Renormalization of stochastic continuity equations on Riemannian manifolds

We consider the initial-value problem for stochastic continuity equations of the form $$ \partial_t \rho + \text{div}_h \left[\rho \left(u(t,x) + \sum_{i=1}^N a_i(x)\circ \frac{dW^i}{dt}\right)\right] = 0, $$ defined on a smooth closed Riemanian manifold $M$ with metric $h$, where the Sobolev regular velocity field $u$ is perturbed by Gaussian noise terms $\dot{W}_i(t)$ driven by smooth spatially dependent vector fields $a_i(x)$ on $M$. Our main result is that weak ($L^2$) solutions are renormalized solutions, that is, if $\rho$ is a weak solution, then the nonlinear composition $S(\rho)$ is a weak solution as well, for any"reasonable"function $S:\mathbb{R}\to\mathbb{R}$. The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna-Lions type commutators $\mathcal{C}_\varepsilon (\rho,D)$ between (first/second order) geometric differential operators $D$ and the regularization device ($\varepsilon$ is the scaling parameter). This work, which is related to the"Euclidean"result in Punshon-Smith (2017), reveals some structural effects that noise and nonlinear domains have on the dynamics of weak solutions.


Introduction
For a number of years many researchers appended new effects and features to partial differential equations (PDEs) in fluid mechanics in order to better account for various physical phenomena. An interesting example arises when a hyperbolic PDE is posed on a curved manifold instead of a flat Euclidean domain, in which case the curvature of the domain makes nontrivial alterations to the solution dynamics [3,7,35]. Relevant applications include geophysical flows and general relativity. Another example is the rapid rise in the use of stochastic processes to extend the scope of hyperbolic PDEs (on Euclidean domains) in an attempt to achieve better understanding of turbulence. Randomness can enter the PDEs in different ways, such as through stochastic forcing or in uncertain system parameters (fluxes). Generally speaking, the mathematical literature for stochastic partial differential equations (SPDEs) on manifolds is at the moment in short supply [13,18,21,22]. In this paper we consider stochastic continuity equations with a non-regular velocity field that is perturbed by Gaussian noise terms powered by spatially dependent vector fields. In contrast to the existing literature, the main novelty is indeed that we pose these equations on a curved manifold, being specifically interested in the combined effect of noise and nonlinear domains on the dynamics of weak solutions.
Fix a d-dimensional (d ≥ 1) smooth Riemannian manifold M , endowed with a metric h. We assume M to be compact, connected, oriented, and without boundary. We are interested in the initial-value problem for the stochastic continuity equation where W 1 , . . . , W N are independent Wiener processes, a 1 , . . . , a N are smooth vector fields on M (i.e., first order differential operators on M ), the symbol • refers to the Stratonovich interpretation of stochastic integrals, u : [0, T ] × M → T M is a timedependent W 1,2 vector field on M (a rough velocity field), div h is the divergence operator linked to the manifold (M, h), and ρ = ρ(ω, t, x) is the unknown (density of a mass distribution) that is sought up to a fixed final time T > 0. The equation (1.1) is supplemented with initial data ρ(0) = ρ 0 ∈ L 2 on M .
In the deterministic case (a i ≡ 0, M = R d ), the well-posedness of weak solutions follows from the theory of renormalized solutions due to DiPerna and Lions [10]. A key step in this theory relies on showing that weak solutions are renormalized solutions, i.e., if ρ is a weak solution, then S(ρ) is a weak solution as well, for any "reasonable" nonlinear function S : R → R. The validity of this chain rule property depends on the regularity of the velocity field u. DiPerna and Lions proved it in the case that u is W 1,p -regular in the spatial variable, while Ambrosio [1] proved it for BV velocity fields. An extension of the DiPerna-Lions theory to a class of Riemannian manifolds can be found in [14] (we will return to this paper below).
The well-posedness of stochastic transport/continuity equations with "Lipschitz" coefficients (defined on Euclidean domains) is classical in the literature and has been deeply analyzed in Kunita's works [9,23]. In [2] the renormalization property is established for stochastic transport equations with irregular (BV ) velocity field u and "constant" noise coefficients (a i ≡ 1). Moreover, they proved that the renormalization property implies uniqueness without the usual L ∞ assumption on the divergence of u, thereby providing an example of the so-called "regularization by noise" phenomenon. In recent years "regularization by noise" has been a recurring theme in many papers on the analysis of stochastic transport/continuity equations, a significant part of it motivated by [17], see e.g. [6,11,15,16,28,29,30,36].
Recently [33,35] the renormalization property was established for stochastic continuity equations with spatially dependent noise coefficients, written in Itô form and defined on an Euclidean domain. In the one-dimensional case and without a "deterministic" drift term, the equations analyzed in [33] take the form where σ = σ(x) is an irregular coefficient that belongs to W 1, 2p , while ρ is an L p weak solution (p ≥ 2). The derivation of the (renormalized) equation satisfied by F (ρ), for any sufficiently smooth F : R → R, is based on regularizing (in x) the weak solution ρ by convolution with a standard mollifier sequence {J ε (x)} ε>0 , ρ ε := J ε ⋆ ρ, using the Itô (temporal) and classical (spatial) chain rules to compute F (ρ ε ), and deriving commutator estimates to control the regularization error. A key insight in [33], also needed in one of the steps in our renormalization proof for (1.1), is the identification of a "second order" commutator, which is crucial to conclude that the regularization error converges to zero, without having to assume some kind of "parabolic" regularity like σ∂ x ρ ∈ L 2 -the nature of the SPDE (1.2) is hyperbolic not parabolic, so this regularity is not available (at variance with [25]). To be a bit more precise, the "second order" commutator in [33] takes the form where ̺ ∈ L p loc (R) and σ = σ(x) ∈ W 1,q loc (R), p, q ∈ [1, ∞]. It is proved in [33] that, as ε → 0, C 2 (ε; ̺, σ) → (∂ x σ) 2 ̺ in L r loc (R) with 1 r = 1 p + 2 q . Modulo a deterministic drift term (which we do not include), the equation (1.2) can also be written in the form This particular equation is similar to the equation studied in [18], which arises in the kinetic formulation of stochastically forced hyperbolic conservation laws (on manifolds). The uniqueness proof in [18] relies on writing the equation satisfied by F (ρ) = ρ 2 . In the Euclidean setting, one is lead to control the following error term, linked to the second order differential operator in (1.3) and the "Itô correction": again without imposing a condition like σ∂ x ̺ ∈ L 2 . Nevertheless, in the kinetic formulation of conservation laws one has access to additional structural information, namely that ∂ x ρ is a bounded measure. In [18] we use this, and the observation to establish that R(ε) → 0 as ε → 0. The detailed handling of error terms like R(ε) becomes significantly more complicated on a curved manifold, cf. [18] for details. Let us return to the equation (1.1). Our main result is the renormalization property for weak L 2 solutions, roughly speaking under the assumption that u(t, ·) is a W 1,2 vector field on M , whereas a 1 , . . . , a N are smooth vector fields on M . As corollaries, we deduce the uniqueness of weak solutions and an a priori estimate, under the additional (usual) condition that div h u ∈ L 1 t L ∞ . The complete renormalization proof is long and technical, with the "Euclidean" discussion above shedding some light on one part of the argument in a simplified situation. A key technical part of the proof concerns the regularization of functions via convolution using a mollifier. In the Euclidean case mollification commutes with differential operators and the regularization error (linked to a commutator between the derivative and the convolution operator) converges as the mollification radius tends to zero. These properties are not easy to engineer if the function in question is defined on a manifold. On a Riemannian manifold there exist different approaches for smoothing functions, including (i) the use of partition of unity combined with Euclidean convolution in local charts (see e.g. [12]), (ii) the so-called "Riemannian convolution smoothing" [20] that is better at preserving geometric properties, and (iii) the heat semigroup method (see e.g. [14]). In [14], the authors employ the heat semigroup to regularize functions as well as vector fields on manifolds. As an application, they extend the DiPerna-Lions theory (deterministic equations) to a class of Riemannian manifolds. One of the results in [14] says that the DiPerna-Lions commutator converges in L 1 . It is not clear to us how to improve this to L 2 convergence, which is required by our argument to handle the regularization error coming from the second order differential operators (arising when passing from Stratonovich to Itô integrals), cf. the discussion above.
In the present work we need to regularize functions as well as tensor fields. We will make use of an approach based on "pullback, Euclidean smoothing, and then extension", in the spirit of [18]. When applied to functions our approach reduces to (i). Our regularizing procedure consists of three main steps: (I ) a localization step based on a partition of unity; (II ) transportation of tensor fields from M to R d and vice versa via pushforwards and pullbacks to produce "intrinsic" geometric objects; (III ) a convenient choice of atlas that allows us to work (locally) with the standard d-dimensional volume element dx instead of the Riemannian volume element dV h , which in local coordinates equals |h| 1 2 dx 1 · · · dx d (presumably not essential, but it dramatically simplifies some computations). Although our approach shares some similarities with the mollifier smoothing method found in Nash's celebrated work [27] on embeddings of manifolds into Euclidean spaces, there are essential differences. The most important one is that Nash regularizes tensor fields on Riemannian manifolds by embedding the manifold into an Euclidean space and then convolve the tensor field with a mollifier defined on the ambient space. Since the mollifier lives in the larger Euclidean space, we cannot easily use it as a test function in the weak formulation of (1.1) to derive a similar SPDE for ρ ε , the regularized version of the weak solution ρ.
Roughly speaking, our proof starts off from the following Itô form of (1.1) (cf. Section 3 for details): Recall that for a vector field X (locally given by X j ∂ j ), the divergence of X is given by div h X = ∂ j X j + Γ j ij X i , where Γ k ij are the Christoffel symbols associated with the Levi-Civita connection ∇ of the metric h (the Einstein summation convention over repeated indices is used throughout the paper). For a smooth function f : where ∇ 2 f is the covariant Hessian of f and ∇ X X is the covariant derivative of X in the direction X. In the Itô SPDE (1.4) we denote by Λ i (·) := div h (div h (ρa i )a i ) the formal adjoint of a i a i (·) . Later we prove that the second order differential operator Λ i (f ) may be recast into the form div 2 h (fâ i ) − div h (f ∇ ai a i ), where div 2 h (S) is defined by div h div h (S) for any symmetric (0, 2)-tensor field S. Further,â i is the symmetric (0, 2)-tensor field whose components are locally given byâ kl i = a k i a l i . We refer to an upcoming section for relevant background material in differential geometry.
Fixing a smooth partition of the unity {U κ } κ∈A subordinate to a conveniently chosen atlas A, cf. (III ) above, we utilize our regularization device to derive a rather involved equation for each piece ρ(t)U κ ε . A global SPDE for ρ ε := κ ρ(t)U κ ε is then obtained by summing up the local equations. We subsequently use the Itô and classical chain rules to arrive at an equation for F (ρ ε ), F ∈ C 2 with F, F ′ , F ′′ bounded, which contains numerous remainder terms coming from the regularization procedure, some of which can be analyzed in terms of first order commutators related to the differential operators div h (·u), div h (·∇ ai a i ) and second order commutators related to div 2 h (·â i ). In addition, we must exploit specific cancellations coming from some quadratic terms linked to the covariation of the martingale part of the equation (1.4) and the second order operators Λ i . The localization part of the regularization procedure generates a number of error terms as well, some of which are easy to control whereas others rely on the identification of specific cancellations. At long last, after sending the regularization parameter ε to zero, we arrive at the renormalized equation . The remaining part of this paper is organized as follows: In Section 2 we collect the assumptions that are imposed on the "data" of the problem, and present background material from differential geometry and stochastic analysis. The definitions of solution and the main results are stated in Section 3. Section 4 is dedicated to an informal outline of the proof of the renormalization property, while a rigorous proof is developed in Section 5. Corollaries of the main result (uniqueness and a priori estimate) are proved in Section 6. Finally, in Section A we bring together a few basic results used throughout the paper.

Background material and hypotheses
In an attempt to make this paper more self-contained and to fix relevant notation, we briefly review some basic aspects of differential geometry and stochastic analysis. Furthermore, we collect the precise assumptions imposed on the coefficients u, a i appearing in the stochastic continuity equation (1.1).
2.1. Geometric framework. We refer to [4,24] for basic definitions and facts concerning manifolds. Consider a d-dimensional smooth Riemannian manifold M , which is closed, connected, and oriented (for instance, the d-dimensional sphere). Moreover, M is endowed with a smooth (Riemannian) metric h. By this we mean that h is a positive-definite 2-covariant tensor field, which thus determines for every x ∈ M an inner product h x on T x M . Here, T x M denotes the tangent space at x, whereas T M = x∈M T x M denotes the tangent bundle. For two arbitrary vectors X 1 , X 2 ∈ T x M , we will henceforth write h x (X 1 , X 2 ) =: (X 1 , X 2 ) hx or even (X 1 , X 2 ) h if the context is clear. We set |X| h := (X, X) 1/2 h . Recall that in local coordinates x = (x i ), the partial derivatives ∂ i := ∂ ∂x i form a basis for T x M , while the differential forms dx i determine a basis for the cotangent space T * x M . Therefore, in local coordinates, h reads We will denote by (h ij ) the inverse of the matrix (h ij ). We denote by dV h the Riemannian density associated to h, which in local coordinates takes the form where |h| is the determinant of h. Integration with respect to dV h is done in the following way: if f ∈ C 0 (M ) has support contained in the domain of a single chart Φ : where (x i ) are the coordinates associated to Φ. If supp f is not contained in a single chart domain, then the integral is defined as where (α i ) i∈I is a partition of unity subordinate to some atlas A. Throughout the paper, we will assume for convenience that For p ∈ [1, ∞], we denote by L p (M ) the usual Lebesgue spaces on (M, h). Always in local coordinates, the gradient of a function f : M → R is the vector field given by the following expression A smooth k-dimensional real vector bundle is a pair of smooth manifolds E (the total space) and V (the base), together with a surjective map π : E → V (the projection), satisfying the following three conditions: (i) each set E x := π −1 (x) (called the fiber of E over x) is endowed with the structure of a real vector space; (ii) for each x ∈ V , there exists a neighborhood U of x and a diffeomorphism φ : π −1 (U ) → U × R k , called a local trivialization of E, such that π 1 • φ = π on π −1 (U ), where π 1 is the projection onto the first factor; (iii) the restriction of φ to each fiber, φ : E x → {x} × R k , is a linear isomorphism.
Given a smooth vector bundle π : E → V over a smooth manifold V , a section of E is a section of the map π, i.e., a map σ : For an arbitrary finite-dimensional real vector space H, we use T m (H), T l (H), and T m l (H) to denote the spaces of covariant m-tensors, contravariant l-tensors, and mixed tensors of type (m, l) on H, respectively. For an arbitrary smooth manifold V , we define the bundles of covariant m-tensors, contravariant l-tensors, and mixed tensors of type (m, l) on V respectively by Note the natural identifications T 1 (V ) = T * V and T 1 (V ) = T V . Let F : V →V be a diffeomorphism between two smooth manifolds V ,V . The symbols F * , F * denote the smooth bundle isomorphisms F * : T m l (V ) → T m l V and F * : T m l V → T m l (V ) satisfying F * S X 1 , . . . , X m , ω 1 , . . . , ω l = S F −1 * X 1 , . . . , F −1 * X m , F * ω 1 , . . . , F * ω l , for S ∈ T m l (V ), X i ∈ TV , ω j ∈ T * V , and F * S X 1 , . . . , X m , ω 1 , . . . , ω l = S F * X 1 , . . . , F * X m , F −1 * ω 1 , . . . , F −1 * ω l , for S ∈ T m l V , X i ∈ T V , ω j ∈ T * V (for further details see [24,Chapter 11]). The symbol ∇ refers to the Levi-Civita connection of h, namely the unique linear connection on M that is compatible with h and is symmetric. The Christoffel symbols associated to ∇ are given by In particular, the covariant derivative of a vector field X = X α ∂ α is the (1, 1)-tensor field which in local coordinates reads (∇X) α j := ∂ j X α + Γ α kj X k . The divergence of a vector field X = X j ∂ j is the function defined by div h X := ∂ j X j + Γ j kj X k . For any vector field X and f ∈ C 1 (M ), we have X(f ) = (X, grad h f ) h , which locally takes the form X j ∂ j f . We recall that for a (smooth) vector field X, the following integration by parts formula holds: recalling that M is closed (all functions are compactly supported).
Given a smooth vector field X on M , we consider the norms Given a smooth vector field X, consider the second order differential operator X(X(·)). We have Lemma 2.1 (geometric identity). For any smooth vector field X and ψ ∈ C 2 (M ), where ∇ 2 ψ denotes the covariant Hessian of ψ and ∇ X X denotes the covariant derivative of X in the direction X.
Proof. In any coordinate system, we have On the other hand, X(X(ψ)) = ∂ lm ψX l X m + X m ∂ m X l ∂ l ψ.
In the following, we will consistently write (∇ 2 ·)(a i , a i ) + (∇ ai a i )(·) instead of a i (a i (·)), thereby highlighting the presence of the Hessian.
Let us introduce the following second order differential operators associated to the vector fields We will need to write these operators in a more appropriate form. To this end, we will first make a short digression into some concepts from differential geometry. Given a smooth symmetric (0, 2)-tensor field S on M , we can compute div h S, which is the smooth vector field whose local expression is given by where, obviously, S = S ij ∂ i ⊗ ∂ j (since S is symmetric, it is irrelevant which index we contract). Because div h S is a vector field, it can operate on functions by differentiation. Moreover, we can compute its divergence. Henceforth, we set . Given any vector field X on M , we can canonically construct a symmetric (0, 2)tensor field on M in the following fashion: we consider the endomorphism induced by X on the tangent bundle T M , This endomorphism can be canonically identified with a (1, 1)-tensor field. Besides, rising an index via the metric h produces a symmetric (0, 2)-tensor fieldX, whose components are locally given byX jk = X j X k .

Remark 2.1.
In what follows, we use the symbolsâ 1 , . . . ,â N to denote the smooth symmetric (0, 2)-tensor fields obtained by applying the procedure defined above to the vector fields a 1 , . . . , a N .
We may now state Lemma 2.2 (alternative expression for Λ i ). For ψ ∈ C 2 (M ), (2.4) Λ i (ψ) = div 2 h ψâ i − div h ψ∇ ai a i , i = 1, . . . , N. Proof. In any coordinates, from the definition of the divergence of a vector field, Therefore, recalling that locally (2.2) and ∇ ai a i = a ℓ i ∂ ℓ a β i + Γ β jkâ jk i ∂ β hold, we obtain the following identity between vector fields: We apply div to this equation to obtain (2.4).
Remark 2.2 (adjoint of Λ i ). The adjoint of Λ i (·) is a i (a i (·)), i.e. ∀ψ, φ ∈ C 2 (M ), The following lemma turns out to be an extremely useful instrument in the proof of Theorem 3.2. It allows us to introduce a special atlas on M , in whose charts the determinant of the metric h will be constant. It turns out that this atlas significantly simplifies several terms in some already long computations; in broad strokes, the underlying reason is we can work locally with the standard d-dimensional Lebesgue measure dz instead of the Riemannian volume element dV h . Lemma 2.3 (convenient choice of atlas). On the manifold M there exists a finite atlas A = {κ : X κ ⊂ M →X κ ⊂ R d } such that, for any κ ∈ A, the determinant of the metric written in that chart is equal to one: |h κ | ≡ 1. In particular, we have (2.5) Γ m mj = 0 on X κ , for any j = 1, · · · , d. Proof. Fix x ∈ M and consider a chart Φ around x, whose induced coordinates are named (u i ) and whose range is the open unit cube in R d , (0, 1) d . Then, * is defined in Section 2.1, and ∧ denotes the wedge product between forms. Without loss of generality, we can assume from the beginning that f ∈ C ∞ ([0, 1] d ). Consider the following map from (0, 1) d to R d : Ψ : One can check that Ψ is smooth and invertible onto its image (recall f > 0). Moreover, |Ψ ′ | = f (u 1 , . . . , u d ) > 0. By the inverse function theorem and the fact that Ψ admits a global inverse, we infer that Ψ is a diffeomorphism of (0, 1) d onto its image, and We set κ x := Ψ • Φ. We repeat this procedure for any x ∈ M , and by compactness of M we end up with a finite atlas A = {κ : X κ ⊂ M →X κ ⊂ R d } with the desired property. In general, Γ m mj = ∂ j log |h κ |  [5].
Finally, we discuss the conditions imposed on the vector field u. Firstly, In particular, we have dV h ds is absolutely continuous, P-a.s., and hence is not contributing to cross-variations against W i . These cross-variations appear when passing from Stratonovich to Itô integrals in the SPDE (1.1), consult the upcoming Lemma 3.1.
For the uniqueness result (cf. Corollary 3.3), we must also assume Remark 2.4. In the following, for a function f : M → R and a vector field X, we will freely jump between the different notations for the vector field obtained by pointwise scalar multiplication of f and X.

Stochastic framework.
We refer to [31,34] for relevant notation, concepts, and basic results in stochastic analysis. From beginning to end, we fix a complete probability space (Ω, F , P) and a complete right-continuous filtration Without loss of generality, we assume that the σ-algebra F is countably generated.
It is important to pick good modifications of stochastic processes. Right (or left) continuous modifications are often used (they are known to exist for rather general processes), since any two such modifications of the same process are indistinguishable (with probability one they have the same sample paths). Besides, they necessarily have left-limits everywhere. Right-continuous processes with left-limits are referred to as càdlàg.
where F is a finite variation process and M is a local martingale. In this paper we will only be concerned with continuous semimartingales. The quantifier "local" refers to the existence of a sequence {τ n } n≥1 of stopping times increasing to infinity such that the stopped processes 1 {τn>0} M t∧τn are martingales.
Given two continuous semimartingales Y and Z, we can define the Fisk-Stratonovich where t 0 Y (s)dZ(s) is the Itô integral of Y with respect to Z and Y, Z denotes the quadratic cross-variation process of Y and Z. Let us recall Itô's formula for a continuous semimartingale Y . Let F ∈ C 2 (R). Then F (Y ) is again a continuous semimartingale and the following chain rule formula holds: Martingale inequalities are generally important for several reasons. For us they will be used to bound Itô stochastic integrals in terms of their quadratic variation (which is easy to compute). One of the most important martingale inequalities is the Burkholder-Davis-Gundy inequality. Let Y = {Y t } t∈[0,T ] be a continuous local martingale with Y 0 = 0. Then, for any stopping time τ ≤ T , where C p is a universal constant. We use (2.8) with p = 1, in which case C p = 3. Finally, the vector fields driving the noise in (1.1) satisfy

Weak solutions and main results
Inspired by [17], we work with the following concept of solution for (1.1).
The first result brings (1.1) into its equivalent Itô form. The result is analogous to Lemma 13 in [17].
Proof. Let us commence from (3.1). The Stratonovich integrals can be written as where ·, · denotes the cross-variation between stochastic processes. Using where we have exploited that the time-integral is absolutely continuous and thus not contributing to the cross-variation against W i , which follows from (2.6) and the fact that ρ belongs P-a.s. to Since a j (a i (ψ)) ∈ C ∞ (M ), the stochastic process (ω, t) → M ρ a j (a i (ψ)) dV h is a continuous semimartingale by assumption. It follows from [23, Theorem 2.2.14] that the variation process M ρ a j (a i (ψ)) dV h , W j is continuous and of bounded variation. Hence, ·, · , · = 0. Therefore, and the sought equation (3.2) follows. Finally, we can repeat this argument, starting with (3.2) and working our way back to (3.1). This concludes the proof.
In view of Lemma 3.1, we have an equivalent concept of solution.

Definition 3.3 (renormalization property).
Let ρ be a weak L 2 solution of (1.1) with initial datum ρ| t=0 = ρ 0 ∈ L 2 (M ). We say that ρ is renormalizable if, for any F ∈ C 2 (R) with F, F ′ , F ′′ bounded, and for any ψ ∈ C ∞ (M ), the stochastic process (ω, t) → M F (ρ(t))ψ dV h has a continuous modification that is {F t } t∈[0,T ] -adapted and satisfies the following SPDE weakly (in x) P-a.s.: where the second order differential operator Λ i is defined in (2.4), and The equation (3.3) is understood in the space-weak sense, that is, for all test functions ψ ∈ C ∞ (M ) and for all t ∈ [0, T ], P-a.s., if we apply the product rule to the divergence of the scalar G F (ρ) times the vector fieldā i , remembering that Λ i (1) = div hāi , cf. (2.1) and (3.4). We will make use of this expression for J in the upcoming computations.
We can now state the main result of this paper. As an application of this result, we obtain a uniqueness result for (1.1), if we further assume that div h u ∈ L 1 t L ∞ x , cf. (2.7). More precisely, we have Corollary 3.3 (uniqueness). Suppose conditions (2.6), (2.7), and (2.9) hold. Then the initial-value problem for (1.1) possesses at most one weak L 2 solution ρ in the sense of Definition 3.2.
According to Definition 3.2, a weak solution ρ belongs to the space L ∞ t L 2 ω,x . Combining the proof of Corollary 3.3 and a standard martingale argument, we can strengthen this through "shifting" ess sup t inside the expectation operator E[·], so that ρ ∈ L 2 ω L ∞ t L 2 x and consequently, P-a.s., ρ ∈ L ∞ t L 2 x . Corollary 3.4 (a priori estimate). Suppose the assumptions of Corollary 3.3 are satisfied. Consider a weak L 2 solution ρ of (1.1) with initial datum ρ 0 ∈ L 2 (M ).
where the constant C depends on div h u L 1 Remark 3.3. Throughout the paper, we assume that the vector fields driving the noise are smooth, a i ∈ C ∞ . In the Euclidian setting [32], the renormalization property holds under appropriate Sobolev smoothness, say a i ∈ W 1,p with p ≥ 4. A conceivable but quite technical extension of our work would allow for a i ∈ − −−−−− → W 1,p (M ). We leave this extension for future work. We refer to [19] for proof of the existence of weak solutions. Beyond the existence result, in that paper, we identify a delicate "regularization by noise" effect for carefully chosen noise vector fields (these vector fields must be linked to the geometry of the underlying domain). Consequently, we obtain existence without an L ∞ assumption on the divergence of the velocity u.

Informal proof of Theorem 3.2
In this section, we give a motivational account of the proof of our main result, assuming simply that all considered functions have the necessary smoothness for the operations we perform on them. To this end, consider a solution ρ of (1.1), which in Itô form reads (1.4), cf. Lemma 3.1. An application of Itô's formula with By the product and chain rules, To take care of the term F ′ (ρ)Λ i (ρ), we need Lemma 4.1. Let S be a smooth symmetric (0, 2)-tensor field on M , f ∈ C 1 (M ), and F ∈ C 1 (R). Then, as vector fields, Proof. In any coordinates, by the product and chain rules, In view of Lemmas 2.2 and 4.1, On the other hand, Therefore, subtracting (4.2) from (4.3), where G F is defined in (3.5) and Λ i (1) is defined in (3.4).
In view of the above computations, we can write (4.1) as where we need to take a closer look at the (potentially) problematic term Q, which contains the difference between some quadratic terms linked to the covariation of the martingale part of the equation (1.4) and the second order operators Λ i . We apply the product rule to write div h (ρa i ) = ρ div h a i +a i (ρ), and then expand whereā is defined in (3.4). As a result of this and the chain/product rules forā i , and so Q becomes We note here that the problematic term a(ρ) 2 has cancelled out in the final expression for Q. This is similar to what happens in the Euclidean setting [32]. On a curved manifold we must in addition exploit "cancellations" to control some error terms coming from the localization part of our regularization procedure, i.e., terms related to the geometry of the underlying domain (cf. Section 5 for details). This concludes the informal argument for (3.3).

Rigorous proof of Theorem 3.2
The aim of this section is to develop a rigorous proof of Theorem 3.2. The proof will involve a series of long computations, which will be scattered over seven subsections. We begin with the procedure for regularizing tensor fields on a manifold, along with several commutator estimates for controlling the regularization error. 5.1. Pullback and extension of tensor fields. We first recall and extend some concepts from Section 2.1. Let V be an arbitrary (boundaryless) smooth manifold of dimension d. Consider an arbitrary chart κ : X κ →X κ for V , where X κ and X κ are open subsets of V and R d respectively. Let RS(T m l (X κ )) denote the space of m covariant and l contravariant tensor fields onX κ ⊂ R d , and define similarly RS(T m l (X κ )) and RS(T m l (V )). Observe that we do not impose any assumptions on the regularity of the coefficients of the tensor fields; RS is an acronym for Rough Sections. Let SS(T m l (X κ )) be the subspace of smooth sections, and define similarly SS(T m l (X κ )), SS(T m l (V )). We are going to define a procedure for pulling an element of RS(T m l (X κ )) back to V . Indeed, given σ ∈ RS(T m l (X κ )), we may transport it on X κ ⊂ V via the diffeomorphism κ, and we call the result κ * σ, which will belong to RS(T m l (X κ )). We refer to [24, and Section 2.1 for details (the fact that κ is a diffeomorphism is crucial). Moreover, we may trivially extend it to the whole of V , by simply declaring that it is the null (m, l)-tensor field outside X κ . Let us name the resulting object (κ * σ) ext . Let us give a name to the entire procedure: where L κ will be referred to as a "pullback-extension" operator.
Assuming in addition that ) and supp L κ σ ⊂ X κ . In the following, starting from objects defined on open Euclidean subsets, we are going to use this procedure repeatedly to build global objects on M .

5.2.
Regularization & commutator estimates. From now on we are going to use the atlas A provided by Lemma 2.3. For fixed κ ∈ A, the induced coordinates will be typically denoted by z orz. We need a smooth partition of the unity supp U κ ⊂ X κ (and compact). Let ρ be a weak L 2 -solution of (1.1) with initial datum ρ 0 ∈ L 2 (M ). In what follows, we introduce a series of local objects that appear later in a localized version of (1.1), and establish their main properties. For κ ∈ A, fix a standard mollifier φ on R d with support in B 1 (0), and define the rescaled mollifier Remark 5.1. As is customary in differential geometry, we will use the convention of not explicitly writing the chart, in order to alleviate the notation. For example, if f : M → R, then we write f (z) instead of f (κ −1 (z)).
Remark 5.2. By writing a i (z) we mean the vector field evaluated at the point z, not differentiation. It is a minor abuse of notation that should not provoke too much confusion. We will use this convention in the following also for other objects.
We regularize ρ κ (ω, t, z)a l i (z) l componentwise via the mollifier φ ε (as above). The result is an object in SS(T 0 1 (X κ )) that is compactly supported, uniformly in ε. We denote this regularized object by We apply the pullback-extension operator L κ defined in (5.1), yielding . Finally, we define the companion vector field (ρ κ ) ε (t)a i ∈ SS T 0 1 (M ) . We need the following version of a well-known result found in [10].
, for any q ∈ [1, ∞). Furthermore, for x ∈ X κ (the only relevant case), in the coordinates induced by κ, we have the representations Then, in the coordinates induced by κ, we have by definition that (ρ κ a i ) ε l = ρ κ a l i ε (l = 1, . . . , d) and Γ a ab = 0; therefore, div h coincides with the Euclidean divergence and thus Differentiating this expression according to a i leads to the claimed representation for a i (r κ,ε,i ). The convergence claim is also clear. Indeed, with "ρa i ∈ L ∞ t L 2 x ", repeated applications of Lemma A.1 lead to r κ,ε,i → 0 in L q ([0, T ]; L 2 (Ω ×X κ )), for any q ∈ [1, ∞). Because the support of r κ,ε,i is compactly contained in X κ and |h κ |

5.2.3.
Localization & smoothing of ρ∇ ai a i . Consider for ω ∈ Ω, t ∈ [0, T ], and z ∈X κ , the vector field which is an object in RS T 0 1 (X κ ) that is compactly supported in κ (supp U κ ) ⊂ X κ , uniformly in ω, t. Observe that We regularize ρ κ ∇ ai a i componentwise using the mollifier φ ε , denoting the result by . We apply the pullback-extension operator L κ , arriving at the compactly supported vector field . Also in this case, we define the companion vector field (ρ κ ) ε (t)∇ ai a i ∈ SS T 0 1 (M ) .
Furthermore, for x ∈ X κ (the only relevant case), in the coordinates induced by κ, we have the representation The proof is identical to the one of Lemma 5.2, since the vector fields ∇ a1 a 1 , . . . , ∇ aN a N are smooth.
Let us compute div 2 h L κ ((ρ κ (t)â i ) ε ) in the local coordinates given by Lemma 2.3. The only relevant case is x ∈ X κ , where we use the coordinates induced by κ.
Recall that in these coordinates we have Γ α jα = 0 for all j. Hence, To be able to control the regularization error linked to div 2 h (ρâ i ), we need first to consider some additional terms appearing in the definition of div 2 h that is related to the Christoffel symbols Γ of the Levi Civita connection. To this end, consider which belongs to RS T 0 1 (X κ ) and is compactly supported in κ (supp U κ ) ⊂X κ , uniformly in ω, t. The regularized counterpart of V κ,i is denoted by V κ,i,ε . Clearly, Applying the pullback-extension operator L κ yields . We multiply the components of (ρ κ (t)â i ) ε by the Christoffel symbols Γ (written in the coordinates induced by κ) and then add them together. The result is an object in SS(T 0 1 (X κ )) that is compactly supported, uniformly in ε. Pushing forwardV κ,i,ε to M via L κ , we obtain L κVκ,i,ε ∈ SS T 0 1 (M ) . For x ∈ X κ , in the coordinates given by κ (z = κ(x)), we have We also need the globally defined objects Finally, consider the smooth (in x) and compactly supported (inX κ ) vector field , which is compactly supported in X κ , uniformly in ε.
As before, we introduce the global function Fix κ ∈ A, cf. Lemma 2.3. The quantity R κ,ε,i (t, ·) is supported in X κ ⊂ M , and there we are going to use the coordinates induced by κ, z = κ(x). From the definition of div 2 h , cf. (2.3), and by means of formulas (5.8) and (5.11), we deduce where the reminderr κ,ε,i is defined in (5.12) Remark 5.3. Be mindful of the fact that we have computed R ε,i in the (convenient) coordinates provided by Lemma 2.3. With a different choice of coordinates, we would need to handle some additional terms involving the Christoffel symbols Γ b ab , further complicating the analysis.
and making use of Lemma 5.2, we arrive at We recognize C ε [ρ κ (t), a i ] as a "second order" commutator, first identified in [33] (cf. the appendix herein for more details). Using again the product rule, where r κ,ε,i is defined in (5.5). For convenience, let us set g κ,i := ∂ l a m i ∂ m a l i . In our coordinates (cf. Lemma 2.3), div h a i = ∂ m a m i . By compactness of the supports of the involved functions, the quantities (ρ κ ) ε ∂ l a m i ∂ m a l i and C ε [ρ κ (t), a i ] can be thought of as globally defined (on M ). In other words, (5.16) may be seen as a global identity on M , that is, for x ∈ M , If x / ∈ X κ for some κ, then (5.17) reduces to the trivial statement "0 = 0". Referring to the appendix, a simple application of Lemma A.3 shows that G κ,ε,i Let us summarize our findings.
Lemma 5.5 (Second order commutator; "div 2 h (ρâ)"). For (ω, t, x) ∈ Ω×[0, T ]×M and, cf. (5.4), ε < ε 0 , the remainder R ε,i defined in (5.14) takes the form wherer ε,i is defined in (5.12) and G ε,i := κ∈A G κ,ε,i with G κ,ε,i defined in (5.18). Remark 5.4. Regarding the error term G ε,i , in general we cannot "sum away" κ due to the nonlinearity of the domain M , which is manifested in the local nature of the commutator C ε [ρ κ (t), a i ], cf. (5.15), and its dependence on different mollifiers! Remark 5.5. In view of (5.19), we do not expect the remainder term to converge to zero as ε → 0, althoughr ε,i and G ε,i do! Indeed, there is no reason to expect a i (r ε,i ) to have a limit as ε → 0. By good fortune, it turns out that this quantity is going to cancel out with a term that appears when applying the Itô formula, see the upcoming equation (5.38). This cancellation is the reason why the renormalization property holds for weak L 2 solutions without having to assume some kind of "parabolic" regularity (cf. the discussion in Section 1).

5.2.6.
Localization & smoothing of partition of unity terms. For reasons that will become apparent later, we need to apply the machinery developed so far to some additional terms that are related to the partition of unity {U κ } κ and its derivatives. These terms are linked to the nonlinear geometry of the manifold M .
For ω ∈ Ω, t ∈ [0, T ], and z ∈X κ , we introduce the functions (5.21) cautioning the reader that the superscripts do not mean exponentiation. Observe that these functions have their supports contained in supp U κ , uniformly in ω, t. Besides, recalling the local expressions for ∇ 2 U κ and (∇ ai a i ), . We regularize these functions using the mollifier φ ε and then apply the pullbackextension operator (5.1). The next lemma is analogous to Lemma 5.1, with the proof being evident at this stage.
(3) For any p ∈ [1,2] and (ω, t) ∈ Ω × [0, T ], where the convergences taking place in L p (M ), and where the pullback-extension operator L κ is defined in (5.1). Finally, for ω ∈ Ω, t ∈ [0, T ], and z ∈X κ , consider the vector field which belongs to RS T 0 1 (X κ ) and is compactly supported in κ (supp U κ ) ⊂X κ , uniformly in ω, t. Following our (by now) standard procedure, we regularize A 4 κ,i componentwise via the mollifier φ ε , cf. (5.2). We denote the resulting object by κ,i ε , and observe that, by definition, A 4 κ,i ε l = ρ a i (U κ )a l i ε . We apply the pullback-extension operator L κ to obtain the compactly supported vector field L κ A 4 κ,i ε ∈ SS T 0 1 (M ) . Summing over κ ∈ A, we obtain the global object From the very definitions of A 1 i,ε and L κ (A 4 κ,i ) ε , we have, for x ∈ M , Observe that for x ∈ X κ in the coordinates given by κ, z = κ(x), we have the representation By now the following lemma should be easy to prove.
We define a global remainder function by summing over κ: , for all q ∈ [1, ∞). Finally, we introduce the function which has support contained in supp U κ , uniformly in ω, t. We regularize A κ,u in z using the mollifier φ ε , and then apply the pullback-extension operator (5.1) to produce a function defined on M . We state the following lemma (without proof), noting that it is related to Lemmas 5.1 and 5.7.
We also introduce the globally defined function
We make use of ψ z,ε as test function in (5.27), which results in valid for z ∈X κ and ε < ε κ /4. Note that for the terms involving the covariant Hessian we have used the geometric identities appearing in Lemma 2.1. We can rewrite (5.28) in the following (pointwise) form: where V κ,i,ε ,V κ,i,ε are respectively the regularized versions of V κ,i , cf. (5.9), and V κ,i , cf. (5.10). Next, we turn these regularized local SPDEs, one equation for each chart κ ∈ A, into a globally defined SPDE.
In a nutshell, to prove Theorem 3.2, i.e., the renormalized equation (3.6), we need to send ε → 0 (after integrating in x). The task is rather nontrivial, and, before we can accomplish that, we need several intermediate results, which will involve crucial cancellations among some of the terms in (5.31).
Proof. By definition, in a coordinate-free notation, Therefore, by the product and chain rules, recalling (3.5), the lemma follows.
where r ε,i is defined in (5.6),ā i = (div h a i ) a i , and G F is defined in (3.5).
Proof. By inspecting the integrands of H 2 and H 6 , we recognize that the claim follows from the "second order" commutator Lemma 5.5.
Thus the claim follows by noting that F ′′ (ρ ε (s)) a i (ρ ε (s)) = a i F ′ (ρ ε (s)) . Lemma 5.17 ("H 9 + H 11 "). Consider H 9 = I 10 and H 11 = I 13 with I 10 and I 13 defined in (5.33). Then Combining (5.37) and Lemmas 5.13, 5.14, and 5.17, we arrive at the following expression for the remainder term R ε , which is a function of (ω, t, x) ∈ Ω×[0, T ]×M : where the difficult terms involving a i r ε,i cancel out: H 2+6,1 + H 5,2 = 0. 5.6. Passing to the limit in SPDE for F (ρ ε ); u ≡ 0. We wish to send ε → 0 in the x-weak form of (5.36), analyzing the limiting behavior of the remainder R ε , which is going to vanish, separately from the other terms in (5.36), which are going to converge to their respective terms in (3.6). Denote by ·, · the canonical pairing between functions on M . In the following, we will make repeated (unannounced) use of Lemma A.6 and the (stochastic) Fubini theorem.
We begin with the convergence analysis of the remainder term.
As of now, we have proved our main result (Theorem 3.2) under the additional assumption that u ≡ 0. 5.7. The general case u ≡ 0. Let us adapt the prior proof to the general case. First, (5.31) continues to hold provided we add to the right-hand side the terms where the first term is (by now) easily seen to be equal to where G F is defined in (3.5) and the remainder r ε,u is defined in (5.20). In other words, the equation (5.36) for F (ρ ε ) continues to hold with − J u and − J A added to the left-hand side of the equality sign.
Some tedious computations will reveal that and lim µ→∞ G Fµ (ξ) = ξ 2 , for ξ ∈ R, µ > 0. (6.2) Finally, to prove Corollary 3.4, we will also make use of the bounds for some constant C χ that does not depend on µ.
In view of the bounds on G Fµ , cf. (6.2), it is clear that the stochastic integral in (6.4) is a zero-mean martingale, and taking the expectation leads then to E M F µ (ρ(t)) dV h = E F µ (ρ(r)) dV h , for t > 0, and f µ (0) = M F µ (ρ 0 ) dV h . By the boundedness of F µ , note that f µ ∈ L ∞ (for a forthcoming application of Grönwall's inequality, we simply need f µ ∈ L 1 ). Set where we have taken advantage of the assumption ρ ∈ L ∞ t L 2 ω,x to conclude that ρ 2 ∈ L 1 (Ω × [0, T ] × M ) and thus The constant hidden in " " depends on max i a i C 2 and χ.
By the Burkholder-Davis-Gundy inequality (2.8), where the constant C 1 is independent of µ, t (but it depends on max i a i C 1 ). On that account, we obtain We collect here several results that have been used throughout the paper (often unannounced), starting with a minor generalization of a well-known commutator estimate, see [ We fix a standard Friedrichs mollifier φ ε (= ε −d φ(x/ε)) on R d . In what follows, we will consider functions and vector fields defined on an open (bounded or unbounded) subset of the Euclidean space R d .
Our next lemma is about the convergence of a "second order" commutator. The lemma is taken from Punshon-Smith's preprint [32].
We say that a triple (α 1 , α 2 , β) is (2)-admissible if α 1 , α 2 ∈ [1, ∞], 1 α1 + 2 α2 ≤ 1, Lemma A.2 (Punshon-Smith). Suppose g ∈ L p1 loc (G), V ∈ W 1,p2 loc (G; R d ), for some (2)-admissible triple (p 1 , p 2 , p), and define For any compact subset K ⊂ G, Furthermore, there is a constant C independent of ε, p, g, V such that Proof. By [32, Lemma 3.2], ≤ C V 2 W 1,p 2 (K;R d ) g L p 1 (K) , for some constant C independent of ε, p, g, V . The lemma follows from this, the triangle inequality, and the bound ( 1 Let us also state the following generalization of Lemma A.2, which is analogous to Lemma A.1 (the proof is also the same). Lemma A.3. Let (Z, µ) be a finite measure space. Suppose g ∈ L q1 (Z; L p1 loc (G)) , V ∈ L q2 Z; W 1,p2 loc (G; R d ) , for some (2)-admissible triples (p 1 , p 2 , p), (q 1 , q 2 , q). Then, for any compact K ⊂ G, ≤ C g L q 1 (Z;L p 1 (K)) V 2 L q 2 (Z;W 1,p 2 (K;R d )) , for some constant C that does not depend on ε, p, g, V . Furthermore, On several occasions we use the following basic convergence lemma: Proof. We can assume r < ∞, as the result is trivial for r = ∞. Fix an arbitrary subsequence {f jn } n≥1 ⊂ {f j } j≥1 . Then, by the "inverse dominated convergence" theorem, there exists a sub-subsequence f jn k k≥1 ⊂ {f jn } n≥1 which converges a.e. to f , and there exists a function g ∈ L r that dominates f jn k k≥1 , see [8, By the arbitrariness of the fixed subsequence and the uniqueness of the limit, the original sequence must converge as well.
We will also need an easy variant of the previous lemma. Finally, we recall (without proof) a simple result that has been utilized several times when passing to the limit in the space-weak formulation of the SPDE.  Then F j → F in L r (Ω × [0, T ]) as j → ∞.