, hence the above integral cannot be defined as Riemann-Stieltjes integral; other extensions, for example using Young integration theory, also do not work here. However, using probability theory, we can make sense of this integral by approximation via Riemann sums and an isometry property. Here we recall the construction of this integral, called the Itô integral. We follow mostly [DPZ14, Section 4.2].
In the following, W is a Q-Wiener process on E (with respect to F ) as in A.2. For simplicity of exposition, we assume for the moment that Q is actually positive definite, explaining later how to extend the results to Q only positive semidefinite.
A.5 Definition For Q positive definite, we define a norm on L(E, H): It is easy to show, using the property tr[(AQ 1/2 )(AQ 1/2 ) * ] = tr[Q 1/2 A * AQ 1/2 ], that Note also that L(E, H) is not complete with respect to the L 2,Q norm.
A.6 (Operator valued processes) We denote by M Q ([a, b]) the space of progressively measurable processes ξ : It is a normed space endowed with the norm from Definition A.5, but it is not complete. The completion will include unbounded operators, but in view of Stratonovich integration to be defined below, we do not need this.
Here and in what follows, a L(E, H)-valued map is (Borel) measurable if it is measurable to B (L(E, H)). This applies also to the definition of progressively measurable processes A.1 (G = B(L(E, H)) here). Any such L(E, H)-valued Borel measurable map is also measurable with respect to the Borel σ-algebra generated by the L 2,Q norm (because the L E,H topology is stronger than the L 2,Q topology); see Remark A.8 later for more information on measurability for L(E, H)-valued maps.
We say that a process ξ : [a, b] × Ω → L(E, H) is elementary if it can be written as where a = t 0 ≤ t 1 < . . . < t n = b and, for every j, ξ j is F tj -measurable.
Hence we can extend the Itô integral I a,b to a linear isometry on M Q ([a, b]), such that (28) and (29) hold for I a,b (ξ) for all ξ in M Q ([a, b]). We use the notation (sometimes omitting a) The Itô integral is additive on time intervals, that is For a process ξ in M Q ([a, b]), we can consider the stochastic process on [a, b] defined by the Itô integral I a,t (ξ) for t in [a, b]. There exists a version, still denoted by I t (ξ), of this process which is progressively measurable and is a continuous martingale, that is, for P -a.e. ω, t → I t (ξ) is continuous as E-valued map and, for every s < t, [DPZ14] this is a consequence of the completeness of the space M 2 T (H) of continuous square integrable martingales). Moreover, the Burkholder-Davis-Gundi (BDG) inequality holds ([DPZ14, Theorem 4.36]): for every 1 ≤ p < ∞, We also mention a property of later use: for every F a -measurable random variable as it can be verified easily for elementary processes. Finally, all the above-mentioned facts hold for Q which is only positive semidefinite, with the following change: one has to take the quotient space of L(E, H) with respect to the L 2,Q seminorm (which is not a norm in this general case).

A.7 Remark
The book [DPZ14] uses a slightly different class of integrands, namely predictable processes (see [DPZ14, Section 3.3]) rather than progressively measurable ones. However one can show that, for every progressively measurable process ξ, there exists a predictable processξ with ξ t (ω) =ξ t (ω) for dt ⊗ P -a.e. (t, ω) (see [Kun90, Section 2.3, page 60] for a proof in the finite-dimensional case, which works also in our case). With this identification, the results in [DPZ14] hold in our context.
The paper [BE00] uses a more general setting for Banach space-valued processes and with abstract Wiener space. However, as known classically, one can reduce that setting to our case by setting (i,H, E) as abstract Wiener space withH = Q 1/2 E, with the norm induced by Q 1/2 , and i :H → E the embedding. Note that, if (e k ) k is a complete orthonormal basis on E of eigenvectors of Q with eigenvalues (λ k ) k , Q 1/2 e k = λ 1/2 k e k is a complete orthonormal basis onH and W is a cylindrical Wiener process onH.
A.8 Remark Concerning measurability for L(E, H)-valued maps, one can actually show that the Borel σ-algebra B (L(E, H)) on L(E, H) generated by the L E,H norm and the Borel σ-algebra on L(E, H) generated by the L 2,Q norm coincide (more precisely, when Q is not strictly positive, the Borel σ-algebras are taken on the quotient space and are generated by the quotient norms): the main point of the proof is to show that the L E,H norm is lower semi-continuous in the L 2,Q topology. However we do not need, and we do not use, this fact here.

Itô processes and Itô formula
An Itô process is the sum of a deterministic integral and a stochastic integral. For an Itô process, a stochastic chain rule, called the Itô formula, holds. It differs from the classical chain rule (for smooth objects) by a second order term.
A.9 Definition Let H be a separable Hilbert space. An H-valued process ξ on [a, b] is called an Itô process if there exist progressively measurable processes B : such that, P -a.s., The processes B and S are called drift part and diffusion part of ξ. We write in short For Itô process a chain rule is available, known as Itô formula, which contains a second order correction. The following version of Itô formula is taken from [BNVW08, Theorem 2.4], see also [BE00, Theorem 2.16]: A.10 Theorem (Itô formula) Assume that ξ is an H-valued Itô process with drift part and diffusion part B and S respectively. LetH be another separable Hilbert space and let f : x f are continuous and bounded. Then it holds, P -a.s. and for all t ∈ [a, b], In short, the Itô formula reads Here, for a complete orthonormal basis (e k ) k of E,

Localization up to random times
As we will see the solution of an SDE may live only up to a finite time, which depends on the specific realization of the Brownian motion, hence this time is random. Also exit times from a given set for a process are random. Here we introduce a suitable notion of random times and its basic properties (see for example [Bal17, Section 3.5]).
A.11 Definition A stopping time τ is a map τ : Ω → I ∪ {+∞} such that, for every t, the set {τ ≤ t} belongs to F t (that is, we can decide if τ is ≤ t or not looking only up to time t). We set the σ-algebra associated to τ . A stopping time is called accessible if there exists a sequence of stopping times τ n , non-decreasing, with τ n < τ P -a.s. for every n, such that τ n ր τ P -a.s..
Deterministic times are stopping times, the minimum and the maximum of two stopping times are also stopping times, the pointwise non-decreasing limit of stopping times is also a stopping time. A special example of stopping time is the exit time τ U of a progressively measurable process ξ from an open set U in a metric space G (endowed with its Borel σ-algebra), namely If ξ has continuous paths and ξ 0 is in U P -a.s., then τ U is accessible, taking τ n = τ Un the first exit times from U n , where U n = {x ∈ G | dist(x, U c ) > 1/n}.  G) is a measurable space, such that, for every t in I, the mapξ t := ξ t 1 {t<τ } +y 0 1 {t≥τ } is a random variable on (Ω, A, P ), where y 0 is some element of G (this definition of process on [0, τ ) and the next definitions are independent of the choice of y 0 ); see the notation paragraph in [BE00] for a similar definition. It is called adapted, resp. progressively measurable if the processξ is adapted, resp. progressively measurable. Analogous definitions can be given for [0, τ ] × Ω (assuming that τ is finite P -a.s.) and processes defined on this space.
A.13 In the situation of A.12 we define spaces of processes up to stopping time.
• If τ is a stopping time with τ < ∞ P -a.s., we let N Q ([0, τ ]) be the space of progressively measurable processes ξ : , we can take stopping times then τ n ր τ and τ n = τ definitely P -a.s.. Moreover the process ξ n s = ξ s 1 {s≤τn} , s ∈ I, is in M Q (I) and hence we can define I 0,t (ξ n ), t ∈ I. By (32), P -a.s., for every t in [0, τ ], I 0,t (ξ n ) is definitely constant in n, hence we define the Itô integral for ξ as the process on [0, τ ] given by (setting I t (ξ) = 0 in the P -null set where the limit does not exists for some t).
For ξ in N Q loc ([0, τ )), we can define I t (ξ) for t ≤ τ n , for all t, hence we can define the Itô integral before τ , I t (ξ) for t < τ . By (32), this integral is well-defined (P -a.s. for all t in [0, τ )) and independent of the sequence τ n chosen.
The definition of Itô process and Itô formula can be extended to processes defined up to a P -a.s. finite stopping time τ . An H-valued process ξ on [0, τ ] is called an Itô process if there exist progressively measurable processes B : such that, P -a.s., Also for such ξ, Itô formula holds as in Theorem A.10, replacing [a, b] with [0, τ ) and removing the boundedness assumption on f and its derivatives. This follows applying Itô formula to ξ t 1 {t≤τn} and letting n → ∞.

Stochastic differential equations in Itô form
In this subsection we give the main local well-posedness result for stochastic differential equations in Itô form, under locally Lipschitz coefficients. We consider the SDEs only on an open set where the coefficients are Lipschitz.
Having an E-valued Q-Brownian motion W as in A.2 (under the setting in 1.1), we consider the Itô SDE Here the coefficients b : [0, +∞) × H → H, σ : [0, +∞) × H → L(E, H), called resp. drift and diffusion coefficient, are given Borel functions, the initial datum ζ : Ω → H is a F 0 -measurable random variable. A progressively measurable process X, defined on [0, τ ) for some accessible stopping time τ with τ > 0 P -a.s., is called a local strong solution of (33), if b(t, X t ), σ(t, X t ) are resp. in L 1 P −a.s.,loc ([0, τ )) and in N Q loc ([0, τ )) and there holds, P -a.s., This definition extends to progressively measurable processes X, defined on the closed interval [0, τ ] for some P -a.s. finite stopping time τ with τ > 0 P -a.s., requiring that b(t, X t ), σ(t, X t ) are resp. in L 1 P −a.s. ([0, τ ]) and in N Q ([0, τ ]) and that (34) holds for all t in [0, τ ]. For notational simplicity, we also consider solutions where τ can be zero on a non-zero probability set (obviously a solution defined only at the initial time does not carry useful information).
A.14 Remark The word "strong" here has to be intended in the probabilistic sense and refers to the fact that the solution is defined on a given filtered probability space, with a given Brownian motion; it is not to be confused with the notion of strong solutions in the analytic sense (roughly speaking, regular solutions).
A.15 Theorem Let U be a bounded open set in H and assume that, locally uniformly in t, b(t, ·) is bounded and Lipschitz onŪ and σ(t, ·) is L 2,Q -bounded and Lipschitz on U , that is, for every T > 0, Then existence and uniqueness up to the first exit time from U hold, that is, for every T > 0: there exists a solution X on [0, τ U ∧ T ], where τ U is the exit time of X from U , of the SDE (33) and, ifX is another solution defined on [0,τ ], thenX = X on [0,τ ∧ τ U ∧ T ] P -a.s.; moreover τ U > 0 P -a.s. on the set {ζ ∈ U }.
The uniqueness part of the Theorem A.15 can be extended for two SDEs which coincide on U . For this, we consider another SDE dX t =b(t,X t )dt +σ(t,X t )dW t , X 0 =ζ, withb andσ given Borel functions andζ F 0 -measurable.
Proof. We call ρ = τ ∧τ ∧ τ U . The difference (X t∧ρ −X t∧ρ ) 1 Ω0 satisfies the equation where we have used (31) and (32). We take the expectation of the supremum of the squared norm: by Hölder inequality for the drift and BDG inequality (30) for the diffusion, we get By the Lipschitz property of b and σ on U , we obtain (the constant C possibly depending on T ). We apply Gronwall inequality and get E sup that is uniqueness.
Proof of Theorem A.15. Uniqueness is a particular case of Lemma A.16. For existence, we take, for any positive integer n, For each n we deduce from [KM97,15.9 (2)] that there exists a map ϕ n : H → R, C 1 with both ϕ n and its derivative globally Lipschitz continuous (whence bounded), such that ϕ n ≡ 1 on U n and ϕ n ≡ 0 on U c . Therefore the coefficients are globally Lipschitz and globally bounded on H, uniformly in t in [0, T ]. Hence the SDE on H dX n t = b n (t, X n t )dt + σ n (t, X n t )dW t admits a (unique) solution X n , by classical well-posedness theory (see e.g. [DPZ14,Theorem 7.2], applied with A = 0 there). For any m > n, since b m = b n and σ n = σ m on U n , by Lemma A.16 X m and X n coincide up to the first exit time τ Un from U n . Hence, calling τ U the limit of the non-decreasing sequence τ Un , P -a.s., for each t < τ U , the sequence X n t 1 {t≤τn} is definitely constant, in particular the limit X t = lim n X n t , t < τ U is well-defined P -a.s. and τ U is the (possibly infinite) exit time of X from U . Using (32), one can check easily that X is a solution to (33) on [0, τ U ). Finally, since U is bounded and b and σ are bounded, if τ U is finite, then X is extended by continuity to τ U and this extension is still a solution to (33). This shows existence for the SDE (33). The fact that τ U > 0 P -a.s. on {ζ ∈ U } follows from the continuity of paths of X. The proof is complete.
Note that, by (27), if the diffusion coefficient σ is Lipschitz in the L(E, H) norm on some set U (locally uniformly in t), then it is also Lipschitz in the L 2,Q norm (locally uniformly in t), hence the previous results apply.
A.17 Remark The definition of solution to an SDE and the existence and uniqueness Theorem A.15 can be extended to the case of a random drift, under the following assumption: given an accessible stopping time τ , the drift b : [0, τ ) × Ω × H → H is such that • for every x inŪ , b(·, ·, x) is progressively measurable, and • it holds P -a.s.: for every t in [0, τ ), b(t, ω, ·) is Lipschitz continuous onŪ , uniformly with respect to (t, ω).
Then existence and uniqueness for the SDE (33) hold on [0, τ U ∧τ ). The proof proceeds as in the proof of Theorem A.15 (noting that [DPZ14, Theorem 7.2] applies also to random drifts).

Stratonovich integral and Stratonovich differential equations
Here we recall the Stratonovich stochastic integral and Stratonovich SDEs. While not a martingale, the Stratonovich integral has the advantage of a classical chain rule, without second order terms, hence it is more suited to be extended to the manifold case. The price to pay is an additional degree of regularity for well-posedness of SDEs driven by Stratonovich integrals.
A.19 Definition Let ξ be as before and let g : H → L(E, H ′ ) be a C 1 map, where H ′ is another separable Hilbert space. The Stratonovich stochastic integral of g(ξ) in dW is the H ′ -valued progressively measurable process on [0, τ ] defined by where, for (e k ) k complete orthonormal basis of E, For x, v in H, g ′ (x).v is in L(E, H ′ ), so the Itô-Stratonovich correction, that is the last term in (35), makes sense. It is well-defined and progressively measurable as A.20 Remark By this very definition, the Stratonovich integral is defined in terms of ξ, g and S separately: even if g(ξ) = h(η) for another function h and Itô process η, the definition for the Stratonovich integral of h(η) may give a different object. Actually one can show that the integral depends only on g(ξ), regardless the way it is written, using the concept of quadratic variation (see also [BE00, Theorem 3.7] without using quadratic variation). But we will not use this fact and understand the Stratonovich integral as function of ξ, g and S.
For τ accessible stopping time, the above definition can be extended to Itô processes ξ on [0, τ ) in the form where B is progressively measurable and in L 1 P −a.s.,loc ([0, τ )) and S is in N loc ([0, τ )), the space of progressively measurable processes with τn 0 S t 2 L(E,H) dt < ∞ for every n, P − a.s.. Now we consider Stratonovich stochastic differential equations, in this case autonomous, namely Also here the drift b : H → H and the diffusion coefficient σ : H → L(E, H) are given functions assumed continuous and in C 1 resp., the initial datum ζ : Ω → H is a F 0measurable random variable. A progressively measurable process X, defined on [0, τ ) for some accessible stopping time τ with τ > 0 P -a.s., is called a local strong solution if it has P -a.s. continuous paths and there holds, P -a.s. for all t ∈ [0, τ ) This definition extends to progressively measurable processes X, defined on the closed interval [0, τ ] for some P -a.s. finite stopping time τ with τ > 0 P -a.s., requiring that (37) holds for all t in [0, τ ]. Moreover, if X takes values P -a.s. in an open subset U of H, it is enough that b and σ are defined and in C 0 , in C 1 resp. on U . Note that, since b and σ are resp. in C and in C 1 , all the integrals are well-defined. Note also that, if X is a solution, then X is an Itô process and the last two terms in (35) are the Stratonovich integral We can extend the above definitions and properties to Stratonovich differentials, namely, on [0, τ ) (or [0, τ ]), where σ is C 1 and B is progressively measurable and in L 1 P −a.s.,loc ([0, τ )) (or in L 1 P −a.s. ([0, τ ])): that is, X satisfies A.21 Theorem (Itô formula for Stratonovich differentials) Assume that X satisfies (38), with B, σ as above. Let f : H →H be a C 2 function, whereH is another Hilbert space. Then it holds, P -a.s., Proof. We apply Itô formula A.10, extended to possibly unbounded C 2 functions and to processes defined up to τ , as seen in the localization subsection A.13, to the Itô process X in (38) and to f : To prove (39), we have to show that Now we note that D x f σ is in C 1 (H, L(E, H)) and, for any x, v in H, it holds as equality in L(E, H). We take x = X t , v = σ(X t )Q 1/2 e k and compose with Q 1/2 e k : we obtain, for each k, P -a.s. for every t < τ , Summing over k, we get (40). The proof is complete.
The local well-posedness result requires the diffusion coefficient σ to have one more degree of regularity (with respect to the analogue result in the Itô context): A.22 Theorem Let U be a bounded open set in H and assume that b is in C 0,1 (Ū , H), that is Lipschitz onŪ , and σ is in C 1,1 (Ū , L(E, H)), that is C 1 fromŪ to L(E, H), with σ ′ Lipschitz fromŪ to L(H, L(E, H)). Then existence and uniqueness up to the first exit time from U hold, that is, for every T > 0: there exists a solution X on [0, τ U ∧ T ], where τ U is the exit time of X from U , of the SDE (36) and, ifX is another solution defined on [0,τ ], thenX = X on [0,τ ∧ τ U ∧ T ] P -a.s.; moreover τ U > 0 P -a.s. on the set {ζ ∈ U }.
As for the Itô SDE, we can extend uniqueness to two SDEs which coincide on U . For this, we consider the Stratonovich SDE A.23 Lemma Let U be a bounded open set in H. Assume that b,b coincide onŪ and are in C 0,1 onŪ and σ,σ coincide onŪ and are C 1,1 (as L(E, H)-valued maps) onŪ ; assume also that ζ andζ coincide on some F 0 -measurable set Ω 0 . Let X,X be two solutions, on [0, τ ] and [0,τ ] resp., of the SDEs driven by b, σ andb,σ resp.. Then X andX coincide on [0, τ ∧τ ∧ τ U ] × Ω 0 P -a.s., where τ U is the first exit time of X (orX) from U .
Proof of Theorem A.22 and Lemma A.23. The equation (36) reads in Itô form: Hence the results follow from Theorem A.15 and Lemma A.16 once we show that the above coefficients satisfy the Lipschitz assumptions for the Itô SDEs. The diffusion coefficient σ is C 1,1 onŪ , in particular σ ′ is Lipschitz, hence bounded onŪ becauseŪ is bounded; therefore σ is also Lipschitz onŪ and so bounded, in the L(E, H) topology.
Since the L(E, H)-topology is stronger than the L 2,Q topology, σ is also bounded and Lipschitz in the L 2,Q topology. The function b is Lipschitz, and so bounded, onŪ , by assumption. It remains to show that the Itô-Stratonovich correction tr[σ ′ σQ] is Lipschitz onŪ . For every x, y inŪ , taking (e k ) k basis of eigenvectors of Q with eigenvalues λ k , we have where in the last inequality we used that σ and σ ′ are Lipschitz and bounded onŪ . Hence the Itô-Stratonovich correction is Lipschitz, and so bounded, onŪ .

B. Essentials on spaces of Sobolev maps as infinite-dimensional manifolds
In this appendix we recall the construction of manifolds of mappings. Here we follow the classical exposition of [EM70,Pal68] and recall the relevant constructions. Let us begin by recalling the construction of Sobolev type sections of vector bundles B.1 In addition to our usual requirements we set d := dim K and fix a Riemannian metric g K on K. Its associated volume form is denoted by µ. Now π E : E → K will be a smooth vector bundle over K together with a Riemannian structure g E on E. Further, we let N be a finite-dimensional manifold without boundary together with a Riemannian metric g N and Riemannian exponential map exp N : T N ⊇ Ω → N . Again H s (K, N ) denotes the space of H s -maps f : K → N (cf. Definition 2.2).
Albeit we chose auxiliary Riemannian structures, the constructions in this appendix do not depend on the specific choice. In particular, every choice yields the same space of Sobolev sections (see e.g. [Pal68,§8]) we are about to define now.

Sobolev sections of a vector bundle
We recommend the survey in [Weh04, Appendix B] on spaces of Sobolev sections. For the readers convenience, the necessary definitions and main results are repeated now.
B.2 (Spaces of Sobolev sections) Denote by L 2 (E) the set of all Borel-measurable sections X of π E : E → M such that X L 2 = g E (X(k), X(k))dµ It is clear from the construction, that H s (E) is a Banach space (whose norm sums up the L 2 -norms (41) of the jets j s (X)) and even a Hilbert space (whose inner product is induced by the L 2 inner product on the jet spaces).
Contrary to our treatment of H s -morphisms between manifolds (which were defined as being H s in suitable charts), we defined the Sobolev sections as a completion with respect to an L 2 -inner product. This immediately establishes the Hilbert space property, but lacks a convenient description in local charts. It is well known [EM70, Section 2], [Weh04, Appendix B] that instead one can also define Sobolev sections using local trivialisations, being mindful of the warning 2.3 we need to take some care in showing that indeed H s (E) ⊆ H s (K, E).
that the sum of the inner products (42) is equivalent to the inner product inducing the Hilbert space structure of H s (E). As a consequence of (42), cf. [IKT13,Eq. (69) and Section 4.1], the mapping induces a Hilbert space isomorphism of H s (E) onto its image.
B.4 Lemma For s > d 2 , K compact (possibly with smooth boundary) and π : E → K a finite rank bundle. We have Moreover, H s (E) is separable.
Proof. We have already seen in B.3 that a section is in H s (E) if and only if localises in charts an H s -Sobolev map. Hence H s (E) ⊆ H s (K, E). For the converse take an element X in H s (K, E) which is also a section of π E . Restricting the bundle trivialisations to a relatively compact neighbourhood O of the image of X we obtain a fine cover ((U k ), (dom Ψ k ∩ O) k , X) of X (see [IKT13,Definition 3.2]. Then [IKT13, Lemma 3.2] implies that X localises in these charts to an H s -mapping, whence X ∈ H s (E) by the above. This proves the first statement of the Lemma.
To prove that H s (E) is separable, let us assume first that ∂K = ∅. Then the isomorphism (43)  B.5 Remark In [IKT13] Sobolev sections are described via the characterisation in Lemma B.4. This leads to a natural notion of Sobolev mappings allowing to treat fractional Sobolev exponents on manifolds without boundary (see e.g. [BV17] for a discussion). Though we followed the older approach in [Pal68], the proofs of Lemma B.4 and the approach in B.3 follows the characterisation in local charts.

Manifolds of H s -mappings
We now endow the set H s (K, N ) with a manifold structure. To this end let us first consider spaces of sections covering an H s -map.
B.6 Definition Consider f ∈ H s (K, N ) and define is only a a C ℓ -map. Using the identification T H s (K, N ) ∼ = H s (K, T N ) in (44) the derivatives can be identified as Thus Diff s (K) is only a topological group, but not a Lie group. In particular, the inversion map ι : Diff s (K) → Diff s (K) is only continuous. Considering inversion as a mapping inv : Diff s+ℓ (K) → Diff s (K) it is of class C ℓ with first derivative (for ℓ ≥ 1) given by the formula [EM70, p. 108] (cf. [Mic80,Proposition 11.13]) The topological group Diff s (K) is a Hilbert manifold such that right multiplication is smooth, whence Diff s (K) is a so called half Lie group, see [MN18].

Case 2: The underlying manifold has smooth boundary
We discuss now the case of K having smooth boundary and follow [EM70] (though a direct approach as in [Mic80] yields the same).
B.10 Embed K into its doubleK [Mun66, Theorem 6.3]. We can endowK with a Riemannian metricg such that ∂K ⊆K becomes a totally geodesic submanifold [EM70, Lemma 6.4]. Constructing the canonical manifold structure on H s (K,K) with respect tog, the canonical H s -chart around Φ ∈ Diff s (K) induces a submanifold chart for Diff s (K) mapping the closed subspace X s Φ,∂ (K) := {X ∈ H s (K, TK) | π • X = Φ, X(k) ∈ T k ∂K, ∀k ∈ ∂K} to Diff s (K) ⊆ H s (K,K) [EM70, Lemma 6.6]. We note the following important facts: If K has smooth boundary, Diff s (K) is a closed submanifold of H s (K,K) and the canonical charts with respect to the Riemannian metricg restrict to submanifold charts. As a consequence a direct calculation shows 1. The differentiability properties of the composition maps from B.9 carry over. In particular, Diff s (K) is a half Lie group.
2. Applying the identification (44) to the composition map on the submanifold Diff s (K), the identities from B.9 for the derivatives of the composition mappings are available.
The point of the above definition is that we can use the Hodge theory developed in [EM70, Section 7] to translate the Euler equation on manifolds with boundary to our infinite-dimensional setting. To this end consider the group of volume preserving diffeomorphisms (with respect to the volume form µ, which for K with boundary is assumed to be induced by the restriction of the Riemannian metricg in B.10).
B.11 (Volume preserving H s diffeomorphisms) The H s -diffeomorphisms for s > d 2 + 1 act by pullback on the differential forms on K. Hence we can consider the subgroup Diff s µ (K) := {Φ ∈ Diff s (K) | Φ * µ = µ} of volume preserving H s -diffeomorphisms. It is well known that Diff s µ (K) is a closed submanifold of Diff s (K) [EM70, Theorem 4.2 and Theorem 8.1], whence it is a half Lie group and in particular, a topological group. Furthermore, Diff s µ (K) is a Hilbert manifold modelled on a separable Hilbert space. We can thus apply the Birkhoff-Kakutani theorem [HR79,Theorem 8.3 • is open and separable, the same holds for T Diff s µ (K) • ⊆ T Diff s µ (K) (Note that T Diff s µ (K) is second countable as in any trivialisation T Diff s µ (K) • is homeomorphic to a product of second countable spaces (the model space is separable!) and we can cover T Diff s µ (K) with a countable cover of such neighbourhoods as Diff s µ (K) is second countable.). The tangent space at the identity can be identified as the space X s µ (K) := {X ∈ X s (K) | div(X) = 0, X(k) ∈ T k ∂K, ∀k ∈ ∂K} of divergence free H s -vector fields (where we suppress in the notation the condition of being tangential to the boundary). Due to the loss of derivatives in forming the Lie bracket of vector fields, X s µ (K) is not a Lie algebra. For s ∈ N the manifold Diff s µ (K) possess a strong right invariant Riemannian metric 16 . To this end, one defines an inner product on X s µ (K) via X, Y H s ,L := g K (X, L • Y )dµ, where L = (id +∆ s ) and ∆u = (δdu ♭ + dδu ♭ ) ♯ is the positive definite Hodge Laplacian. Then g Hs (X ϕ , Y ϕ ) := X ϕ • ϕ −1 , Y ϕ • ϕ −1 H s ,L , X ϕ , Y ϕ ∈ T ϕ Diff s µ (K) s (K) 15 Our proofs for the separability and metrizability of the manifold Diff s µ (K) (resp. its components) heavily exploited that Diff s µ (K) is a topological group. If one wants to avoid using such arguments, a direct proof using either the strong Riemannian metric (see below) or working with the Sobolev H s -topology on H s (K, K) will yield similar results. 16 Recall that a Riemannian metric is strong, if it induces the topology on every tangent space. While strong metrics retain properties of finite-dimensional Riemannian metrics, also new phenomena occur, see [Lan99,Kli95] for introductions to strong Riemannian metrics in infinite dimensions.
is a smooth right invariant Riemannian metric [EM70,p.140]. Note that smoothness of the metric is actually quite surprising as the inversion map is only continuous (see [BV17,Section 6.1] for a detailed discussion).