The need for robust and computationally efficient Parametrization Schemes (PS) which model the effects of fast sub-grid scale physics and other unresolved processes is well understood in Geophysical Fluid Dynamics (GFD). See, e.g., [GL19] for a recent overview. For example, the inclusion of stochastic schemes to represent model uncertainties has led to a new class of Stochastic Parametrization Schemes (SPS), surveyed, e.g., in [BAB + 17]. The SPS have improved the probabilistic skill of the ensemble weather forecasts, by increasing reliability and reducing the error of the ensemble mean. The coming years are likely to see a further increase in the use of SPS in ensemble methods in forecasts and assimilation. This, however, will put increasing demands on the methods used to represent computational model uncertainty in the dynamical core and other components of the Earth system, while maintaining overall computational efficiency [LLO + 17].
The preservation of geometrical structure and physicality of fluid dynamics can serve as a guiding principle in designing the desired robust PS for GFD. The PS are meant to preserve predictive power, accuracy and computational efficiency in modelling the effects of both: (i) unresolved phenomena due to the known but unresolved rapid sub-grid scale physics; as well as (ii) uncertainty due to unknown bias in the data. Thus, in ensemble computations PS face a daunting combination of tasks.
In this paper, we propose to design a structured parametrization of the rapid scales of fluid motion, by using a temporally rough vector field in the framework of geometric rough paths (GRP) [FH14] , which we call Geometric Rough Path Parametrization Schemes (GRPPS). Namely, we will develop a new class of variational principles for fluids which represents resolved and unresolved motions of fluid advection as GRPPS. Critical points of these rough-path constrained variational principles yield rough partial differential equations whose dynamics incorporates both the resolved-scale fluid velocity and the effects of the unresolved fluctuations.
In the special case when the generating rough path is just a straight line (or a smooth curve), the GRPPS approach introduced here reduces to the PS approach obtained through classical/deterministic variational principles. Similarly, when the generating rough path is a suitable realization of a Brownian motion (or, more generally, of a semimartingale process), GRPPS specializes to a pathwise interpretation of the SPS characterized through the stochastic variational principles first introduced in [Hol15] . In other words, this work enhances the mathematical framework of Stochastic Advection by Lie Transport (SALT), in which the Lagrangian trajectories are treated as time-dependent Stratonovich stochastic processes [Hol15] .
The present approach is based on the premise that Euler's fluid equations arise from Hamilton's variational principle for geodesic flow on the manifold of diffeomorphisms with respect to the metric defined by the kinetic energy of the fluid, [Arn66, EM70] . The variations are constrained by the condition of right-invariance of the velocity vector field. Hamilton's principle for fluids is modified when advection by the fluid motion under the action of the diffeomorphisms carries fluid properties such as mass and heat, whose contribution to the thermodynamic equation of state affects the motion [HMR98] . These advected fluid quantities are said to follow Lagrangian trajectories of fluid parcels in the flow. Since the Lagrangian trajectories for Euler's ideal fluid equations are pushed forward by time-dependent diffeomorphic maps, these trajectories may be regarded as curves parametrized by time on the manifold of smooth invertible maps (diffeomorphisms) [EM70] .
The aims of this paper. The overall goal of the present paper is to formulate rigorously in Theorems 2.5 and 3.12 variational principles for ideal fluid dynamics with advection of fluid quantities along Geometric Rough Paths (GRP). To achieve this goal, our first aim is to derive a rough version of the Lie chain rule in Theorem 3.3 leading to the GRP version of the classical Reynolds transport formula in Corollary 3.5. The Reynolds transport formula for momentum density encapsulates the force law which governs fluid motion. Mathematically, the formula describes the rate of change of the integral of the fluid momentum density over a moving control volume that is being transported by the rough (fluid) flow along a GRP. Thus, our key result is the Lie chain rule formula (3.4) in Theorem 3.3 for the rough differential (or increment in time) of the pull-back and push-forward of a tensor-field-valued GRP by a rough flow.
Formula (3.4) is intuitive and natural because it follows from the extension of ordinary calculus to GRP. The formula leads to a unified, stable, and flexible framework for modelling fluids whose Lagrangian parcels move along temporally rough paths. Theorem 3.3 for the Lie chain rule is the foundation on which the other contributions of the paper rest.
The paper is structured as follows:
• Section 2 formulates the first variational principle for fluid dynamics on geometric rough paths in Theorem 2.5, by imposing the Clebsch constraint for the advection fluid quantities along rough paths.
• Section 3 formulates the Lie Chain Rule Theorem 3.3 and the Reynolds Transport Corollary 3.5 for geometric rough paths. The Reynolds transport formula in the special case of one-forms yields the rough Kelvin-Noether Theorem 3.6. Section 3 also formulates the Hamilton-Pontryagin variational principle for rough paths in Theorem 3.12, which imposes the constraint that the vector fields which generate the Lagrangian trajectories are right-invariant under diffeomorphisms whose time dependence is rough. The Clebsch and Hamilton-Pontryagin variational principles for rough paths correspond to those derived in the SALT approach in [Hol15] and [GBH18] , respectively. Next we formulate the Euler-Poincaré constrained variational principle for ideal fluid motion on GRP in Theorem 3.14. Here • Finally, Section 5 contains the proofs of the main results formulated in Section 3.
In addition, the paper contains five Appendices which are meant to provide notation and background information, including proofs of key technical results invoked in the main text, a simple example of our procedure in the setting of smooth paths, and additional history and motivation. The first two Appendices are essential and contain together the key relationships and definitions needed in both geometric mechanics and the theory of rough paths for the present work, which as far as we know are found together nowhere else. The latter three Appendices provide additional information and motivation for the theory of rough paths. Appendix A defines the notation we use and summarises the essential background and results for both rough paths and geometric mechanics that we use in the text. We choose to put this section in the appendix rather than in the main text since different classes of readers may be familiar with at least some of our notation, and might wish to see the statements of the main results presented first. The main text will refer to sections in Appendix A as needed if we think a notation is not standard. Appendix B contains proofs of selected technical results which facilitate the proofs in the main text. Appendix C illustrates the variational principles we use in the example of a homogeneous incompressible fluid flow perturbed by spatially and temporally smooth noise. This example serves as a guide for introducing rough perturbations into the variational principles for more general fluid theories. Appendix D provides a short history and motivation in the development of the theory of rough paths and Appendix E discusses the concrete example of Gaussian rough paths and provides additional references to this important class of rough paths.
Contributions of this paper. This paper introduces a GRPPS framework which transcends the scope of either deterministic or stochastic parametrization schemes, by allowing GRP with Hölder index α ∈ 1 3 , 1 . 1 The case α = 1 recovers deterministic fluid dynamics. The case α = 1/2 − ǫ (ǫ ≪ 1) gives a pathwise characterization of SPS. Widening the choice of Hölder index provides a broader scope for modelling with PS. Indeed, one may also include models which are non-Markovian (for example, by choosing the rough path as a realization of a fractional Brownian motion, or of a more general Gaussian process with suitably chosen time dependence). Non-Markovian models include models with memory which are of interest in ocean dynamics, see e.g. [Ins19, Dun19, Sci, Bor19] [Woo81, LBDB + 13, PH06] .
The GRPPS presented here possess the following fundamental properties:
• Being derived from Hamilton's variational principle, they preserve the geometric structure of fluid dynamics [HMR98] .
• They satisfy a Kelvin circulation theorem, which is the classical essence of fluid flow.
• They are consistent with the modern mathematical formulation of fluid flow as geodesic flows on the manifold of smooth invertible transformations, with respect to the metric associated with the fluid's kinetic energy [Arn66] .
• They accommodate Pontryagin's maximum principle for control in taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls [BHCM00] .
Our work provides a bridge between two previously un-related areas of Mathematics; namely, Geophysical Fluid Dynamics and Rough Path Theory. We believe crossing this bridge will be highly beneficial for both areas. On one hand it will offer sound theoretical support for developing models in applications such as Numerical Weather Prediction, Data Assimilation, Ocean Dynamics, Atmospheric Science, perhaps even Turbulence. These models will benefit from a plethora of theoretical results (stability results, support theorems, large deviation principles, splitting schemes) resulting from their characterization as random dynamical systems generated by the solutions of the rough partial differential equations. In turn, the new connection between GFD and geometric rough paths (GRP) can be expected to become a fruitful source of open problems in mathematics. We enumerate a few of these open problems:
• Completeness of the constrained velocity variations in formulating the RPDEs in the Euler-Poincaré Theorem 3.14.
• Well-posedness of the RPDEs resulting in each section, starting with the 3D Euler fluid equation on GRP derived in Sec.4.1 and Sec. 4.2. 2
• Estimation of the rough path properties and calibration of the GRPPS model from observed or simulated data.
• The development of pathwise data assimilation methods for the incorporation of data into GRPPS.
• Uncertainty quantification and forecast analysis using GRPPS.
All of the authors are grateful to our friends and colleagues who have generously offered their time, thoughts and encouragement in the course of this work during the time of COVID-19. We are particularly grateful to T. D. Drivas and S. Takao for thoughtful discussions. DC and DH are grateful for partial support from ERC Synergy Grant 856408 -STUOD (Stochastic Transport in Upper Ocean Dynamics). JML is grateful for partial support from US AFOSR Grant FA9550-19-1-7043 -FDGRP (Fluid Dynamics of Geometric Rough Paths) awarded to DH as PI. TN is grateful for partial support from the DFG via the Research Unit FOR 2402.
To streamline the presentation of our results and to provide a ready reference for the reader, we have assembled the notation and background of geometric mechanics and rough paths theory needed for this paper into one place -Appendix A -rather than dispersing it sequentially in the main text.
Let Z ∈ C α T (R K ) be a given geometric rough path with Hölder index α ∈ ( 1 3 , 1] defined in the time interval t ∈ [0, T ]. Let X = X F 1 denote a function space of vector fields. Let X ∨ = X ∨ F 2 denote a function space of one-form densities such that the canonical pairing ·, · X : X ∨ C ∞ × X C ∞ → R defined in (A.13) extends to a continuous pairing on X ∨ ×X. For incompressible fluids, we implicitly use the constructions of the divergencefree, or both divergence-and harmonic-free vector fields and their canonical 'duals' (see Definition A.20). We write all spaces in brief notation without including Riemannian measure µ g or other extraneous adornment for a unified treatment of the compressible and incompressible case. That is to say, for incompressible fluid flows, all vector fields (and variations of vector fields) and one-form densities are constrained. Using Cartan's formula (A.12) and the Stokes theorem, one can show that for all u ∈ X C ∞ , the adjoint (see (A.14)) of the vector-field operation ad u = − £ u : X C ∞ → X C ∞ relative to the canonical pairing ·, · X is given by ad
Let A be a direct summand of alternating form bundles and tensor bundles such that the first component of A is the density bundle Λ d T * M. Let A ∨ denote the canonical dual in Section A.2.2. Define ·, · A : 2 The solution properties of the corresponding equations of motion derived here will be studied in a subsequent work using PDE methods similar to [HLN19a, HLN19b] .
be function spaces such that the pairing ·, · A extends to a continuous pairing on
Definition 2.1 (Diamond operator (⋄)). We define the bilinear diamond operator ⋄ :
We refer the reader to [HMR98] and Section 4 (esp. Section 4.4) for explicit computations with the diamond operator in fluid dynamics. We assume that ⋄ extends to a continuous operator ⋄ :
(iii) the mapping δℓ δu (·, a) : X → X ∨ is an isomorphism.
Let ξ ∈ X K D ′ denote a fixed collection of vector fields. 3
Remark 2.4. LetX,Ã, andÃ ∨ be function spaces (see Section A.2.2) such thatX ֒→ X,Ã ֒→ A, and A ∨ ֒→ A ∨ , and £ ∈ L(X ×Ã, A) and £ * ∈ L(X ×Ã ∨ , A ∨ ). If (u, a, λ) ∈ C T (X) ×D Z,T (Ã) × D Z,T (Ã ∨ ) and ξ ∈X K , then (i) and (ii) hold.
Clebsch variational principle. The Clebsch action functional S Clb Z : Clb Z → R is defined by
A variation of (u, a, λ) ∈ Clb Z is a curve {(u ǫ , a ǫ , λ ǫ )} ǫ∈(−1,1) ⊂ Clb Z of the form (u ǫ , a ǫ , λ ǫ ) = (u + ǫδu, a + ǫδa, λ + ǫδλ),
which vanishes at t = 0 and t = T . We say (u, a, λ) ∈ Clb Z is a critical point of the action functional S Clb Z , if for all variations one has
By virtue of the controlled rough path calculus and, in particular, Lemmas A.13 and B.4, we obtain the following Clebsch variational principle. 4
Theorem 2.5 (Clebsch variational principle on geometric rough paths). A curve (u, a, λ) ∈ Clb Z is a critical point of S Clb Z in (2.1) iff for all t ∈ [0, T ], the following equations hold.
Proof. See Section 5.1.
Remark 2.6. By Remark A.9, the integral T 0 λ t , da t A in the Clebsch action functional in (2.1) is welldefined. Indeed, the extra structure provided by the Gubinelli derivative in the controlled rough path space (see Appendix subsection A.1.1) allows one to construct this integration.
Remark 2.7. The Lagrange multiplier λ enforces the constraint that 'a' satisfies
That is, the quantity 'a' is (formally) advected by the integral curves of the vector field dx t = udt + ξdZ t . The Lie chain rule (Theorem 3.3) and Hamilton-Pontryagin variational principle in Section 3.3 explain the nature of this differential notation (see Remarks 3.4 and 3.9) which we will use freely. It follows that a = (a, −£ ξ a) ∈ D Z,T (A) and (£ ξ a,
For more information about rough partial differential equations (RPDEs) and their solutions, we refer the reader to [FH14, BG17, DGHT19, HLN19a, HLN19b] . We mention that to prove well-posedness and, in particular, to show that 'a' is controlled, one must obtain a priori estimates of a remainder term which contains third-order Lie derivatives, see equation (A.11 ., projections) , we can easily impose that u is both divergence and harmonic-free and derive the corresponding momentum equation with enough 'free-variables' to impose the divergence-free and harmonic-free constraints. In particular, we shall explain how the pressure and constant harmonic terms naturally decompose into a smooth and rough part, and how they can be recovered from u as was done in, for example, [MR04, MR05] and [HLN19a, HLN19b] ).
For an incompressible ideal fluid evolving on a compact oriented Riemannian manifold (M, g) with associated volume-form µ g , the Lagrangian flow map η : [0, T ] → Diff µ g may be regarded as a curve in the group G := Diff µ g of volume-preserving diffeomorphisms on M endowed with some appropriate topology, initiated from the identity η 0 = id and parametrized by time, t ∈ [0, T ]. In his seminal paper [Arn66] , V. I. Arnold showed that the configuration space for incompressible hydrodynamics is the space of volume preserving diffeomorphisms and that Euler's equation for the Eulerian velocity field u (i.e.,η t = u t • η t ) is equivalent to the path η being a critical point of the kinetic energy action functional. Otherwise said, Euler's equation can be recast as the geodesic equation on the diffeomorphism group endowed with the right-invariant weak L 2 -metric.
However, various geometric-analytic challenges arise if one wishes to make this viewpoint constructive and solve the geodesic equation as an ODE (and show there is no derivative loss). The crux of the matter is that composition from the right is not smooth if one wants to endow G with a Banach topology and work with a standard functional analytic tool-set [EM70] . The variational principles developed in this paper can be seen as extensions of the geodesic principle in [Arn66] or, more generally, the overarching EPDiff theory [HMR98] .
Theorem B.1 can be extended via a coordinate chart or approximate flow argument (see, e.g., [Wei18, Dri18, Bai19] ) to obtain the following theorem concerning smooth rough flows on the closed manifold M. We assume smoothness in the spatial variable and compactness of our manifolds for simplicity. More relaxed conditions can be found in, for example, in [Wei18] . 
such that η ts = Flow(u, ξ, Z) ts , (s, t) ∈ [0, T ] 2 , satisfies the following properties:
(i) for all (s, θ, t) ∈ [0, T ] 3 , η tt = Id and η tθ • η θs = η ts ;
(ii) for all (s, t) ∈ ∆ and f ∈ C ∞ ,
and
Proof. See Section 5.2.
Remark 3.2. Let us recall that η * ts and η ts * denote the pullback and push-forward, respectively (see (A.10)). Item (ii) in (3.1) means that for all X ∈ M, the quantity η ·s X is the unique solution of the RDE dη ts X = u t (η ts X)dt + ξ t (η ts X)dZ t , t ∈ (s, T ], η ss X = X. We refer to the following theorem as the Lie chain rule. The proof of the theorem is given in Section 5.2.
Theorem 3.3 (Rough Lie chain rule). For given
Then for all (s, t) ∈ ∆ T ,
where the time-dependent vector fields u and ξ are given in equation (3.3).
Remark 3.4. By (3.5), for an arbitrary τ 0 ∈ T r,s C ∞ , it follows that τ · = η ·0 * τ 0 is a classical solution of
Notice that if we introduce the notation dx t := dη t0 • η −1 t0 := u t dt + ξ t dZ t , then we may write
which generalizes the dynamic definition of the Lie-derivative to the rough case.
The following corollary is an extension of the Reynolds transport theorem. It is an immediate application of the definition of the integral on manifolds (see, e.g., Sec. 8.1 and 8.2 of [AMR12] ), the global change of variables formula, the Lie chain rule (Theorem 3.3) and the rough Fubini theorem (Lemma B.3. We will use this formula next in the case k = 1 for the proof of the Kelvin circulation theorem (see Section 3.2).
Corollary 3.5 (Rough Reynolds transport theorem). For given
Then for all k-dimensional smooth submanifolds Γ embedded in M and (s, t) ∈ ∆ T , we have
where η ts (Γ) denotes the image of Γ under the action of the flow η.
Assume that for all t ∈ [0, T ],
where all the paths and integrands are assumed to be smooth. By virtue of Theorem 3.1, there exists a flow
We obtain the following rough version of the Kelvin-Noether theorem in [HMR98] as an application of the Reynolds transport theorem in Corollary 3.5, Theorem 3.6 (Rough Kelvin-Noether Theorem). Let γ denote a compact embedded one-dimensional smooth submanifold of M and denote γ t = η t (γ) for all t ∈ [0, T ]. If D 0 is non-vanishing, then
Proof. See Section 5.3.
In this section, in addition to the assumptions in Section 2, we require that A = A C ∞ , X = X C ∞ , ξ ∈ {X C ∞ } K , and Z ∈ C α g,T (R K ), α ∈ 1 3 , 1 2 , is truly rough as in Definition A.14. We define the space of rough diffeomorphisms by
For a given a 0 ∈ A C ∞ , the Hamilton-Pontryagin action integral S HP Z a 0 : HP Z → R is defined by
(3.7)
Remark 3.9. By Theorem A.15, the Lagrange multiplier λ in (3.7) enforces 
where the notation for dx t is explained in Remark 3.4. That is, the quantity a is advected by the flow η ∈ Diff Z,C ∞ . This advection equation is used directly as the constraint in the Clebsch variational principal in Theorem 2.5.
Definition 3.10. A variation of (u, η, λ) ∈ HP Z is a curve {(u ǫ , η ǫ , λ ǫ )} ǫ∈(−1,1) ⊂ HP Z of the form
which vanishes at t = 0 and t = T , Remark 3.11 (Variation η ǫ ). The type of variation we use for the rough diffeomorphism is common in the geometric mechanics community (see, e.g., Lemma 3.1 of [ACC14] ). Notice that for all t ∈ [0, T ] and f ∈ C ∞ ,
Applying Theorem 3.3 and using the natural property of the Lie derivative leads to
where v ǫ t = ψ ǫ t * v and σ ǫ t = ψ ǫ t * σ. Thus, for a given η = Flow(v, σ, Z) ·0 , it follows that
The proof of the following theorem is given in Section 5.4.
Remark 3.13. The corresponding Hamilton-Pontryagin principle was derived for SALT in [GBH18] .
Proof. See Section 5.4.
In this section, we assume all stated quantities exist and are smooth, and work formally (see Remark 3.16).
Theorem 3.14 (Euler-Poincaré variational principle). Consider a path η = Flow(u, ξ, Z) ∈ Diff Z,C ∞ . The following are equivalent:
(ii) The Euler-Poincaré equations on geometric rough paths hold: that is, for all t ∈ [0, T ],
Proof. See Section 5.5.
Remark 3.15. Recalling that for all u ∈ X ∞ the adjoint of ad u = −£ u :
(3.9)
Remark 3.16. This is not strictly a variational principle in the same sense as the standard Hamilton's principle. It is akin to the classic Lagrange d'Alembert principle for dynamics with nonholonomic constraints, because the variations of of δu and δa in (3.8) are restricted in terms of δw. These restrictions are discussed next.
and for all t ∈ [0, T ],
(ii)
.
Using the equality of mixed derivatives, we find
to deduce δu t dt = ∂ t δw t − ad u t δw t dt − ad ξ δw t dZ t .
where u denotes the inverse of δℓ δu (·, a) : X → X ∨ applied to m; that is, m = δℓ δu (u, a). The rough Hamiltonian h t (m, a) is the sum of the deterministic Hamiltonian defined by Legendre transformation associated with the Lagrangian ℓ and given by
Let us take variations of h(m, a) in m and a. For arbitrary δm ∈ X ∨ C ∞ and δa ∈ A C ∞ , we find
where in the second equality we have used m := δℓ δu and set dx t := u t dt + ξdZ t . Thus,
which is to say
in which the boxes ( ) represent substitution in the operator. For arbitrary f :
in which the last equality adopts the notation for the semidirect-product Lie-Poisson bracket given in [HMR98] .
In differential notation, we find
Remark 3.19. Stochastic Hamilton equations were introduced along parallel lines with the deterministic canonical theory in [Bis82] . These results were later extended to include reduction by symmetry in [LCO08] . Reduction by symmetry of expected-value stochastic variational principles for EulerPoincar equations was developed in [ACC14] and [CCR15] . Stochastic variational principles were also used in constructing stochastic variational integrators in Bou-Rabee and Owhadi [BRO09] .
Let (M, g) denote a smooth, compact, connected, oriented d-dimensional Riemannian manifold without boundary. Denote by µ g ∈ Dens C ∞ the associated volume form, which is given in local coordinates by
In this example, we will explain how to impose incompressibility through Lagrangian constraints. In the following example, we will explain how to impose incompressibility through projections and spatial constraints. Towards this end, we introduce an additional Lagrangian multiplier π ∈ D Z,T (F 3 ∩ F 4 ) to enforce incompressibility. We consider the Clebsch action functional
where dx t = u t dt + ξdZ t and the Lagrangian ℓ : X × A → R is defined by
where the ♭ operation is defined in Section A.2.3. We take variations of (u, a, λ) as defined in Section 2. A variation of π is defined to be π ǫ = π + ǫδπ for δπ ∈ C ∞ T (C ∞ ) such that δπ 0 = δπ T = 0. It follows that all (u, a) ∈ X × A ,
With minor modifications of the Proof of Theorem 2.5, we find that (u, a, λ, π) is a critical point of S Clb Z = 0, if and only if for all t ∈ [0, T ]:
Substituting D = ρµ g into the equation for D and applying the diffeomorphism 1 µ g (see A.15), we find
Since ρ is advected, we obtain for all t ∈ [0, T ],
In order to conclude that div µ g u ≡ 0, we need to assume either div µ g ξ ≡ 0, or true roughness of Z. In the next example, where u is constrained to be divergence-free, we do not require additional assumptions to conclude that u is divergence-free since we will impose this directly. Substituting m = u ♭ ⊗ D into the momentum equation and recalling that D is advected, that the Lie derivative is a derivation, and that the product rule (Lemma A.13) holds, we find
Assuming ρ is non-vanishing and applying the diffeomorphism 1 D (see A.15) yields
in which the pressure is identified in terms of the Lagrange multiplier dπ t as dp t := −ρ t dπ t . We will elaborate more on this equation in the following example.
Let (M, g) and µ g be as in the previous example. Let A = Λ d T * M and A ∨ = Λ 0 T * M in this example, and notice that for all D ∈ A = Dens F 3 , there exists ρ ∈ Ω 0 F 3 such that D = ρµ g . Let X µ g = X µ g ,F 1 denote the space of incompressible vector fields and X ∨ µ g = X µ g ,F 2 denote the dual space of one-form densities modulo the kernel of the divergence-free projection as defined in Definition A.20. Denote by ·, · X µg :
The square brackets denote an equivalence class, where we have modded out by elements of the form
where the diffeomorphism ♯ D is defined in Section A.2.2. Using the diamond operation computed in the previous example, we find δℓ δD
Clebsch critical points (Theorem 2.5) thus satisfy
Critical points of the Hamilton-Pontryagin action functional (Theorem 3.12) also satisfy the first two equations. Since D is Lie-advected and the Lie derivative is a derivation, using the product rule (Lemma A.13), we find
Upon invoking the definition of X ∨ µ g in Definition A.20, we find
where dp ∈ C α T (Ω 0 ) and dq ∈ D Z,T ((Ω 0 ) K ). Applying 1 D (as defined in A.15) and recalling that div µ g u t ≡ 0 yields 
The following identity is well-known
, [DR78] [Section 3]). 5 Thus, applying the ♯ operator to the equation for u ♭ yields
where in a local coordinate chart (see (A.11)),
It is worth noting that for all u ∈ X µ g ,C ∞ and v, w ∈ X C ∞ ,
Remark 4.2. When ξ ≡ 0, the corresponding equation is the usual deterministic incompressible nonhomogeneous Euler fluid equation (see, e.g., [BF12] [Ch. VI] or [Mar76] ).
Let us now take the fluid to be homogeneous ρ ≡ 1; that is,
or equivalently,
In [HLN19b] , two of the authors proved the existence of a local strong solution
Vorticity dynamics Let us now discuss vorticity dynamics in both the inhomogeneous and homogeneous case. We assume div µ g ξ ≡ 0 and that all quantities are regular enough subsequently to perform each calculation. First, notice that for every f ∈ C ∞ ,
That is, f (ρ t ) is advected by dx t (see Remark 3.4). Let ω = du ♭ ∈ Ω 2 be the vorticity two-form. Since the exterior derivative d commutes with the Lie derivative, we obtain
Thus, by the produt rule (Lemma A.13), we get
In particular, in dimension three, using that dρ t ∧ dp
Moreover, in dimension three, applying Stokes theorem we get
We also have that
where G is the anti-derivative of g(ρ) = f (ρ) ρ 2 . Thus, in dimension two, we obtain
In the homogeneous setting, the vorticity equation is given by
Using the Hodge-star operator ⋆ : Ω 2 → Ω d−2 and settingω = ⋆ω ∈ Ω 0 in dimension two andω = ♯ ⋆ ω ∈ X µ g in dimension three, we find
Here, we have used that ♯⋆ and the Lie derivative commute (see, e.g., Section A.6 of [BF17] .) In three dimensions, the helicity, defined as
measures the linkage of field lines of the divergence-free vector fieldω [AK99] . It follows that
Therefore, the linkage number of the vorticity vector field Λ(ω) is preserved by the 3D Euler fluid equations. In dimension two, for any smooth
which implies that in dimension two enstrophy is conserved.
Divergence and harmonic-free It is worth noting that in the homogeneous setting, one can only recover u directly from ω if H 1 ∆ = ∅ via the Biot-Savart law (see Section A.2.3). Otherwise one needs to keep track of the harmonic constant. Nevertheless, one can repeat the above analysis with the harmonic and divergence-free
Here, u is constrained to be both divergence free and harmonic free. For this equation, u ♭ (and u) can be recovered directly from ω via the Biot-Savart operator. This equation has been studied in [CFH19, BFM16, BM19, CL19b] . See, also, the discussion in [FL19] .
Let M = S be the flat one-dimensional torus (i.e., the circle). Denote the standard normalized volume form by µ ∈ Dens C ∞ and coordinates by x. In this example, we take A = ∅ = A ∨ . We define the Lagrangian ℓ : X → R by
It follows that
If (u, η) is a critical point of the Hamilton-Pontryagin action functional (Theorem 3.12), then
which we may interpret as Lie transport of the momentum in the Camassa-Holm equation along rough paths.
Since the Lie derivative is a derivation and we have the explicit formula (A.11) for one-forms, we find
Thus, identifying u with a scalar-valued function, we get
After some simplification, we find
written as a nonlocal Cauchy problem with the pseudo-differential operator Λ −2 . Indeed, notice that if one substitutes ξ with u in the dZ t -term, then one obtains the same operator in dt-term. Now, if α = 0, we obtain the Burgers equation on rough paths,
The properties of the stochastic Burgers equation with stochastic transport noise have been investigated, e.g., in [AOdLT19] , and the properties of the Burgers equation with rough transport noise have been studied in [HNS19] . It follows that
Let A = Λ d T * M ⊕ Λ 0 T * M and A ∨ = Λ 0 T * M ⊕ Λ d T * M. Denote the advected variables by a = (D, s) ∈ A F 3 = Dens F 3 ⊕Ω 0 F 3
where h(p, s) = e(ρ, s) + p/ρ is the specific enthalpy/mass, which satisfies dh(p, s) = 1 ρ dp + T ds.
Applying the calculations with the diamond operation (⋄) in Section 4.1 yields
Critical points of the Clebsch and Hamilton-Pontryagin action functionals satisfy
Since D is Lie-advected and the Lie derivative is a derivation, using the product rule (Lemma A.13) and applying the diffeomorphism 1 D yields
Restricting to dimension three and working with enough regularity to perform the subsequent calculations implies three advected quantities,
Let ω = du ♭ denote the vorticity two-form and letω denote the corresponding divergence-free vector field. From the three quantities in (4.2), one may construct the following advected scalar quantity known as the potential vorticity
Consequently, the following functional is conserved for the adiabatic compressible Euler equations on GRPs
for any smooth function Φ : R 2 → R.
Proof. It is worth noting that this proof closely mirrors the proof in [Hol15] for stochastic variational principles. Nonetheless, we repeat the proof for the convenience of the reader. If (u, a, λ) ∈ Clb Z is a critical point of the action functional, then it satisfies
where
Here, we have used the definition of the diamond operator and (2.2) to exchange the order of derivative in ǫ and the time-integral for the Lagrangian terms. Since we may always take δu ≡ 0, δa ≡ 0, and δλ ≡ 0, we conclude that I(δu) = 0, II(δa) = 0, and III(δλ) = 0 for all smooth (δu, δλ, δa) that vanish at t = 0 and t = T . 
Subtracting t 0 δℓ δa (u r , a r )dr from both sides of the above, testing against a smooth φ ∈ X C ∞ and working in differential notation yields
Consequently,
The converse can be obtained by reversing the above proof.
Proof. We will prove the statement in the following steps.
1. We prove the formula for scalar functions by working in a local chart;
2. We prove the formula for vectors by reducing to step 1 and using the product formula;
3. We prove the formula for one-forms from steps 1 and 2 and the product formula;
4. Using steps 3 and 4, we apply an induction argument to prove the general formula.
For simplicity, we will drop the dt-terms and time-dependence on ξ, and assume s = 0. That is, we
Since we are working in C ∞ , we will simply write D Z for the controlled spaces.
Step 1. Assume that f ∈ C α T (C ∞ ) has the decomposition
We aim to show that
and
where both integrals are understood in the sense of controlled calculus. We focus only on the pull-back formula (5.1) as the equation for the push-forward can be shown in a similar way.
Towards this end, let us fix a coordinate chart (U, φ) with coordinates denoted by x. Let ρ := x • η, 1 and b(t, ·) = π t (x −1 (·)). We will now show that
which is (5.1) written in local coordinates. Since (φ, U) was arbitrary, proving (5.2) completes step 1. To see this, it will be convenient to spell out the expansion in terms of scalars. We identify Ξ(·) as an operator on L(R K , R d ) acting on Z with Ξ k (·)δZ k st and write the Davie's expansions of ρ and F:
and
To prove (5.2), we Taylor expand F s , b k (s, ·), and b ′ k,l (s, ·), and use (5.4) to write
Since Z is geometric (i.e., Z lk st + Z kl st = Z l st Z k st ), plugging in the expansion (5.3), we find
Straightforward, but tedious, computations show that the local expansion
satisfies |δ 2 Ψ sθt | |t − s| 3α . By the uniqueness in Lemma A.1, we get (5.2).
Step 2.
for some (π, π ′ ) ∈ D Z . For any f ∈ C ∞ , we have
Recall that η t = η t * ∈ C α T (C ∞ ) satisfies (3.2) with s = 0. Applying Lemma A.13 with the continuous bilinear map B :
Because f was arbitrarily chosen, we conclude that (η * V, η * £ ξ V + η * π) ∈ D Z and
Noting that
, and following a similar proof, we find that (η * V, −£ ξ η * V + η * π) ∈ D Z and
(5.5)
Step 3. Assume that α ∈ C α T (Ω 1 C ∞ ) has the decomposition
Fix an arbitrary vector V ∈ X C ∞ independent of t. Using (5.5) and Lemma A.13, we get
Applying (5.1), we obtain
The derivation property of the Lie derivative implies
Noting that η * t (π t (η t * V)) = (η * t π t )(V) and that V was arbitrary, we obtain
Following a similar argument, we get η t * α t = α 0 + t 0 η r * π r − £ ξ (η r * α r ) dZ r , which completes step 3.
Step 4. Let us show how to extend to T lk C ∞ . Let V 1 , . . . , V k ∈ X C ∞ , and α 1 , . . . , α l ∈ Ω 1 C ∞ . We recall that (η * t τ t )(α 1 , . . . , α l , X 1 , . . . , X k ) = η * t τ t (η t * α 1 , . . . , η t * α l , η t * X 1 , . . . , η t * X k )
Using induction in the product formula, we obtain τ t (η t * α 1 , . . . , η t * α l , η t * V 1 , . . . , η t * V k ) = τ 0 (α 1 , . . . , α l , V 1 , . . . , V k )
τ r (η r * α 1 , . . . , £ ξ η r * α j , . . . , η r * α l , η r * V 1 , . . . , η r * V k ) − k j=1 τ r (η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , £ ξ η r * V j , . . . , η r * V k ) dZ r which from (5.1) yields
τ r (η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , £ ξ η r * V j , . . . , η r * V k ) dZ r + t 0 η * r (ξ[τ r (η r * V 1 , . . . , η r * V k )])dZ r .
By the derivation property of the Lie derivative, we get ξ[τ r (η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , η r * V k )] = (£ ξ τ r )(η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , η r * V k ) + l j=1 τ r (η r * α 1 , . . . , £ ξ η r * α j , . . . , η r * α l , η r * V 1 , . . . , η r * V k ) + k j=1 τ r (η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , £ ξ η r * V j , . . . , η r * V k ), and hence η * t τ t (α 1 , . . . , α l , V 1 , . . . , V k ) = τ 0 (V 1 , . . . , V k ) + t 0 η * r γ r (η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , η r * V k ) + η * r (£ ξ τ r )(η r * α 1 , . . . , η r * α l , η r * V 1 , . . . , η r * V k ) dZ r = τ 0 (α 1 , . . . , α l , V 1 , . . . , V k ) + t 0 (η * r γ r )(α 1 , . . . , α l , V 1 , . . . , V k ) + (η * r £ ξ τ r )(α 1 , . . . , α l , V 1 , . . . , V k )dZ r .
Since α 1 , . . . , α l and V 1 , . . . , V k were arbitrary, the result follows.
Proof. Let µ ∈ Dens C ∞ be an arbitrary non-vanishing density and set ρ = dD dµ ∈ C ∞ so that D = ρµ. Recall that for all w ∈ X C ∞ , £ w D = (£ w ρ + div µ w)µ. It follows that for all t ∈ [0, T ],
Using the Lemma A.13 and the identity £ w
For all m = α ⊗ ν ∈ X ∨ C ∞ and w ∈ X C ∞ , we have
Therefore,
Applying the Lemma A.13 and the identity m
We then complete the proof by applying Corollary 3.5 with α = m/D.
Proof. If (u, g, λ) ∈ HP Z is a critical point of the action functional, then 0 = d dǫ ǫ=0 S HP Z a 0 (u ǫ , η ǫ , λ ǫ ) = I(δu) + II(δw) + III(δλ),
where We now focus on II(δw) = 0. By the equality of mixed derivatives (see, also, Lemma 3.1 in [ACC14] ), we have
Using the above relation and that ψ 0 t X = X, we find ∂ψ ǫ ∂ǫ ǫ=0 = δw, and hence
Therefore,
where we have exchanged the order of d dǫ and the rough integral using Theorem A.8. Moreover,
The proof is completed by splitting the variations of δw in space and time and applying Lemma B.4.
Proof. Using the definitions of δu and δa in (3.8), integrating by parts, and taking the endpoint conditions w 0 = w T = 0 into account, we find
We conclude with the corresponding momentum equation by splitting up variations in space and time and applying Lemma B.4. In addition, the advection equation da t + £ dx t a t = 0 follows from the push-forward relation a t = (η t ) * a 0 by the Lie chain rule in Theorem 3.3. These two results complete the proof of Theorem 3.14.
In this section, we will provide an overview of the theory of geometric rough paths. We invite the reader to consult Appendix D for a historical account motivating the use of rough paths and [LCL07, FV10b, FH14, Bai14] for more thorough expositions. Let T > 0, ∆ 2 T = {(s, t) ∈ [0, T ] 2 : s ≤ t} and ∆ 3 T = {(s, θ, t) ∈ [0, T ] 3 : s ≤ θ ≤ t}. Let E denote an arbitrary Fréchet space E with family of seminorms P. Elements of family of seminorms P will be denoted by p. For a given α ∈ (0, 1], let C α T (E) denote the space of Hölder continuous paths; in particular, C 1 T (E) is the space of Lipschitz paths. Moreover, for a given m ∈ {2, 3} 6 and α ∈ R + , denote by C α m,T (E) the space of functions that satisfy
Define δ :
For a given Ξ ∈ C α 2,T (E), β ∈ R + , and p ∈ P, the quantity [δ 2 Ξ] β,p , defined above, may be regarded as a measure of the extent to which Ξ is an increment δ f for some f ∈ C α T (E). The following lemma, proved in [HH18] [Proposition A.1], is referred to as the sewing lemma. The lemma says that if β > 1, one can construct a "unique" f ∈ C α T (E) such that Ξ is close to δ f in C 
Remark A.2. Notice that ifΞ ∈ C α,β 2,T (E) and Ξ −Ξ ∈ C β 2,T (E), then I(Ξ) = I(Ξ). For a given Fréchet space E and K ∈ N, let E K denote the direct sum of E with itself K-times. By virtue of the Sewing Lemma, one can construct an integral of Y ∈ C β
This integral construction coincides with the integral that L.C. Young [You36] constructed. In particular, for Z ∈ C α T (R K ) with α ∈ 1 2 , ∞ , we may define Z ∈ C 2 2,T (R K×K ) by
where we have used the δ notation defined above in the second equality. One can easily verify that (Z,
and
The condition (A.3) is a geometric property which encodes the usual chain and product rules, upon which our variational theory is based. Paths Z ∈ C α T (R K ) with α ∈ ( 1 2 , 1] are referred to as Young paths. Young paths are distinguished from rough paths Z = (Z, Z) ∈ C α T (R K ) × C α T (R K×K ), α ∈ ( 1 3 , 1 2 ], which are defined to be paths such that an a priori postulated two-parameter path Z ∈ C 2α T (R K×K ) satisfies (A.2). A subclass of rough paths are the geometric rough paths, for which a classical calculus can be developed. In particular, (A.3) holds. For a few more words of motivation about rough paths see Appendix D.
Definition A.3. For a given α ∈ 1 3 , 1 2 , define the set C α g,T (R K ) of geometric K-dimensional α-Hölder rough paths on the interval [0, T ] to be the closure of
It follows that both (A.2) and (A.3) hold for all Z = (Z, Z) ∈ C α g,T (R K ) by a limiting argument. For a given α ∈ 1 2 , 1 , we denote C α g,
Remark A.4. To have a uniform notation for all α ∈ 1 3 , 1 , we write
It is possible to consider infinite-dimensional geometric rough paths, but for simplicity we restrict ourselves to finite-dimensional paths. However, we consider controlled rough paths (defined in the next section) in Fréchet spaces. We also remark that our theory can be extended to more irregular paths α < 1 3 , but this requires higher-order iterated integrals and more cumbersome notation.
There are a large class of Gaussian processes that belong to C α g,T (R K ) for α ∈ 1 3 , 1 2 . We refer the reader to Appendix E for a slightly more in-depth discussion of Gaussian rough paths. The present discussion will be brief.
: Ω × [0, T ] → R K on a complete probability space (Ω, F , P) and let B be the stochastic iterated integral constructed from Stratonovich integration theory. By virtue of the Kolmogorov continuity theorem, one can find a eventΩ ∈ F with P(Ω) = 1 such that B(ω) = (B(ω), B(ω) ∈ C α g,T (R K ) for any α ∈ 1 3 , 1 2 . This the Stratonovich lift of Brownian motion. Indeed, the Stratonovich integral is a limit of integrals of piecewise-linear approximations of Brownian motion.
Example A.6 (Gaussian rough paths). More broadly, a Gaussian process Z : Ω × [0, T ] → R K can be lifted to a geometric rough path Z(ω) = (Z(ω), Z(ω) ∈ C α g,T (R K ) for α ∈ 1 3 ,α , provided the correlation in time of the process is fast enough depending onα (see Appendix E). In particular, fractional Brownian motion B H : Ω × [0, T ] → R K can be lifted to a strong geometric rough path B H (ω) = (B H (ω), B H (ω) ∈ C α g,T (R K ) for all α ∈ 1 3 , 1 4H for all ω in some event of probability one.
Let us first describe the integration theory for paths Z = (Z, Z) ∈ C α g,T (R K ) such that α ∈ 1 3 , 1 2 .
satisfies R Y ∈ C 2α 2,T (E). For α ∈ 1 3 , 1 2 , we define the linear space D Z,T (E) of controlled rough paths to be
The function Y ′ is referred to as the Gubinelli derivative [Gub04] . The space D Z,T (E) is a Fréchet space with seminorms
We note that any Y ∈ C 2α T (E) satisfies (A.4) with Y ′ ≡ 0. Moreover, Z itself is controlled with Z ′ ≡ id. However, the additional structure provided by Y ′ is natural in the context of rough differential equations (see Remark A.17). It is worth mentioning that Y ′ is not uniquely specified unless the path is truly rough (see Definition A.14 and Theorem A.15 below). The integration of controlled rough paths is an immediate consequence of Lemma A.1 Theorem A.8. Let Z = (Z, Z) ∈ C α g,T (R K ) for a given α ∈ 1 3 , 1 2 . There exists a linear continuous map
and I is as in Lemma A.1. We write
Indeed, for all (s, θ, t) ∈ ∆ 3 T ,
which implies in Ξ ∈ C 3α 2,T (G), so that we may apply Lemma A.1. Notice that if Y ∈ C 2α T (E) and Y ′ ≡ 0, then (A.6) agrees with the Young integral. This definition is used in the Clebsch variational principle in order to define the integral of the Lagrange multiplier against an advected quantity (see Remark 2.7).
For Young paths, the extra structure provided by the Gubinelli derivatives is not needed.
Definition A.10 (Controlled paths in the Young case). We define D Z,T (E) = C α T (E) if α ∈ 1 2 , 1 .
Remark A.11. To have a uniform notation for all α ∈ 1 3 , 1 , we write Y = Y ∈ D Z,T (E) if α ∈ 1 2 , 1 . We also remark that obviously the controlled space does not depend on Z in this case.
Let E and F be Fréchet spaces and C(E; F) denote the space of continuous maps. Let C 1 b (E; F) denote the space of bounded functions Φ : E → F such that the limit
exists for all e, h ∈ E and DΦ : E × E → F is continuous and bounded. We define C m b (E; F) for m ≥ 2 analogously (see [Ham82] [Def. 3.1.1 & Section I.3.6]). Let N α = 0 if α = 1, N α = 1 if α ∈ 1 2 , 1 and N α = 2 if α ∈ 1 3 , 1 2 . The following lemma says that controlled rough paths are stable under composition and products. Their proof can be found in Lemma 7.3 and Corollary 7.4 of [FH14] .
Lemma A.12.
In order to construct the integration theory given above, we have actually not needed the geometric nature of the path (i.e., (A.3) ). However, to obtain an extension of the ordinary chain and product rule, we require (A.3) to hold (see [FH14] [Section 7.5]).
Lemma A.13.
(ii) For a given,
In Section 3.3, we need the decomposition of Y in (A.7) to be unique. The decomposition is unique if the rough path Z is truly rough (Theorem 6.5 of [FH14] ). Examples of truly rough paths include fractional Brownian motion B H with H ∈ 1 3 , 1 2 .
Definition A.14 (Truly rough path). Let α ∈ 1 3 , 1 2 and Z ∈ C α g,T (R K ). We say Z is truly rough if for all s in a dense set in [0, T ], lim sup t↓s |δZ st | |t − s| 2α = ∞.
Theorem A.15. If Z is truly rough and Y satisfies both (A.7) and (A.7) with β and σ replaced withβ ∈ C T (E) and (σ,σ ′ ) ∈ D Z,T (E), then β ≡β and (σ, σ ′ ) ≡ (σ,σ ′ ).
We will now introduce the definition of solution to an RDE. Let V denote a Banach space.
Remark A.17. The rough integral in (A.9) is well-defined by virtue of Lemma A.12.
The following lemma concerns equivalent notions of solutions. Its proof is a direct application of Theorem A.8 and A.13. The first formulation is referred to as the Davie's formulation [Dav08] and the second naturally extends to the manifold setting.
. The proof of existence and uniqueness for RDEs uses Picard iteration in controlled rough path topology (i.e., A.5). We refer the reader to, e.g., [FH14] [Section 8.5] for a proof. Moreover, in Section B.1 we given more details about flows on Euclidean spaces.
Theorem A.19. There exists a unique continuous solution map Denote Ω k C ∞ = Γ C ∞ (Λ k T * M) as the space of smooth alternating k-forms on M. We let Dens C ∞ := Ω d C ∞ . It is worth remarking that nonorientability and tensor-densities can be easily accommodated by introducing weighted densities (see, e.g., [Spi70, AMR12, vdBC17] ). However, we avoid this extension for brevity in the presentation here.
Denote the wedge and tensor product by ∧ and ⊗, respectively. Let d : Ω k C ∞ → Ω k+1 C ∞ denote the exterior derivative operator and i u : Ω k C ∞ → Ω k−1 C ∞ denote the interior product operator for an arbitrarily given u ∈ X C ∞ . For given F ∈ Diff C ∞ and τ ∈ T lk C ∞ , the push-forward and pull-back are defined by
is the tangent map of F, which extends to an isomorphism (on fibers) (T F) * ∈ C ∞ (T lk T M; T lk T M). For a given time-dependent vector field u ∈ C ∞ (R × M; T M), let η ∈ C ∞ (R 2 ; Diff C ∞ ) denote the twoparameter flow of of diffeomorphisms generated by u; 7 that is, for all (s, t) ∈ R 2 , η tθ • η θs = η ts , and η ·s X is the unique integral curveη ts X = u t (η ts X), η ss X = X ∈ M. 7 The time-dependent vector field u may be associated with a time-independent vector fieldū ∈ C ∞ (R × M, T R × T M) on the manifold T R × T M viaū t (p) = {u t (p), 1 t } ∈ T t R × T p M for all (t, p) ∈ R × M. Thus, the two-parameter flow may be defined in terms of the one-parameter flow ofū by η t−s (x, s) = {η ts , t}. It follows from η st • η ts = id that for all s ∈ R
Equivalently, for all f ∈ C ∞ , h · = η ·s * f = (η s· ) * f ∈ C m solves the PDE ∂ t h t + £ ut h t = 0.
For given u ∈ C ∞ (R×; T M) and t ∈ R, the Lie derivative £ u t : T lk C ∞ → T lk C ∞ is defined by
If u is independent of time, we define £ u τ = d dt t=0 η t τ, where η t = η t0 is the corresponding one-parameter flow map. It is well-known that the Lie derivative (see, e.g., [HSS09, AMR12] ) is the unique operator on the tensor algebra ⊕ l,k T lk C ∞ that i) commutes with tensor contractions, ii) is natural with respect to restrictions, and iii) satisfies for a given local chart (U, φ) and all f ∈ C ∞ | U , u, v ∈ X C ∞ | U ,
where, again, d denotes the exterior derivative and i u denotes the interior product operator. It follows that for
Thus, for given u ∈ C ∞ (R × M, T M), the Lie derivative £ u t is a first-order differential operator on the bundles Λ k T * M and T lk T M. For non-vanishing µ ∈ Ω d C ∞ , the operator div µ : X C ∞ → C ∞ is defined by the relation
Moreover, for all u ∈ X C ∞ and α ∈ Ω k C ∞ , we have
which is referred to as Cartan's formula.
For a given vector bundle E, denote by E * the dual bundle and by E ∨ = E * ⊗ T * M the functional dual bundle. Noting that ·, · E : Γ C ∞ (E ∨ ) × Γ C ∞ (E) → Dens C ∞ , we define the canonical pairing ·, · Γ(E) :
The quantity s ′ , s E ∈ Dens C ∞ in the integrand is a volume form and it is being integrated over the manifold M. The distributional sections of E and E ∨ are defined by
respectively. The canonical pairing (A.13) induces the following dense embeddings:
The pairing and definitions of distributions are canonical in the sense that no metric or volume form are needed to define them. We extend the pairing to ·, · Γ(E) to Γ D ′ (E ∨ ) × Γ C ∞ (E) and Γ C ∞ (E ∨ ) × Γ D ′ (E) in the usual way. The adjoint of a linear differential operator L :
14)
It follows that L * restricts to L * : Γ D ′ (C ∞ ) → Γ C ∞ (E ∨ ), and we write L = L * * :
For a normal, local, and invariant Fréchet, Banach, or Hilbert function space F on R d , 8 we define a Fréchet, Banach, or Hilbert, respectively, of sections Γ F (E) via a cover of total trivializations. 9 It follows that
where the embedding Γ C ∞ (E) ֒→ Γ F (E) is dense. We refer the reader to [vdBC17] [Ch. 3] for more details. Exactly the same construction applies to obtain a function space Γ F (E ∨ ):
In the present work, we assume that all function spaces F are normal, local, and invariant. In particular, we let
. We also set Dens ∨ F = F . Any strong bundle pseudo-metric denoted (·, ·) E :
for an arbitrarily given function space F . Moreover, a non-vanishing volume form µ ∈ Dens C ∞ induces an isomorphism id ⊗µ :
To describe the inverse, note that for all densities ν ∈ Dens C ∞ , there exists dν dµ ∈ C ∞ such that ν = dν dµ µ. The inverse is then given by
Upon composing these isomorphisms, we obtain an isomorphism ♭ ⊗ µ :
. In particular, we may define a pairing (·, ·) Γ L 2 (E) :
If (·, ·) E is a metric, we obtain a Hilbert structure on Γ L 2 (E), the space of square-integrable equivalence classes of measurable sections. Furthermore, for every s ∈ R, there exists an order s elliptic operator A satisfying A :
where W s 2 = (I − ∆) − s 2 L 2 denotes the Besselpotential spaces, which provides a Hilbert structure to Γ W s 2 (E) [Hin09, Mel08] . Moreover, if L :
is the adjoint of L relative to the pairing (·, ·) Γ L 2 (E) . 8 Let F denote a locally convex topological vector space of functions f :
, and invariant if any smooth diffeomorphism χ ∈ Diff C ∞ induces an topological isomorphism on F via push-forward. 9 A total trivialization is a triple (U, φ, ψ) such that (U, φ) is a local chart of M and ψ : E U → U × R rank(E) is a trivialization of E over U. Any such trivialization induces an isomorphism h φ,ψ : Γ D ′ (E| U ) → Γ D ′ (φ(U)). A section s ∈ Γ D ′ (E) belongs to Γ F (E) if for every total trivialization (U, φ, ψ), h φ,ψ (s| U ) ∈ F (φ(U)) rank(E) .
Any Riemannian metric g on M gives rise to a volume form µ g defined in a local coordinate chart (U, φ) by
Furthermore, the metric g extends to bundle metrics on T lk T M and Λ k T * M in the usual way. In particular, we obtain the diffeomorphisms ♭, id ⊗µ g and ♭ ⊗ µ g discussed in the previous section. In particular, for every k ∈ {0, 1, . . . , d}, the metric and orientation gives rise to the inner product on Ω k L 2 (a Hilbert space) defined by
where we have used the Hodge-star diffeomorphism ⋆ :
The adjoint of d :
The Hodge decomposition plays an essential role in incompressible fluids on manifolds. We now briefly describe the decomposition and the canonical pairing we use in the incompressible case.
Let
denote the Hodge Laplacian, which is formally self-adjoint and non-negative with respect to the inner product d k=0 (·, ·) Ω k . Let
denote the finite-dimensional space of harmonic-k forms, which are constant and belong to Ω k C ∞ . Let F denote either the smooth F = C ∞ , the Bessel-potential F = W s p , s ≥ 0, p ∈ [1, ∞), or the Hölder functions F = C m,α , m ≥ 0, α ∈ (0, 1). The Hodge decomposition of Ω k F is given by War13, MJ09, Sco95, MR04] , and F +1 and F +2 are the one and two-more regular spaces (in the non-smooth case). That is, F +1 = W s+1 p and F +2 = W s+2 p , and similarly for Hölder spaces. Letting k = 1 in A.16, applying the diffeomorphism ♯ : Ω 1 F → X F , and defining
we obtain an extension of the Helmholtz decomposition of (possibly non-smooth) vector fields to manifolds:
which is an orthogonal decomposition with respect to the inner product (·, ·) X L 2 :
Using i u µ g = ⋆u ♭ and Cartan's formula, we find div µ g u = −d * u ♭ = 0 for all u ∈ X F ,µ g . Thus, X F ,µ g consists of divergence-free vector fields andẊ F ,µ g consists of harmonic-free and divergence-free vector fields. Let us recall the canonical pairing (A.13) ·, · X : X ∨ C ∞ × X C ∞ → R:
, u X = (v, u) X L 2 for all u, v ∈ X C ∞ . Applying the diffeomorphism ♭ ⊗ µ g to (A.17), we get
Define the 'projection' operators P :
, which act only on the one-form component. Clearly, if we restrict the canonical pairing ·, · X to X ∨ C ∞ × X C ∞ ,µ g and X ∨ C ∞ ×Ẋ C ∞ ,µ g , then the pairing is degenerate; indeed,
Notice that the kernel of P is (id ⊗µ g )dC ∞ and the kernel ofṖ is (id ⊗µ g )H 1 ∆ ⊕ (id ⊗µ g )dC ∞ . To restore non-degeneracy, we mod out by the kernel; the following definition is standard [AK99, KW08, KMM20] .
Moreover, we define the canonical pairings ·, · X µg :
where the [α ⊗ µ] denotes an equivalence class with representative α ⊗ µ. It follows that ♭ ⊗ µ g :
It can easily be checked the definition is well-defined in the sense that the right-hand-sides of A.18 are independent of the representative. Indeed, for any two given representatives α ⊗ µ and β ⊗ ν of an equivalence class of X ∨ F ,µ g , we have
and for any two given representatives α ⊗ µ and β ⊗ ν of an equivalence class ofẊ ∨ 
such that the flow η ts = Flow(u, ξ, Z) ts , (s, t) ∈ [0, T ] 2 satisfies the following properties:
(i) for all (s, θ, t) ∈ [0, T ] 3 , η tt = Id and η tθ • η θs = η ts ;
(iii) η is the unique two-parameter flow satisfying (i) and
for a constant C, where µ ∈ C α 2,T (Diff C ∞ (R d )) is the C ∞ -approximate flow given by
or equivalently by µ ts (X) := Y 1 such thaṫ
Remark B.2. Claims (i-iii) are a direct extension of Corollary 11.14 of [FV10b] ; one can easily verify the Davie's estimates (Corollary 11.14 of [FV10b] ). We do not impose that our drift coefficient is Lipschitz in time because it will be the solution of a rough partial differential equation driven by the path Z, and hence it can only be expected to be α-Hölder continuous. We also allow for time dependence in the vector field ξ since this is used in Section 3.3 to take variations. Claim (iv) is a minor extension of Theorem 16 of [Bai14, BR19] (or Theorem 1.27 of [Dri18] ), which uses the method of approximate flows. We remark that it is certainly possible to weaken the required regularity assumption in space and time of the coefficients. Claim (iv) is the initial-value equation for the inverse flow; it is a transport rough partial differential equation (RPDE) . We understand g to be classical solution in the spatial variable and in the sense of controlled rough paths in time. In [CF09] [Corollary 8], a method of characteristics solution theory for forward RPDEs (in the case u ≡ 0) is established. However, the solutions are only characterized as being a limit point of g n = f (X n t ), where X n is the solution of the time-reversal along a sequence of smooth paths Z n = (Z n , Z n ) converging to Z in the rough path topology. It is not immediately clear that one can deduce a stronger notion of solution (in the sense of controlled rough paths) from this result in a simple manner (see, also, Remark 2.10 of [DFS17] ). The paper [DFS17] and its extension [BDFT20] , also, do not seem to directly state this result.
Nevertheless, the solution of the RPDE can be derived using theory of unbounded rough drivers ( [BG17, DGHT19] ), which is analogous to the energy method in deterministic PDE. Indeed, one may first derive a solution g ∈ C([s, T ]; W n 2 (R d )) under the assumption u ∈ C T (C m (R d ; R d )), ξ ∈ C m+3 (R d ; R d ), and f ∈ W m 2 (R d ) for any m ∈ N 0 by adapting Theorem 2 of [HH18] and Section 5.2 of [CN19] . Then one may obtain a solution u ∈ C α T (C ∞ (R d ; R)) by applying the Sobolev embedding. Finally, one can apply the pull-back version of the Lie chain rule Theorem 3.3 to show that g(η ·s ) = f .
Let T > 0, α ∈ 1 a Banach space. Denote by L 1 (X; W) the Banach space of equivalent classes of Bochner integrable functions f : X → W endowed with the norm
Recall that for an arbitrary Banach space V and linear map L ∈ L(W, V),
The following lemma is then a straightforward application of (B.1), Theorem A.8, and
which itself follows from Fatou's lemma.
Let T > 0, α ∈ 1 3 , 1 , and Z ∈ C α g,T (R K ).
Remark B.5. On the right-hand-side of (B.2), we have used that (φ, 0) ∈ D Z,T (R), and thus that φY = (φY, φY ′ ) ∈ D Z,T (R K ) by Lemma A.12.
Proof.
Step 1. We will begin by showing that equality (B.2) must hold for any Lipschitz φ ∈ C 1 T (R) such that φ 0 = φ T = 0, whereφ on the left hand side is the bounded weak derivative (which exists by Rademacher's theorem [Hei12] [Theorem 6.15]). Consider a mollifier on R defined by ρ n (θ) := nρ(nθ), n ∈ N, where R ρ(θ)dθ = 1 and supp ρ ⊂ [0, T ]. Because φ vanishes at the end points, we can extend φ by zerõ
and note thatφ ∈ C 1 T (R) has the same Lipschitz constant as φ. For a given n ∈ N, define
which is clearly bounded in C T (R). For all n ∈ N and s, t ∈ [0, T ], we find
By Arzela-Ascoli's theorem, φ n → φ uniformly, and, in fact, in C β T (R) for all β < 1. A classical argument shows that
For fixed Y ∈ D Z,T (R K ), the mapping ψ → ψY := (ψY, ψY ′ ) is a linear and continuous operation from C β T (R) to D Z,T (R K ) for all β ≥ 2α. Moreover,
Thus, by the continuity of the rough path integral (Theorem A.8), we obtain
which completes step 1.
Step 2. We will now construct a sequence of Lipschitz functions {φ n } n∈N ⊂ C 1 T (R) converging to the characteristic function 1 [s,t] , for s, t ∈ R such that 0 < s < t < T , and then pass to the limit on both sides of (B.2) to obtain (B.3). We then extend the equality to (s, t) ∈ ∆ 2 T by continuity. Towards this end, for large enough n ∈ N and r ∈ [0, T ], define
otherwise, so that |φ n | ∞ = 1 and |φ n | ∞ = n whereφ n is the weak derivative defined bẏ
otherwise.
A classical argument shows that
Since the rough integral is an increment, we have
If we can show that the first and last integrals converge to zero as n → ∞, then we are finished. We will only show that the last term converges to zero because the argument for the first integral is easier. Let C denote a constant that is independent of n and may vary from line to line. By Theorem A.8 and the fact that |φ n Y| Z ≤ Cn by (B.4), we find
as n → ∞. Moreover,
as n → ∞, which completes the proof.
The purpose of this section is to explain the variational principles we formulate in this paper in the simplified setting of an incompressible homogeneous ideal fluid evolving on the the torus with a smooth perturbation. The beginning of the section can be read with no knowledge of differential geometry. The rest of the section assumes some basic knowledge of differential geometry (see Section A.2). We also explain the presence of the so-called line-element stretching term in our main equation. The presence of this term distinguishes our equations from a pure transport perturbation of the deterministic Euler equation on flat space in velocity form. In particular, we show that the stretching term arises as a direct consequence of the variational principle and not by how momentum is characterized; that is to say, our variational principle indirectly enforces a covariant formulation, which naturally leads to a Kelvin's circulation theorem, helicity conservation in dimension three, and enstrophy conservation in dimension two.
By appealing to the Helmholtz decomposition (Hodge decomposition), we explicitly show the decomposition of the pressure terms into the unperturbed and perturbed part, which motivates the corresponding decomposition in the rough case. As a result of the presence of the stretching term, our equations do not preserve mean-freeness (i.e., harmonic-freeness) unless we impose an additional constraint in the variational principle. By imposing this constraint, the velocity u can be recovered directly from the vorticityω = ∇×u via the Biot-Savart law. In vorticity form, our equations are a pure transport perturbation of the deterministic Euler equation in dimension two. The associated vorticity equation in the Brownian setting has been studied in the literature with u recovered directly from the vorticity ω via Biot-Savart [BFM16, CFH19, CL19b, BM19].
We consider an incompressible homogeneous fluid moving on the flat d-dimensional torus T d with the standard volume form dV. Denote by X the space of smooth vector fields, X dV the space of smooth divergencefree vector fields andẊ dV the space of smooth divergence and mean-free vector fields. It follows that
where the decomposition is orthogonal with respect to the L 2 -inner product. Let P : X → X dV , Q : X → ∇C ∞ , P : X →Ẋ dV , and H : X → R d denote the corresponding projections (see Section A. 2.3 and (A.17) ). We recall that in dimension three, curl :Ẋ dV →Ẋ dV is an isomorphism, and in dimension two, curl :Ẋ dV → C ∞ is an isomorphism. Denote the inverse of curl by BS (for Biot-Savart).
We assume that the Eulerian velocity field v : [0, T ] → X dV of the fluid admits a decomposition into a sum of a dynamical velocity variable u : [0, T ] →Ẋ dV and a known model vector field ζ :
where the vector field ξ admits the specified decomposition
where ξ ∈Ẋ K dV and Z : [0, T ] → R K in this appendix is a differentiable path, as opposed to the rough paths in the main text.
Using integration by parts in space and time, we get that (u, a, λ) is a critical point of S if and only if
It follows that
Here ((∇v) T · u) i := δ i j u k ∂ x j v k , and we have used the δ i j in order to maintain the geometric index convention even though we are working on flat space. Therefore,
In terms of the projections Q and H, we find
We note that the pressure p is enables us to enforce the constraint that u is incompressible and the the constant (in space) c enables us to enforce that u is mean-free. Substituting in v = u + ξŻ, we find
In dimension two and three, one can readily check an equivalent formulation in terms of the vorticityω = Curl u:
From this point on, we assume the reader is familiar with basic differential geometry (see Section A.2. Let us introduce an arbitrary coordinate system and denote by {dx i } d i=1 a global frame of Ω 1 . Moreover, let the musical notation ♭ : X → Ω 1 denote the isomorphism between vector fields and one-forms. Equation C.5 can be expressed covariantly as
where the Lie-derivative operator £ v t acts on the one-form u ♭ to produce the one-form £ v t u ♭ , given by
Here dp is exterior derivative of the scalar-fieldp.
Let ω = du ♭ ∈ Ω 2 denote the vorticity two-form obtained by applying the exterior derivative operator d. Since the exterior derivative commutes with the Lie derivative, one finds
The two characterizations of the vorticity ω andω satisfying, (C.6) and (C.8), respectively, are related by the Hodge-star operator ⋆ : Ω 2 → Ω d−2 . In dimension two,ω = ⋆ω ∈ Ω 0 , and in dimension three, ω = ♯ ⋆ ω ∈Ẋ dV . In order to obtain equation (C.6) directly from (C.8), one uses that ♯⋆ and the Lie derivative commute (see, e.g., [BF17] [Section A.6]).
Kelvin circulation theorem. The covariant formulation immediately implies a Kelvin circulation theorem. Let γ denote a closed loop in T d . Then using Reynolds transport theorem,
where one transforms the integration around the moving loop η t (γ) to the loop γ in the material frame by applying the pull back η * t to the integrand, then takes the time derivative, applies the dynamic definition of the Lie-derivative, transforms back and substitutes the covariant equation of fluid motion (C.7).
Helicity conservation. In three dimensions, the helicity, defined as
measures the linkage of field lines of the divergence-free vector fieldω [AK99] . Owing to (C.7) and (C.8), we have
Thus, the linkage number of the vorticity vector field Λ(ω) is preserved by the 3D Euler fluid equations (C.7).
Enstrophy conservation in two-dimensions. In two dimensions, for any f ∈ C ∞ , we find
and hence
In particular, taking f (x) = x 2 , we find
which implies that in two-dimensions enstrophy is conserved.
Momentum representation. The Lie derivative of the volume form dV along v is zero since £ v dV = (div v) dV = 0. Thus, we can also write equation (C.5) as
where m t = u ♭ ⊗ dV ∈ X ∨ := Ω 1 ⊗ Dens denotes the space of smooth one-form densities. In Sections 2, 3.3, and 3.4, the momentum will be characterized as a one-form density in order to conveniently incorporate both the inhomogeneous and compressible setting and work canonically. One can always transform between equivalent formulations once a metric and volume form have been fixed. We will now explain how one can directly derive the various equivalent formulations directly from the Clebsch action functional.
Clebsch constrained variational principle revisited. Let us now explain how we can directly derive (C.7) and (C.9) from the Clebsch action functional. The first term on the RHS of (C.3) can be understood in a coordinate-free manner either as:
(i) (u + λ∇a, δu) X L 2 = T d g(u + λ∇a, δu)dV, where (·, ·) X L 2 : X × X → R;
(ii) u ♭ + λda, δu = T d i δu (u ♭ + λda)dV, where ·, · : Ω 1 × X → R,
It follows that
A critical point of the Clebsch action S in (C.2) then satisfieṡ
where we use the same notationṖ for the corresponding projection onto 'divergence and harmonic-free' parts (see Section A.2.3) in all three cases. In all three cases, following a similar calculation to the one given in (C.4), we obtain ∂ t m t +Ṗ£ v t m t = 0.
The first case (i) agrees with the direct calculus computation given above. In general, the main ingredients of this computation (see Section 5.1) are 1) the definition of ⋄, 2) the relation for all v, w ∈ X and a ∈ Ω 0 (i.e., for all tensor fields, a) that
and 3) that m, ad v w = £ v m, w , for all of the above pairings. That is, ad * v m = £ v m. If v is not divergence-free, then ad * v m = £ v m is only true for the pairing ·, · X .
Thus, one may characterize the 'momentum' m in various ways if a metric and volume form are fixed. However, the pairing ·, · X is canonical in that it does not require a metric or volume form to be fixed (see the discussion in Section A.2.2), and we use this pairing above.
As a consequence of this discussion, we see that the line-element stretching term (∇v t ) T · u t in equation (C.5) does not arise because we have characterized momentum in a certain way. This term appears even if we treat m as a vector and work in a fixed standard coordinate system. As derived here, the stretching term tells us that the Clebsch variational principle has produced covariant coordinate-free equations. This is simply the generalized-coordinate theorem for the covariance of variational principles, the first being the Euler-Lagrange equations in classical mechanics, which are valued for precisely this reason.
and invoking | f ♯ st (Y s )(ω)| ≤ C(ω)|t − s| 3α and δ 2 B lk sθt (ω) = δB l sθ (ω)δB k θt (ω), one can directly check that Ξ ∈ C α,3α 2,T (R). Hence, one may apply Lemma A.1 to construct the integral IΞ = ( ξ[ f ]dB)(ω). This integral agrees with the Stratonovich integral t s ξ[ f ](Y s ) • dB s on a set of probability one (see Theorem A.8). The expansion D.3 is called the second-order Chen-Fleiss expansion in the system-identification and control literature. Here, B can be interpreted as a control. Such expansions illustrate that all information of the controls impact on the system is contained in the iterated integrals of the control. The Chen-Fleiss expansion can be obtained directly from (D.2) by formally iterating the integral (Taylor series) with Z = B and and then evaluating at ω. One immediately recognizes the advantage of the Magnus expansion over the Chen-Fleiss series. Namely, the Magnus expansion is an exact solution of an approximating system, while the Chen-Fleiss series is not [Kaw04] . Nevertheless, such expansions are of great utility in the study of controllability and analysis of control systems [Sus87] .
From the above discussion, we have learned that for all ω ∈ Ω ′ , (B(ω), B(ω)) belongs to the class of (Z, Z) ∈ C α T (R K ) × C 2α 2,T (R K×K ) such that for all (s, t) ∈ ∆ T δ 2 Z lk sθt = δZ l sθ δZ k θt , Z lk st + Z kl st = δZ l st δZ k st .
For α ∈ 1 3 , 1 2 , the closure of the set of Lipschitz paths in C α T (R K ) × C 2α 2,T (R K×K ) that satisfy the above properties is called the space of geometric rough paths.
The fundamental idea of T. Lyons is that in the general case of ξ not having vanishing Lie brackets, there is a notion of solution of equations driven by geometric rough paths Z = (Z, Z) and an accompanying well-posedness theory. There are many equivalent notions of solution (see Lemma A.18). For example, the Chen-Fleiss expansion up to level two can be used to define an intrinsic notion of solution by additionally specifying that the remainder f ♯ belongs C 3α 2,T (R) for any f ∈ C ∞ (M) [Dav08] . Higher-order iterated integrals are needed if α < 1 3 . However, one still needs a means of constructing Z, and probability is the main tool used to do so. Effectively, then, the technical ingredient necessary to develop the basic theory of rough paths is the sewing lemma (Lemma A.1) [Gub04] . To wit, the sewing lemma is used to establish the existence of integrals against Z and to obtain bounds on 'remainder' f ♯ st . It is also possible to prove that there exists a unique two-parameter flow associated with the time-one map
even if the third-order Lie-brackets of ξ do not vanish. The main ingredient in this approach is the multiplicative sewing lemma, developed by I. Bailleul [Bai14] . A prophethetical quote of M. Fleiss [Fli81] [pg. 33] translated into English reads,
We know (cf. Schwartz [Sch54] ) that it is generally impossible to multiply the distributions and, in particular, that the powers δ 2 , δ 3 , · · · , of the Dirac impulse are not distributions. Similarly here, we cannot represent the square of a Dirac impulse by a series of Chen. However, it is possible to propose what is called in physics a renormalization (et. Güttinger [Güt55] ) based on natural combinatorial considerations.
T. Lyons showed that by postulating the existence of objects Z which satisfy (D.4), a solution theory can be developed for differential equations driven by rough paths. As explained above, probability is used to construct Z. Thus, probability can be understood as a tool to renormalize through its construction of otherwise analytically ill-defined quantities Z -and it is only this quantity that needs to be defined to construct a solution. M. Hairer extended the T. Lyons program by developing the theory of regularity structures as the basis of a solution theory for stochastic partial differential equations driven by white noise [Hai14] (see, also, [FH14] ). One of the key theorems in M. Hairer's theory is the Reconstruction Theorem, which is a substantial generalization of the sewing lemma. As predicted by M. Fleiss [Fli81] and H. Sussman [Sus78] , this theory has had a transformative impact on renormalization in statistical physics, and of our understanding of stochastic differential equations (in finite and infinite dimensions) and the so-called Wong-Zakai anomalies.
A broad class of geometric rough paths are given by the Gaussian rough paths. Fix a complete probability space (Ω, F , P) supporting a K-dimensional Gaussian process {Z t } t≤T with independent components and zero mean. Let R k (s, t) = E[Z k s Z k t ] denote the corresponding covariance functions and
The existence of a rough path lift for X is contingent upon sufficient rate of decay of the correlation of the increments. If for a given q ∈ [1, 3 2 ), there exist a constant C > 0 such that for all k ∈ {1, . . . , K} and (s, t) ∈ ∆ T , sup 
where the supremum is taken over all finite partitions P([s, t] 2 ) of the interval [s, t] 2 , then there is a random variable Z and setΩ ∈ F for which P(Ω) = 1 and such that for all ω ∈Ω, Z(ω) = (Z(ω), Z(ω))) ∈ C α g,T (R K ) for α ∈ ( 1 3 , 1 2q ). Furthermore, the lift is canonical in the sense that for all (s, t) ∈ ∆ 2 T , lim
We note that B lk st (ω) t s δB l st 2 (ω) • dB k t 1 (ω) because stochastic integrals are defined for non-simple processes via an L 2 (Ω)-closure and there is no pathwise way (i.e., in the sense that it is robust under smooth approximations of the path) to make sense of the right-hand-side other than by simply defining via the left-hand-side.
Example E.2 (Volterra Gaussian processes). A Volterra kernel K : [0, T ] 2 → R is a square integrable function such that K(s, t) = 0 for s ≥ t. One can find conditions on the kernels K : [0, T ] 2 → R such that the corresponding Volterra Gaussian processes can be lifted to a geometric rough path. We refer the reader to [CL19a] for a more in depth discussion of Volterra Gaussian processes and even how to extend the setup to more irregular paths. Fractional Brownian motion, Riemann-Liouville, and more simply, Ornstein-Uhlenbeck processes are all examples of Volterra Gaussian rough paths.