Keywords

1 Introduction

Let \((W_t)\) be a suitably renormalized Brownian motion on the space of vector fields on the two dimensional torus \(\mathbb {T}^2\) with a well chosen Sobolev regularity. In the case where \((u_t)\) is a deterministic vector field, it was shown that equations of the form

$$\begin{aligned} {dg}_t = (\circ dW_t + u_t dt)(g_t) ; g_t=e \end{aligned}$$
(1.1)

could model the Navier-Stokes flows (see for instance the review article [2] and references within). More precisely it was shown that \((u_t)\) solves the Navier-Stokes equation if and only if a certain associated action is stationary. Subsequently, models of the form

$$\begin{aligned} {dg}_t = (\circ dW_t + \dot{v}_t(\omega ) dt)(g_t) ; g_t=e \end{aligned}$$
(1.2)

where considered in Ref. [1], together with a notion of generalized stochastic flows with fixed marginals. In these latter models, the shift \(\dot{v}_t(\omega )\) is allowed to be random: the drift changes from one realization of the noise to another which seems to fit accurately with the microscopic models of the Navier-Stokes equation one encounters in physics. In particular such processes are not necessarily Markovian.

In the case of (1.2) there is no reason why we should hope \(\dot{v}(\omega )\) to solve the Navier-Stokes equation for any \(\omega \) \(a.s.\), and we should focus on the mean velocity field

$$ u: (t,x)\in [0,1] \times \mathbb {T}^2 \rightarrow u(t,x)=E_\eta [\dot{v}_t(x)] \in T_x\mathbb {T}^2 $$

where \(\eta \) is the underlying probability on the canonical path space, and where \(T_x\mathbb {T}^2\) is the tangent space at \(x\).

We extend here the criterion of Ref. [2] from equations with the form (1.1) to equations of type (1.2) for a wide class of stochastic drifts. Namely we focus on drifts \(v\) associated with a probability \(\eta \) with finite entropy with respect to the law \(\mu \) of the renormalized Brownian motion on the corresponding path space. We exhibit a class of such drifts (they will be called admissible) whose mean velocity field solves the Navier-Stokes equation if and only if the associated action, which will be noted \(\mathcal {S}(\eta | \mu )\), is critical. We then prove that this notion naturally extends the variational principle of Ref. [2]. One of the aspects of this model is to allow that the fluctuations of the drift itself may contribute to the pressure. Then we provide an alternative formulation to the least action principle by means of transformation of measure. However in this case, due to the renormalization involved in the definition of the cylindrical Brownian motion, our action for a process with law \(\eta \) is only related to the corresponding relative entropy

$$ \mathcal {H}(\eta | \mu ):= E_\eta \left[ \ln \frac{d\eta }{d\mu } \right] $$

by an inequality. Nevertheless, by introducing a cut-off, the action \(\mathcal {S}(\nu | \eta )\) becomes proportional to the relative entropy, and by cutting the high modes, we provide a least action principle to the Navier-Stokes equation by means of the relative entropy.

The structure of this paper is the following. In Sect. 2 we introduce the general framework as well as the main notations of the paper. In Sect. 3 we provide a characterization of solutions of the Navier-Stokes equation as critical flows of the action. In Sect. 4 this criterion is proved to extend those of Refs. [2, 3]. In Sect. 5 we introduce a cut-off in order to transform variations of the action in variations of the entropy. (Sect. 6).

2 Preliminaries and Notations

2.1 A Basis of Vector Fields on the Two Dimensional Torus

Let \(M:=\mathbb {T}^2\) be the set of pairs \((\theta _1,\theta _2)\) of real numbers modulo \(2 \pi \), and denote \(m_{\mathbb {T}}=\frac{\lambda ^L\otimes \lambda ^L}{4\pi ^2}\) where \(\lambda ^L\) is the Lebesgue measure on \([0,2\pi ]\). Integration will often be noted \(dx\) instead of \(m_{\mathbb {T}}(dx)\). A basis of the tangent space \(T_xM\) at \(x=(\theta _1,\theta _2)\in M\) is given by \((\partial _{i} |_x):= (\frac{\partial }{\partial \theta _i} |_{x=(\theta _1,\theta _2)})\). We define a scalar product \(\langle ., .\rangle _{T_xM}\) on each \(T_xM\) by setting \(\langle \partial _i |_x,\partial _j |_x\rangle _{T_xM} =\delta ^{i,j}\) where \(\delta ^{i,j} =1\) if \(i=j\) and \(0\) if \(i\ne j\). When there is no ambiguity, we will sometimes note \(X.Y\) instead of \(\langle X,Y\rangle _{T_xM}\) for \(X,Y\in T_xM\). If \(\mathcal {X}(M)\) consists of sections of \(TM\), \(\mathcal {X}(M)=\{X:M\rightarrow T(M)\}\), and considering its \(L^2\) equivalence class, we set

$$ \mathcal {G} =\left\{ X\in \mathcal {X}(M) | ~~\hbox {div}(X)=0 \ \hbox {and} \ \int \limits _{M} |X(x)|_{T_xM}^2 dx <\infty \right\} $$

which is a separable Hilbert space with the product

$$ \langle X,Y\rangle _{\mathcal {G}} := \int \limits _M \langle X(x),Y(x)\rangle _{T_xM} dx $$

An Hilbertian basis of \(\mathcal {G}\) is given by a subset \((e_\alpha )_{\alpha =1}^\infty \), whose definition is the following. Let \(k:\alpha \in \mathbb {N} / \left\{ 0\right\} \rightarrow k(\alpha ):= (k_1(\alpha ), k_2(\alpha )) \in (\mathbb {Z} \times \mathbb {Z}) / \left\{ (0,0)\right\} \) be a bijection such that \(|k(\alpha )|:= \sqrt{k_1^2(\alpha )+ k_2(\alpha )^2} \uparrow \infty \); we set

$$ e_\alpha (x) := \sum _{j} a^{\alpha ,j}(x) \partial _j |_x$$

where

$$a^{\alpha , i}(x):= \left\{ \begin{array}{ll} 1&{} \text{ if } (\alpha ,i) \in (1,1)\cup (2,2) \\ 0 &{} \text{ if } (\alpha ,i ) \in (2,1)\cup (1,2) \\ \sqrt{2} \frac{k_2(m)}{|k(m)|}\cos (k(m).x) &{} \text{ if } (\alpha ,i) =(2m+2,1), m\ge 1 \\ -\sqrt{2} \frac{k_1(m)}{|k(m)|}\cos (k(m).x) &{} \text{ if } (\alpha , i)= (2 m+2, 2), m\ge 1 \\ \sqrt{2}\frac{k_2(m)}{|k(m)|}\sin (k(m).x) &{} \text{ if } ~ (\alpha ,i)= (2 m + 1 ,1), m\ge 1 \\ -\sqrt{2}\frac{k_1(m)}{|k(m)|}\sin (k(m).x) &{} \text{ if } ~ (\alpha ,i)=(2 m +1 ,2) , m\ge 1 \end{array} \right. $$

and where, for \(k=(k_1,k_2)\in \mathbb {Z} \times \mathbb {Z}\) and \(x=(\theta _1,\theta _2) \in M\), \(k.x := k_1\theta _1 + k_2\theta _2\). Any \(X\in \mathcal {G}\) can be written

$$X(x) =\sum _j X^j(x) \partial _j |_x$$

where \(X_j(x) =\sum _{\alpha } \langle X,e_\alpha \rangle _{\mathcal {G}} a^{\alpha ,j}(x) \). Let \(Y(x) := \sum _j Y_j(x) \partial _j|_x\) be another vector field: it is straightforward to check that we also have

$$ \langle X,Y\rangle _{\mathcal {G}}= \int \limits _M \sum _j X^j(x) Y^j(x) dx $$

We recall the following formulae

$$ \hbox {div} (X):=\sum _j \partial _j X^j , $$
$$\begin{aligned} \Delta X :=\sum _i (\sum _j \partial ^2_{j,j} X_i) \partial _i |_x \end{aligned}$$

and

$$ (X.\nabla ) Y := \sum _{j} (\sum _i X_i (\partial _i Y_j)) \partial _j |_x $$

2.2 The Group of the Volume Preserving Homeomorphisms

Let \(G\) be the group of the homeomorphisms of \(M\) which leaves \(m_{\mathbb {T}}\) invariant

$$ G: =\left\{ \phi :M \rightarrow M , homeomorphisms , \phi _\star m_{\mathbb {T}} = m_{\mathbb {T}} \right\} $$

We note \(e\) the identity on \(G\) and \(\phi .\psi \) the group operation of \(\phi ,\psi \in G\) (given by the composition of the two maps). We recall [6] that the subset of \(G\) consisting of maps which are, together with their inverses, in the Sobolev class \(H^r\), for \(r>2\) is a Hilbert manifold and a topological group. It is not, strictly speaking, a Lie algebra since left translation is not smooth. \(T_e G\) is given by the set of the vector fields \(X:x\in M \rightarrow X_x\in T_xM\) such that \(\hbox {div} (X)=0\). Let \(X\in T_e G\), and let

$$ c : t\in \mathbb {R} \rightarrow c_t \in G ; c_0= e $$

be a smooth curve on \(G\) to which \(X\) is tangent. We recall that, by setting \(\widehat{c} : (t,x) \in \mathbb {R} \times M \rightarrow c_t(x) \in M \), the value of \(X\) at \(x\in M\) is given by

$$ X(x) = \partial _t \widehat{c}(t,x) |_{t=0} \in T_xM $$

Furthermore \(X\) can be uniquely extended to a right invariant vector field \(\widehat{X}\) on \(G\) by setting

$$\begin{aligned} \widehat{X} : \phi \in G \rightarrow \widehat{X}_{\phi }\in T_{\phi } G \end{aligned}$$

where \(\widehat{X}_\phi \) is given by

$$\begin{aligned} \widehat{X}_\phi : x\in M \rightarrow \widehat{X}_\phi (x) := X(\phi (x)) \in T_{\phi (x)}M \end{aligned}$$

i.e. \(\widehat{X}_\phi \) is tangent to the curve \(c^\phi :t\in \mathbb {R} \rightarrow c_t.\phi \in G\). In particular for any smooth \(f\) on \(M\) and \(x\in M\) denote \(f^x \) the map \( \phi \in G \rightarrow f^x(\phi ) := f(\phi (x)) \in \mathbb {R} \). Then \(f^x\) is smooth on \(G\) and we have

$$\begin{aligned} (\widehat{X} f^x)(\phi ) := \widehat{X}_\phi f^x = \partial _t f(c_t.\phi (x)) |_{t=0} = \partial _t f( \widehat{c}(t,\phi (x))) |_{t=0} = X(\phi (x)) f := (Xf)(\phi (x)) \end{aligned}$$

In the sequel we will simply write \(X\) instead of \(\widehat{X}\) since it will be clear from the context whether we consider \(X\) as an element of the tangent space, or as a right-invariant vector field on \(G\). In order to kill the noise in the higher modes and to control the integrability of the derivatives, we introduce the following Sobolev spaces \((\mathcal {G}_\lambda )_{\lambda >1}\) and the associated abstract Wiener spaces \((W, H_\lambda , \mu _\lambda )\).

2.3 Sobolev Vector Fields

To any positive real number \(\lambda > 1\) we associate a sequence \((\lambda _\alpha )_{\alpha \in \mathbb {N}}\) defined by

$$\begin{aligned} \lambda _\alpha = \frac{|k([\frac{\alpha -1}{2}])|^{2\lambda }}{K(\lambda )} \end{aligned}$$

where \([.]\) is the floor function and where \(K(\lambda )\) is chosen so that

$$\begin{aligned} \sum _\alpha \frac{a^{\alpha ,i}(x)}{\sqrt{\lambda _\alpha }} \frac{a^{\alpha ,j}(x)}{\sqrt{\lambda _\alpha }} =\delta ^{i,j} \end{aligned}$$

Such a \(K(\lambda )\) exists from standard results on Riemann series since \(\lambda >1\), and we have \(K(\lambda ) \uparrow \infty \) as \(\lambda \downarrow 1\). For \(\lambda >1\), let \(S_\lambda \) be the positive, definite, trace class operator defined by

$$\begin{aligned} S_\lambda x :=\sum _i \frac{1}{\lambda _i} \langle x,e_i\rangle _\mathcal {G} e_i \end{aligned}$$

and let

$$\begin{aligned} \mathcal {G}_\lambda := \sqrt{S_\lambda }( \mathcal {G}) \end{aligned}$$

which is an Hilbert space for the scalar product \(\langle .,.\rangle _{\mathcal {G}_\lambda }\) characterized by

$$\begin{aligned} \left\langle \sqrt{S_\lambda } x, \sqrt{S_\lambda } y\right\rangle _{\mathcal {G}_\lambda } = \langle x,y\rangle _{\mathcal {G}}. \end{aligned}$$

A natural Hilbertian basis of \(\mathcal {G}_\lambda \) is given by \((H_\alpha ^\lambda )_{\alpha =1}^\infty \) where

$$\begin{aligned} H^\lambda _\alpha :=\frac{e_\alpha }{\sqrt{\lambda _\alpha }} \end{aligned}$$
(2.3)

We set

$$\begin{aligned} A^\lambda _{\alpha ,j}(x) = \frac{a_{\alpha ,j}}{\sqrt{\lambda _\alpha }} \end{aligned}$$

so that

$$\begin{aligned} \sum _{\alpha } A^\lambda _{\alpha ,i}(x) A^\lambda _{\alpha ,j}(x) = \delta ^{i,j} \end{aligned}$$
(2.4)

and

$$\begin{aligned} H^\lambda _\alpha (x) =\sum _j A^\lambda _{\alpha , j}(x) \partial _j |_x \end{aligned}$$

Since \(\sqrt{S}_\lambda \) is Hilbert-Schmidt, it is well known that \(|.|_{\mathcal {G}}\) is a measurable semi-norm on the Hilbert space \(\mathcal {G}_\lambda \) (see [9]). In particular \((\mathcal {G}_\lambda , \mathcal {G})\) is an abstract Wiener space [9, 12], which allows to regard the cylindrical Brownian motion below as a Brownian sheet (note that we could have defined a Wiener measure directly on the Wiener space \((\mathcal {G}_\lambda , \mathcal {G})\), but we won’t use this in the sequel since we are interested in the path space).

2.4 Associated Wiener Spaces

The space

$$ H_\lambda := \left\{ h : [0,1] \rightarrow \mathcal {G}_\lambda : h:=\int \limits _0^. \dot{h}_s ds, \int \limits _0^1 |\dot{h}_s|^2_{\mathcal {G}_\lambda } ds <\infty \right\} $$

is an Hilbert space whose product will be noted \(\langle .,.\rangle _\lambda \). On the other hand the space

$$\begin{aligned} W :=C_0\left( [0,1], \mathcal {G} \right) \end{aligned}$$

is a separable Banach space for the uniform convergence norm. We denote by \(i_\lambda \) the injection of \(H_\lambda \) in \(W\). Since for \(\lambda >1\) \(|.|_\mathcal {G}\) is a measurable semi-norm on \(\mathcal {G}_\lambda \), it is a classical result on Wiener spaces that \((i_\lambda , W, H_\lambda )\) is also an abstract Wiener space. If \(\mu _\lambda \) is the standard Wiener measure on \(W\) for the A.W.S. \((W, H_\lambda , i_\lambda )\), we recall that under this probability the coordinate process \(t\rightarrow W_t(\omega ) = \omega (t) \in \mathcal {G}\) is an abstract Brownian motion with respect to its own filtration \( \left( \mathcal {F}_t \right) \) (see for instance [10, 12]). From the Itô Nisio theorem, we have \(\mu _\lambda -\)a.s.

$$\begin{aligned} W_t = \sum _\alpha W^\alpha _t H^\lambda _\alpha \end{aligned}$$

with \(W^\alpha _t := \widehat{\delta }H_\alpha (W_t) \), and where \(\left\{ \widehat{\delta }(X), X \in \mathcal {G}_\lambda \right\} \) is the isonormal Gaussian process on \(\mathcal {G}_\lambda \). We recall that under \(\mu _\lambda \), \(\left\{ \widehat{\delta }(X)(W_s), X\in \mathcal {G}_ \lambda ,s\in [0,1] \right\} \) is a Gaussian process with covariance

$$\begin{aligned} E_{\mu _\lambda }[\widehat{\delta }(X)(W_s) \widehat{\delta }(Y)(W_t)] = (s\wedge t) \langle X,Y\rangle _{\mathcal {G}_\lambda } \end{aligned}$$

so that \((W^\alpha _.)\) is a family of real valued independent Brownian motions under \(\mu _\lambda \). Under \(\mu _{\lambda }\), the coordinate process \(t\rightarrow W_t\) is called the cylindrical Brownian motion. The difference with respect to the case where the state space is finite dimensional is that it is a renormalized sum of independent Brownian motions, the renormalization appearing in (2.3). For a measure \(\eta \ll \mu _\lambda \) and a \(u\in L^0_a(\eta ,H_\lambda )\), the stochastic integral \(\delta ^W u := \int _0^1 \dot{u}_s dW_s\) is well defined as an abstract stochastic integral [10, 12]. Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu _\lambda \). Then there is a unique \(v\in L^0_a(\eta , H_\lambda )\) such that \(\eta -a.s.\)

$$\begin{aligned} \frac{d\eta }{d\mu _\lambda } := \exp \left( \delta ^W v -\frac{|v|_\lambda ^2}{2} \right) \end{aligned}$$
(2.5)

Moreover \(W^\eta := I_W - v\) is a \(\left( \mathcal {F}_t\right) \)-Brownian motion on \((W,\mathcal {F},\eta )\). We call \(v\) the velocity field associated to \(\eta \). The famous formula of [7] (which in fact holds in a more general framework: [10, 12]) reads

$$\begin{aligned} 2\mathcal {H}(\eta | \mu _\lambda ) = E_\eta \left[ \int \limits _0^1 |\dot{v}_t |^2_{\mathcal {G}_\lambda } dt\right] \end{aligned}$$
(2.6)

where

$$\mathcal {H}(\eta | \mu _\lambda ):=E_\eta \left[ \ln \frac{d\eta }{d \mu _\lambda }\right] $$

is the relative entropy of \(\eta \) with respect to \(\mu _\lambda \). Note that since \(\mathcal {G}_\lambda \subset \mathcal {G} \subset T_eG\) it makes sense to consider \((Xf)(\phi )\) for \(\phi \in G\), for \(f \) smooth on \(G\) and for \(X \in \mathcal {G}_\lambda \) or \(X\in \mathcal {G}\).

3 Navier-Stokes Flows with Stochastic Drifts

Henceforth and until the end of Sect. 5 we assume that the renormalization sequence is fixed for a \(\lambda \ge 2\), and we drop the indices \(\lambda \) of the notations except for \(\mathcal {G}_\lambda \).

3.1 Constraints on the Kinematics: Regular and Admissible Flows

Definition 1

A probability \(\eta \) which is absolutely continuous with respect to \(\mu \) with finite entropy (\(\mathcal {H}(\eta | \mu ) <\infty \)) is called a regular flow if \( u \in C^1([0,1]\times M)\) and \(dt\)- \(a.s.\) \(\partial _t u \in \mathcal {G} \), where \(u(t,x):=E_\eta \left[ \dot{v}_t(x)\right] \), and where \(v:=\int _0^. \dot{v}_s ds \) is the velocity field of \(\eta \) (see (2.5)). We call \(u\) the mean velocity field of \(\eta \). Moreover we say that a regular flow is admissible if there is a \(C^1 ([0,1]\times M)\) mapping \(p^\star : [0,1]\times M \rightarrow \mathbb {R}\) such that

$$\begin{aligned} Cov(\dot{v}_t(x)) = p^{\star }(t,x) I_d \end{aligned}$$

i.e. for \(i,j\in \mathbb {N} \cap [1,d]\)

$$\begin{aligned} E_\eta \left[ \left( \dot{v}_t^i(x)-u_t^i(x)\right) \left( \dot{v}_t^j(x)-u_t^j(x) \right) \right] = p^\star (x,t) \delta ^{i,j} \end{aligned}$$
(3.7)

where \((\dot{v}^j_t(x))\) denotes the \(j\)th (random) component of \((\dot{v}_t^j)\) at \(x\) i.e. it is such that \(\dot{v}_t(x)= \sum _j \dot{v}_t^j(x) \partial _j |_x\), and where \(u_t^j(x) := E_\eta [\dot{v}_t^j(x)]\).

3.2 Constraints on the Dynamics: Critical Flows

Definition 2

Let \(\eta \) be a regular flow whose velocity field is denoted by \(v^\eta \) (see (2.5)). For any \(k\in C^1([0,1];\mathcal {G})\) we set

$$ L_k\mathcal {S}(\eta | \mu ) := E_\eta \left[ \int \limits _0^1 \left( \,\int \limits _{M} < \dot{v}^\eta _t(x), \partial _t k + (\dot{v}^\eta _t.\nabla )k +\frac{\Delta k}{2}>_{T_xM} dx\right) dt\right] $$

The probability \(\eta \) is said to be critical if and only if for any \(k\in C_{0}^1([0,1],\mathcal {G})\)

$$\begin{aligned} L_k\mathcal {S}(\eta | \mu )=0 \end{aligned}$$

where

$${C_0^1}([0,1],\mathcal {G}) := \left\{ k \in C^1([0,1];\mathcal {G}) : k(0,.)=k(1,.) =0 \right\} $$

The dynamic of the mean velocity field of a critical flow is given by the following theorem

Theorem 1

Let \(\eta \) be a regular flow with a velocity field \(v\) and a mean velocity field \(u\in \mathcal {G}_\lambda \). Then \(\eta \) is critical (Definition 2) if and only if there is a function \(\widehat{p}(t,x)\) such that

$$\begin{aligned} \partial _t u + E_\eta [(\dot{v}_t(x).\nabla )\dot{v}_t(x)] = \frac{\Delta u}{2} - \nabla \widehat{p} (t,x) \end{aligned}$$
(3.8)

In other words, let

$$\begin{aligned} \beta (t,x):= E_\eta [((\dot{v}_t(x)-u_t(x)).\nabla )(\dot{v}_t(x)- u_t(x))] \end{aligned}$$
(3.9)

Then \(u\) solves, in the weak \(L^2\) sense, the following equation :

$$\begin{aligned} \partial _t u + (u_t.\nabla )u = \frac{\Delta u}{2} -\nabla \widehat{p} - \beta \end{aligned}$$
(3.10)

Proof

For any \(k\in {C_0}^1([0,1];\mathcal {G})\) we have \(k(0,.)=k(1,.)=0\), so that an integration by parts yields

$$\begin{aligned} L_k\mathcal {S}(\eta | \mu ) = - \int \limits _M \int \limits _0^1 \left( \partial _t u + E_\eta [ (\dot{v}_t.\nabla )\dot{v}_t] - \frac{\Delta u}{2}\right) (t,x). k(t,x) dx dt \end{aligned}$$
(3.11)

from which we obtain (3.8). Since

$$\begin{aligned} \beta (t,x)&:= E_\eta \left[ [\dot{v}_t(x)-u_t(x)] .\nabla )[\dot{v}_t(x)-u_t(x)]\right] \\&= E_\eta \left[ (\dot{v}_t(x) .\nabla )\dot{v}_t(x)\right] + (u_t(x) .\nabla )u_t(x) - E_\eta \left[ (\dot{v}_t(x) .\nabla )u_t(x)\right] - E_\eta \left[ (u(x) .\nabla )\dot{v}_t(x)\right] \\&= E_\eta \left[ (\dot{v}_t(x) .\nabla )\dot{v}_t(x)\right] + (u_t(x) .\nabla )u_t(x) - (E_\eta \left[ \dot{v}_t(x)\right] .\nabla )u_t(x) - (u(x) .\nabla )E_\eta \left[ \dot{v}_t(x)\right] \\&= E_\eta \left[ (\dot{v}_t(x) .\nabla )\dot{v}_t(x)\right] - (u_t(x) .\nabla )u_t(x) \end{aligned}$$

we obtain (3.10) from (3.8).    \(\square \)

3.3 Navier-Stokes Flows

Definition 3

A regular flow \(\eta \) (see Definition 1) is a Navier-Stokes flow if its mean velocity field \(u\) solves the Navier-Stokes equation, i.e. if and only if there is a function \(p:[0,1]\times M \rightarrow \mathbb {R}\) which is such that \(u\) solves, in the weak \(L^2\) sense, the Navier-Stokes equation

$$\begin{aligned} \partial _t u + u.\nabla u = \frac{\Delta u}{2} - \nabla p \end{aligned}$$

we have:

Corollary 1

An admissible flow is a Navier-Stokes flow if and only if it is critical.

Proof

Let \(\eta \) be an admissible flow. We recall that by definition there exists a mapping \(p^\star \) such that

$$\begin{aligned} Cov(\dot{v}_t(x)) = p^\star (x,t) I_d \end{aligned}$$
(3.12)

where \(v:=\int _0^.\dot{v}_s ds\) is the velocity field of \(\eta \) (see (2.5)). We also recall that

$$u(t,x):= E_\eta [\dot{v}_t(x)]$$

The idea is to apply Theorem 1 and to set

$$p:= p^\star + \widehat{p}$$

We have

$$ \beta ^i(t,x) = \sum _{j}\partial _j Cov(\dot{v}_t(x))^{i,j} $$

Indeed (repeated indices are summed over) we have

$$\begin{aligned} \beta ^i(t,x)&= E_\eta \left[ \left( \dot{v}^j_t(x)-u^j_t(x)\right) \partial _j\left( \dot{v}^i_t(x)- u^i_t(x)\right) \right] \\&= \partial _j E_\eta \left[ \left( \dot{v}^i_t(x)- u^i_t(x)\right) \left( \dot{v}^j_t(x)-u^j_t(x)\right) \right] - E_\eta \left[ \left( \dot{v}^i_t(x)-u^i_t(x)\right) \partial _j\left( \dot{v}^j_t(x)- u^j_t(x)\right) \right] \\&= \partial _j E_\eta \left[ \left( \dot{v}^i_t(x)- u^i_t(x)\right) \left( \dot{v}^j_t(x)-u^j_t(x)\right) \right] - E_\eta \left[ \left( \dot{v}^i_t(x)-u^i_t(x)\right) \hbox {div}\left( \dot{v}_t(x)- u_t(x)\right) \right] \\&= \partial _j E_\eta \left[ \left( \dot{v}^i_t(x)- u^i_t(x)\right) \left( \dot{v}^j_t(x)-u^j_t(x)\right) \right] \\&= \partial _{j} Cov(\dot{v}_t(x))^{i,j} \end{aligned}$$

Assumption (3.12) then yields \(\beta ^i(t,x) = \partial _i p^\star \) i.e.

$$\begin{aligned} \beta = \nabla p^\star \end{aligned}$$
(3.13)

   \(\square \)

Remark 1

Note that by this proof, for critical flows, \(p^\star \) appears as a part of the pressure which is originated from the stochastic model. Specifically it expresses the fluctuations of the drift itself. Indeed by (3.13) and (3.9) for an admissible flow \(\eta \) we have

$$\begin{aligned} \nabla p^\star (t,x) = E_\eta [(\dot{v}_t(x). \nabla ) \dot{v}_t(x)] - (u_t(x). \nabla ) u_t(x) \end{aligned}$$
(3.14)

where \(p^\star \) is the function associated to the admissible flow \(\eta \) by formula (3.7).

4 Interpretation of Critical Flows by Means of the Stochastic Exponential

In this section we prove that the quantities \(L_k\mathcal {S}(\eta | \mu )\) defined in Definition 2 can still be interpreted in terms of certain variations along deterministic paths which extend those of Ref. [3].

4.1 The Stochastic Exponential

Let \(C_G= C_e\left( [0,1],G \right) \) be the space of continuous paths starting from \(e\) and with values in \(G\). The coordinate function \((t,\gamma )\in [0,1]\times C_G \rightarrow \gamma _t(\omega )\) generates a filtration \(\left( \mathcal {F}^G_t\right) \) and we denote \(\mathcal {F}^G:= \mathcal {F}^G_1\).

Proposition 1

The equation

$$\begin{aligned} dX_t = \circ dB_t ; ~ X_0=e \end{aligned}$$
(4.15)

has a continuous strong solution on the space \(\left( W, \mathcal {F}_.^{W}, \mu \right) \) with the canonical Brownian \(t \rightarrow W_t\in \mathcal {G}\). We note \(g\) this solution. By this we mean that for \(\mu {-}a.s.\) \(g \in C_G\) and, for any smooth \(f\) on \(G\),

$$f(g_t) = f(e) + \sum _\alpha \int \limits _0^t (H_\alpha f)(g_t)\circ dW_t$$

where \(\circ \) denotes the Stratonovich integral.

Proof

See [11].    \(\square \)

Girsanov theorem on \((W, H, \mu )\) implies the following:

Proposition 2

Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu \) whose velocity field is noted \(v\), and set \(\widetilde{W}:= I_W-v\). Then \(( g , \widetilde{W})\) is a solution of

$$\begin{aligned} dX_t = (\circ dB_t + \dot{v}_t dt) ; X_0=e \end{aligned}$$
(4.16)

on \((W ,\mathcal {F}_., \eta )\).

Proof

We have

$$ \widetilde{W}_s = \sum _\alpha \widehat{\delta }(H_\alpha )(W_s) H_\alpha - \sum _\alpha \langle v,H_\alpha \rangle _{\lambda } H_\alpha = \sum _\alpha \widehat{\delta }(H_\alpha ) (\widetilde{W}_s) H_\alpha $$

Since \( \widetilde{W}_\star \eta = \mu \), \(\widetilde{W}_.^\alpha := \widehat{\delta }(H_\alpha ) (\widetilde{W}_.)\) are independent Brownian motions on \((W, H, \eta )\), by Itô’s formula we have, \(\eta -a.s.\),

$$f(g_t) = f(e) + \int \limits _0^1 \sum _{\alpha }(H_\alpha f)(g_t) \circ d\widetilde{W}_t^\alpha + \sum _\alpha \int \limits _0^1 (H_\alpha f)(g_s) \langle \dot{v}_s, H_\alpha \rangle _{\mathcal {G}_\lambda } ds $$

i.e.

$$\begin{aligned} f(g_t) = f(e) + \int \limits _0^1 (H_\alpha f)(g_s) \circ d\widetilde{W}_t^\alpha + \int \limits _0^1 (\dot{v}_t(\omega )f)(g_s) ds \\ \end{aligned}$$

   \(\square \)

Proposition 3

Let \(\eta \) be a probability absolutely continuous with respect to \(\mu \), \(v:=\int _0^. \dot{v}_s ds\) the associated velocity field, \(\widetilde{W} = I_W- v\) and \(\widetilde{W}^\alpha _.= \widehat{\delta }(H_\alpha )(\widetilde{W}_.)\). For any smooth function \(f\) on \([0,1]\times M\) we have \(\eta {-}a.s.\)

$$\begin{aligned} f(t, g_t(x)) = f(0,x) + \int \limits _0^t \left( \frac{\Delta }{2} f+ (\dot{v}_\sigma .\nabla ) f+ \partial _\sigma f)(\sigma , g_\sigma (x)\right) d\sigma + \int \limits _0^t \sum _\alpha (H_\alpha f)(\sigma ,g_\sigma (x)) d\widetilde{W}^\alpha _\sigma \end{aligned}$$
(4.17)

and \(\eta {-}a.s.\)

$$\begin{aligned} {\lim _{\delta \rightarrow 0}} E_{\eta }\left[ {\frac{f(t+\delta ,g_{t+\delta }(x)) -f(t,g_t(x))}{\delta }} \Bigg |{\mathcal {F}_t}\right] = \left( \partial _{t} f + ({\dot{v}}_{t}(\omega ).\nabla ) f + {\frac{\Delta f}{2}}\right) {(t,g_{t}(x))} \end{aligned}$$
(4.18)

Proof

Let \(x\in M\), \(f\in C^\infty (M)\). The main part of the proof will be to prove that

$$\begin{aligned} \sum _\alpha (H_\alpha ^2 f^x)(\phi ) = (\Delta f)(\phi (x)) \end{aligned}$$
(4.19)

To see this recall that \(f^x: \phi \in G \rightarrow f(\phi (x))\in \mathbb {R}\). We have

$$\begin{aligned} (H_\alpha f^x) (\phi ):=H_\alpha (\phi ) f^x = H_\alpha (\phi (x)) f =(H_\alpha f)(\phi (x)) = (H_\alpha f)^x(\phi ) \end{aligned}$$
(4.20)

so that by iterating (4.20) we obtain

$$\begin{aligned} \sum _\alpha (H_\alpha ^2 f^x)(\phi ) =\sum _\alpha (H_\alpha ^2f)(\phi (x)) \end{aligned}$$
(4.21)

On the other hand

$$\begin{aligned} \sum _\alpha (H_\alpha ^2f)(\phi (x)) = (\Delta f)(\phi (x)) \end{aligned}$$
(4.22)

Indeed by using the fact that for any \(\alpha \) the vector field \(H^\alpha \) is divergence free together with (2.4) we obtain

$$\begin{aligned} \sum _\alpha H_\alpha ^2f&= \sum _{\alpha , j} A^{\alpha , j } \partial _j (H_\alpha f) \\&= \sum _{\alpha ,i,j} A^{\alpha ,j } A^{\alpha ,i}(\partial _j \partial _i f) + A^{\alpha ,j} (\partial _j A^{\alpha ,i}) (\partial _i f) \\&= \sum _i (\partial ^2_{i,i}f) + \sum _{\alpha ,i,j} A^{\alpha ,j} (\partial _j A^{\alpha ,i}) (\partial _i f) \\&= \Delta f + \sum _{\alpha ,i,j} A^{\alpha ,j}(\partial _j A^{\alpha ,i}) (\partial _i f) \\&= \Delta f + \sum _{\alpha ,i,j} \partial _j( A^{\alpha ,j} A^{\alpha ,i})(\partial _i f) - \sum _{\alpha ,i,j} (\partial _j A^{\alpha ,j}) A^{\alpha ,i} (\partial _i f) \\&= \Delta f+ \sum _{i,j} \partial _j( \sum _\alpha A^{\alpha ,j} A^{\alpha ,i}) (\partial _i f) - \sum _{\alpha ,i} (\hbox {div}( H^\alpha )A^{\alpha ,i} (\partial _i f) \\&= \Delta f \end{aligned}$$

Finally by putting together (4.21) and (4.22) we get (4.19) which yields

$$\begin{aligned} f(t, g_t(x))&= f^x(t, g_t) \\&= f^x(s,g_s) + \int \limits _s^t (H_\alpha f^x)(g_\sigma ) \circ d\widetilde{W}_\sigma ^\alpha + \int \limits _s^t (\partial _\sigma f^x + \dot{v}_\sigma f^x)(g_\sigma )d\sigma \\&= f(s,g_s(x)) + \int \limits _s^t \left( \frac{\Delta }{2} f+ (\dot{v}_\sigma .\nabla ) f+ \partial _\sigma f\right) (\sigma , g_\sigma (x)) d\sigma \\&\quad + \int \limits _s^t \sum _\alpha (H_\alpha f_\sigma )(g_\sigma (x)) d\widetilde{W}^\alpha _\sigma \end{aligned}$$

On the other hand by the Girsanov theorem, \((\widetilde{W}_t)\) is a \((\mathcal {F}_t)\)-Brownian motion on \((W, \eta )\) so that (4.18) follows from (4.17).    \(\square \)

4.2 Perturbations of the Energy Along Deterministic Paths

For \(k\in C^0([0,1], \mathcal {G}_\lambda )\), \(k:=\int _0^. \dot{k}_s ds\), we define \(e(k)\) to be the solution of the ordinary differential equation on \(G\)

$$ d (e_t(k)) = (\dot{k}_t dt)(e_t(k)) ; e_0 = e$$

i.e. for any smooth \(F:G\rightarrow \mathbb R\),

$$\begin{aligned} F(e_t( k))= F(e) + \int \limits _0^t (\dot{k}_s F)(e_s( k)) ds. \end{aligned}$$
(4.23)

Note that \(e_.(0_{H})=e\) i.e. the exponential of the function which is constant and equal to \(0_H\) is constant and equal to \(e\). We denote by \((e^i_t(k))\) the ith component of \((e_t(k))\) in the canonical chart.

Proposition 4

If \(\eta \) is a probability of finite entropy with respect to \(\mu \), for any \(k\in C_0^1([0,1], \mathcal {G}_\lambda )\) we have

$$\begin{aligned} L_k\mathcal {S}(\eta | \mu ) = \frac{d}{d\epsilon } E_\eta \left[ \int \limits _0^1 \left( \int \limits _M \frac{|D^\eta e_t(\epsilon k).g_t(x)|_{T_{g_t(x)}M}^2}{2}dx\right) dt \right] |_{\epsilon =0} \end{aligned}$$
(4.24)

where \(L_k\mathcal {S}(\eta | \mu )\) has been defined in Definition 2 and where \(D^\eta e_t(\epsilon k).g_t(x)\) is defined a.e. by

$$\begin{aligned} D^{\eta } e_t(\epsilon k).g_t(x):= \sum _{i} \lim _{\delta \rightarrow 0} E_{\eta }\left[ \frac{e^{i}_{t+\delta }(\epsilon k). g_{t + \delta }(x) - e^{i}_t(\epsilon k). g_t(x)}{\delta } \Bigg |\mathcal {F}_t\right] \partial _{i} |_{g_{t(x)}} \end{aligned}$$
(4.25)

Proof

By (4.18) of Proposition 3 we first obtain

$$\begin{aligned} D^\eta e_t(\epsilon k).g_t(x) := \sum _i \left( \partial _t e_t^i(\epsilon k) + (\dot{v}_t(\omega ).\nabla ) e_t^i(\epsilon k) + \frac{\Delta e_t^i(\epsilon k)}{2}\right) (g_t(x)) \partial _i|_{g_t(x)} \end{aligned}$$
(4.26)

On the other hand let \(x\in M\) and denote by \(f\) a smooth function on \(M\). Considering \(F:=f^x\) in (4.23) we have

$$f(e_t(\epsilon k)(x))= f(x) + \epsilon \int \limits _0^t (\dot{k}_s f)(e_s(\epsilon k)(x)) ds$$

Since \(e_.(0_{H})(x)=e(x)=x\), we get :

$$\frac{d}{d\epsilon }|_{\epsilon =0}f(e_t(\epsilon k)(x))= \int \limits _0^t (\dot{k}_s f)(x) ds = (k_t f)(x)$$

so that

$$\begin{aligned} \frac{d}{d\epsilon }|_{\epsilon =0} e_t(\epsilon k)(x) = k_t(x) \end{aligned}$$
(4.27)

By (4.26) and (4.27) we obtain

$$\begin{aligned} \frac{d}{d\epsilon }D^\eta e_t(\epsilon k).g_t(x) |_{\epsilon =0}= \left( \partial _t k_t + \dot{v}_t.\nabla k_t + \frac{\Delta k_t}{2}\right) (g_t(x)) \end{aligned}$$
(4.28)

For convenience of notations we denote by \(A\) the right hand term of (4.24). By first differentiating the product, then by applying (4.26) at \(\epsilon =0\), then by applying (4.28), and finally by using that \(g_t\) preserves the measure we obtain

$$\begin{aligned} A&= E_\eta \left[ \int \limits _0^1 \left( \int \limits _M\langle D^\eta g_t(x), \frac{d}{d\epsilon }D^\eta e_t(\epsilon k).g_t(x)|_{\epsilon =0}\rangle _{T_{g_t(x)} M} dx \right) dt \right] \\&= E_\eta \left[ \int \limits _0^1 \left( \int \limits _M\langle \dot{v}_t(g_t(x)) , \frac{d}{d\epsilon }D^\eta e_t(\epsilon k).g_t(x)|_{\epsilon =0}\rangle _{T_{g_t(x)}M} dx \right) dt \right] \\&= E_\eta \left[ \int \limits _0^1 \left( \int \limits _M\langle \dot{v}_t(g_t(x)) , \left( \partial _tk_t + \dot{v}_t.\nabla k_t + \frac{\Delta k_t}{2}\right) (g_t(x))\rangle _{T_{g_t(x)}M} dx \right) dt \right] \\&= E_\eta \left[ \int \limits _0^1 \left( \int \limits _M\langle \dot{v}_t(x) , \partial _tk_t(x) + \dot{v}_t.\nabla k_t(x) + \frac{\Delta k_t}{2}(x)\rangle _{T_{x}M} dx \right) dt \right] \end{aligned}$$

which proves (4.24).    \(\square \)

5 Variations of the Energy Along Translations

Let \(\eta \) be a probability which is absolutely continuous with respect to \(\mu \) (as mentioned in the beginning of Sect. 3 we work with a fixed \(\lambda \ge 2\)) and with velocity field \(v^\eta \). The stochastic action of \(\eta \) is defined by

$$\begin{aligned} \mathcal {S}(\eta | \mu ):= E_\eta \left[ \int \limits _0^1 \frac{|\dot{v}_s^\eta |_\mathcal {G}^2}{2} ds\right] \end{aligned}$$
(5.29)

The motivation for this definition is that, by taking \(\epsilon =0\) in (4.26) and using the fact that \(g_t\) preserves the measure, we also have

$$\mathcal {S}(\eta | \mu ) = E_\eta \left[ \int \limits _0^1 \left( \int \limits _M \frac{|D^\eta g_s(x)|_{T_{g_s(x)}M}^2}{2} dx \right) ds \right] $$

with the notations of Proposition 4. By (2.6), \(\mathcal {G}_\lambda \subset \mathcal {G}\) implies that whenever the entropy is finite we have

$$\mathcal {S}(\eta | \mu ) <\infty $$

as well. More accurately, by a classical result on abstract Wiener spaces together with (2.6), there exists a \(c>0\) such that for any \(\eta \ll \mu \)

$$\mathcal {S}(\eta | \mu ) \le c \mathcal {H}(\eta | \mu )$$

In this section we introduce another kind of variations for the functional \(\mathcal {S}(\eta | \mu )\), namely we study its variations along translations, These variations are generally different from those introduced above; however, when restricted to admissible flows, they are the same. We also investigate similar variations for the relative entropy. Proposition 5 computes the values of the variations of these quantities along deterministic translations.

Proposition 5

Let \(\eta \) be a probability absolutely continuous with respect to \(\mu \) with velocity field \(v^\eta \) and mean velocity \(u_s (x):= E_\eta [\dot{v}^\eta _s (x)]\). If \(\mathcal {S}(\eta | \mu ) <\infty \) we have,

$$\begin{aligned} \frac{d}{d\epsilon } \mathcal {S}(\tau _{\epsilon h} \eta | \mu ) |_{\epsilon =0}= \int \limits _0^1 \langle u_s, \dot{h}_s\rangle _{\mathcal {G}} ds \end{aligned}$$
(5.30)

and if \(\mathcal {H}(\eta | \mu ) <\infty \) we have

$$\begin{aligned} \frac{d}{d\epsilon } \mathcal {H}(\tau _{\epsilon h} \eta | \mu ) |_{\epsilon =0}= \int \limits _{0}^{1} \langle u_s, \dot{h}_s\rangle _{\mathcal {G}_\lambda } ds \end{aligned}$$
(5.31)

where \(\tau _h\eta \) is the image measure of \(\eta \) by the mapping \(\tau _h\) defined by

$$\tau _h : \omega \in W \rightarrow \omega +h\in W$$

Proof

A straightforward application of the Cameron-Martin theorem shows that for any \(h:=\int _0^. \dot{h}_sds \in H\), the velocity field field \(v^{\tau _h\eta }\) of \(\tau _h\eta \) is given by

$$\begin{aligned} v^{\tau _h\eta } = \tau _{h}\circ v^\eta \circ \tau _{-h} = v^\eta \circ \tau _{-h} +h \end{aligned}$$
(5.32)

Hence by (5.29) we have

$$\mathcal {S}(\tau _h\eta | \mu ) = E_\eta \left[ \int \limits _0^1 \frac{|\dot{v}_s^\eta +\dot{h}_s|_\mathcal {G}^2}{2}ds\right] $$

which yields (5.30). Similarly (5.31) follows by (2.6) and (5.32).    \(\square \)

Let

$$\begin{aligned} C^n_{0}([0,1],\mathcal {G}_{\lambda +2}):=\left\{ k\in C^n([0,1],\mathcal {G}_{\lambda +2}) : k(0,.)=k(1,.)=0 \right\} \end{aligned}$$
(5.33)

and let \(\Pi \) be the Helmoltz projection on divergence free vector fields. We set

$$\begin{aligned} \mathcal {K}^\eta _{0}:= \left\{ h := \int \limits _0^. \dot{h}_s (\omega ) ds \Bigg | \exists k \in C^n_{0}([0,1],\mathcal {G}_{\lambda +2}) , ds-a.s., \dot{h}_s = \partial _s k_s +\Pi ( (u_s.\nabla )k_s) + \frac{\Delta k_s}{2} \right\} \end{aligned}$$
(5.34)

so that it makes sense to say that any \(h\in \mathcal {K}_{0}^\eta \) is associated to a \(k\in C^n_0([0,1],\mathcal {G}_{\lambda +2})\). For \(n\) sufficiently large we have \(\mathcal {K}^\eta \subset H\).

Proposition 6

Let \(\eta \) be a smooth flow whose mean velocity field is given by \(u\). Then \(u\) solves the Navier-Stokes equation if and only if for any \(h\in \mathcal {K}_0^\eta \)

$$\frac{d}{d\epsilon } \mathcal {S}(\tau _{\epsilon h}\eta |\mu ) |_{\epsilon =0} = 0$$

Proof

By Proposition 5, and by definition of \(\Pi \), for any \(h\) (which is associated to \(k\)) we have

$$\begin{aligned} \frac{d}{d\epsilon } \mathcal {S}(\tau _{\epsilon h}\eta |\mu ) |_{\epsilon =0} = \int \limits _M \int \limits _0^1 \left( \partial _s k + \Pi ((u.\nabla ) k) + \frac{\Delta k}{2}\right) (s,x). u(s,x) dx ds \end{aligned}$$
(5.35)
$$\begin{aligned} = \int \limits _M \int \limits _0^1 \left( \partial _s k + (u.\nabla ) k + \frac{\Delta k}{2}\right) (s,x). u(s,x) dx ds \end{aligned}$$
(5.36)

and, since \(k(0,.)=k(1,.)=0\), the result directly follows from an integrating by parts.    \(\square \)

We now relate these variations to the ones of Sect. 4. Namely we prove that, for admissible flows, these variations of measure by quasi-invariant transformations yield exactly the same variations as the exponential variations of Sect. 4.

Proposition 7

Let \(\eta \) be an admissible flow. Then, for any \(h\in \mathcal {K}_{0}^\eta \) (see (5.34)) associated with a \(k\in C^n_{0}([0,1],\mathcal {G}_{\lambda +2})\) (see (5.33)) we have

$$ \frac{d}{d\epsilon } \mathcal {S}(\tau _{\epsilon h}\eta |\mu ) |_{\epsilon =0} = L_k\mathcal {S}(\eta |\mu ) $$

Proof

Let \(u\) be the mean velocity field of \(\eta \). Since \(\eta \) is admissible we have, by (3.14)

$$ \langle u_t, (u_t.\nabla )k_t\rangle _{\mathcal {G}} = -\langle (u_t.\nabla ) u_t, k_t\rangle _ {\mathcal {G}} = -E_\eta [\langle (\dot{v}_t.\nabla ) \dot{v}_t, k_t\rangle _{\mathcal {G}} ] = E_\eta [\langle \dot{v}_t ,(\dot{v}_t.\nabla ) k_t\rangle _{\mathcal {G}}] $$

Hence, using (5.36),

$$\begin{aligned} \frac{d}{d\epsilon } \mathcal {S}(\tau _{\epsilon h}\eta |\mu ) |_{\epsilon =0} = E_\eta \left[ \int \limits _0^1 \left( \int \limits _{M} \big \langle {\dot{v}}^{\eta }_{t}(x), \partial _{t} k + (\dot{v}^{\eta }_{t}.\nabla )k +\frac{\Delta k}{2} \big \rangle {_{T_xM}} dx\right) dt\right] \end{aligned}$$

which is exactly the definition of \( L_k\mathcal {S}(\eta |\mu )\) (Definition 2).    \(\square \)

6 Generalized Flows with a Cut-off

In Sect. 5 we have seen that in the infinite dimensional case, the relative entropy was generally not proportional to the action \(\mathcal {S}(\cdot | \mu )\). The reason is that the renormalization procedure gives a different weight to the different modes: hard modes have a weaker weight in the energy than in the relative entropy. However if instead of renormalizing we introduce a cutoff, and rescale the noise accordingly, \(\mathcal {S}(\cdot | \mu )\) becomes proportional to the relative entropy \(\mathcal {H}(\cdot | \mu )\). Within this framework, we investigate the existence of generalized flows with a given marginal.

6.1 General Framework for a Cut-off at Scale \(n\)

We recall that \((e_\alpha )\) denotes the Hilbertian basis of \(\mathcal {G}\) of Sect. 2. By induction we define \((I_l)_{l=1}^\infty \) by \(I_1=1\) and

$$I_{l+1}= \min \left( \left\{ m \ge I_l : |k(m)|>|k(I_l)| \right\} \right) $$

For \(N\in \mathbb {N}, N >1\) we set

$$n:= 2I_N $$

We define \(\mathcal {G}^n = Vect( e_1,\ldots ,e_n) \subset \mathcal {G}\) and recall that we work under the hypothesis

$$e_\alpha (x) = \sum _j a^{\alpha ,j}(x) \partial _j|_x$$

The cut-off has been chosen so that \(\exists S(N)\) such that

$$\sum _{\alpha = 1}^n a^{\alpha ,i}(x) a^{\alpha , j}(x) = \mathcal {S}(N) \delta ^{i,j}$$

where \(\mathcal {S}(N) \uparrow \infty \). We note

$$ H_n :=\left\{ h :[0,1]\rightarrow \mathcal {G}^n, h:=\int \limits _0^. \dot{h}_s ds, \int \limits _0^1 |\dot{h}_s|_{\mathcal {G}}^2 ds <\infty \right\} $$

and \(\langle .,.\rangle _{H_n}\) the associated scalar product. We set \(W_n:= C([0,1], \mathcal {G}^n)\) endowed with the norm of uniform convergence, and \(\mu _n\) the Wiener measure on \((W_n,H_n)\) with a parameter

$$\sigma (N):= \frac{2\nu }{\mathcal {S}(N)}$$

\(t\rightarrow W_t\) is the coordinate process. Define \(g^n\) to be the solution of

$$dg^n_t := ( \circ d W_t )(g^n_t) ; g^n_0=e$$

on the Wiener space \((W_n,H_n,\mu _n)\) i.e., satisfying, for every smooth \(f\),

$$ f(g_t^n) = f(e)+ \int \limits _0^t\sum _{\alpha =1}^n (e_\alpha f)(g^n_s) \circ dW^\alpha _s $$

where \(W^\alpha := \langle W_t,e_\alpha \rangle _{G_n}\). We are now working with the Wiener measure with parameter \(\sigma (N)\). Still by the Girsanov theorem, for any \(\eta \ll \mu _n\) there is a unique \(v\in L^0(\eta , H_n)\) such that

$$\frac{d\eta }{d\mu }= \exp \left( \delta ^W v -\frac{\sigma (N) |v|_{H_n}^2}{2}\right) $$

and \(\widetilde{W}:= I_W -\sigma (N) v \) is a Brownian motion with parameter \(\sigma (N)\) under \(\eta \). We call \(v\) the velocity field of \(\eta \). Furthermore, Föllmer’s formula (c.f. [8]) then reads

$$ \mathcal {H}(\eta | \mu _n) = \sigma (N) E_\eta \left[ \frac{|v|_{H_n}^2}{2}\right] $$

Hence \((g, \widetilde{W})\) is a solution to

$$dg^n_t := \circ (d W^\nu _t + \sigma (N) \dot{v}_t dt) )(g^n_t) ; g^n_0=e$$

on the probability space \((W_n,\eta )\) for the filtration generated by the coordinate process \(t\rightarrow W_t\), i.e., for every smooth \(f\),

$$ f(g_t^n) = f(e)+ \int \limits _0^t \sum _{\alpha =1}^n (e_\alpha f)(g_s) \circ d\widetilde{W}^\alpha _s + \sigma (N) \int \limits _0^t\sum _{\alpha =1}^n (e_\alpha f)(g_s) \langle \dot{v}_s, e_\alpha \rangle ds $$

Within this framework, by an admissible flow we mean a probability \(\eta \) of finite entropy with respect to \(\eta \) satisfying the same conditions as in Definition 1 with \(\mu _n\) (resp. \(\mathcal {G}_n\)) instead of \(\mu \) (resp. of \(\mathcal {G}\)).

6.2 Variations of the Action

We now define the action for the cutoff \(n\in \mathbb {N}\) by

$$ \mathcal {S}(\eta | \mu _n) := E_\eta \left[ \int \limits _0^1\frac{|D^\eta _s g^n_s|_{\mathcal {G}}^2}{2} ds \right] = E_\eta \left[ \int \limits _0^1\frac{|\sigma (N) \dot{v}_s|^2_{\mathcal {G}}}{2}ds\right] =\sigma (N)^2 E_\eta \left[ \int \limits _0^1 \frac{|\dot{v}_s|^2_{\mathcal {G}}}{2}ds\right] $$

Therefore

$$\begin{aligned} \mathcal {S}(\eta | \mu _n) = \sigma (N) \mathcal {H}(\eta | \mu _n) \end{aligned}$$
(6.37)

Similarly to Proposition 7 we note

$$\mathcal {K}^\eta _0(n):=\left\{ h \in H_n : \exists k\in C^1_{0}([0,1],{\mathcal {G}^n}), ds-a.s., \dot{h}_s= \partial _s k + \pi _n \Pi ((\sigma (N)u_s.\nabla ) k) +\nu \Delta k \right\} $$

where \(\pi _n\) is the orthogonal projection \(\pi _n :\mathcal {G} \rightarrow \mathcal {G}_n\) and we say that a \(h\in \mathcal {K}^\eta _0(n)\) is associated to a \(k \in C^1_{0}([0,1],{\mathcal {G}^n})\).

Proposition 8

For any smooth flow \(\eta \)

$$u^n(t,x) := \sigma (N) E_\eta [ \dot{v}_t(x)]$$

solves the Navier-Stokes equation if and only if for any \(h\in \mathcal {K}^\eta _0(n)\) we have

$$\frac{d}{d\epsilon } \mathcal {H}(\tau _{\epsilon h} \eta | \mu _n)] |_{\epsilon =0}= 0$$

for any \(h\) associated with a \(k\in C^1_{0}([0,1],{\mathcal {G}^n})\). Moreover whenever \(\eta \) is an admissible flow, and \(h\in \mathcal {K}^\eta _0(n)\) is associated to \(k\in C^1_{0}([0,1],{\mathcal {G}^n})\) we have

$$\frac{d}{d\epsilon } \mathcal {H}(\tau _{\epsilon h} \eta | \mu _n)] |_{\epsilon =0}= \frac{d}{d\epsilon } E_\eta \left[ \int \limits _0^1 \left( \int \limits _M \frac{|D^\eta e_t(\epsilon k).g^n_t(x)|_{T_{g_t(x)}M}^2}{2}dx\right) dt \right] |_{\epsilon =0}$$

where the notations are those of Sect. 4.

Proof

The first part of the proof is the same as in Proposition 6. We now prove the second part of the claim which is similar to Proposition 7. As in the first subsection we have

$$\sum _{\alpha =1}^n e_\alpha ^2 f = \mathcal {S}(N) \Delta f$$

Therefore by setting

$$A:= \lim _{\epsilon \rightarrow 0} \left( \frac{E_\eta \left[ \int _0^1 \left( \int _M |D^\eta e_t(\epsilon k).g^n_t(x)|_{T_{g_t(x)}M}^2dx\right) dt \right] - E_\eta \left[ \int _0^1 \left( \int _M |D^\eta g^n_t(x)|_{T_{g_t(x)}M}^2dx\right) dt \right] }{2 \epsilon } \right) $$

and using the fact \(g_t\) preserves the measure we get

$$A =E_\eta \left[ \int _0^1\langle \dot{v}_t, \partial _t k + \sigma (N)\dot{v}_t. \nabla k + \nu \Delta k \rangle _\mathcal {G}dt \right] $$

If \(\eta \) is assumed to be admissible, then similarly to the proof of Proposition 7 we obtain

$$\begin{aligned} A= \frac{d}{d\epsilon } \mathcal {H}(\tau _{\epsilon h} \eta | \mu _n)] |_{\epsilon =0} \\ \end{aligned}$$

   \(\square \)

Concerning existence of Lagrangian Navier-Stokes flows with a cut-off they have been shown to exist in Ref. [4] for deterministic \(L^2\) drifts. Examples of random solutions of Navier-Stokes equations were constructed in Ref. [5] but we did not prove existence of the corresponding flows.