Time-reversibility and nonvanishing Lévy area

We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.


Introduction
The classical Wong-Zakai question [23] from 1965 is concerned with weak convergence of smooth processes W n to Brownian motion W and the consequences for the interpretation of the corresponding stochastic integral W dW .
It is by now well-understood (if not always well-known) that the Lévy area is an important and nontrivial correction to the Wong-Zakai question.However, there are few investigations of whether this formula for E as displayed in Section 2 (which looks like it may be nonzero) is actually nonzero in the presence of additional constraints such as time-reversibility.
The Lévy area is skew-symmetric (E T = −E) and hence vanishes in the scalar case.In higher dimensions, Lévy area corrections vanish as a consequence of exactness, or commutativity of the defining vector fields, but such conditions are atypical outside the scalar case.On the other hand, an example where it is proved that E = 0 is given by Hairer, Pavliotis & Stuart, see [21,Section 11.7.7].
From time to time, we have been asked whether time-reversibility can force E = 0. (The example of Hairer et al. in [21] is not time-reversible.)In dispersing billiard examples considered by Chernov & Dolgopyat [3], the Lévy area is indeed zero as a consequence of time-reversibility and the structure of the equations.For Markov processes, there is a related condition, detailed balance, which forces E = 0, see for example [12,Section 3.3.2]or [20,Section 5.1].See also [17,Section 1.4.2] for further comments on E being zero in certain time-reversible situations.
The recent work [7] gives numerical evidence that E = 0 in certain examples, but it is written (at the end of Section 4 of that paper) that E could be negligible in many situations.On the other hand, it has recently been conjectured that in numerical simulations of certain stochastic systems which were obtained via some stochastic parametrisation it is numerically advantageous to neglect the Lévy area and set E = 0 [9,2].
In this paper, we offer a complete description and clarification of the structure of the Lévy area.In particular, we classify the constraints on E imposed by timereversibility.The cases where E is forced to vanish are easily described.Outside of these rare situations, we find that E is typically far from negligible.
Remark 1.1 Since our conclusions are contrary to those for systems satisfying detailed balance, our results can be viewed as advising caution on when to invoke detailed balance as a modelling assumption.
The remainder of this paper is organised as follows.Section 2 contains the setup in this paper and gives a rough description of our main result, namely Theorem 4.3.In Section 3, we introduce time-reversibility and determine the constraints imposed by the time-reversibility on the Lévy area.In Section 4, we state and prove Theorem 4.3, showing that there are no further constraints on the Lévy area beyond those determined in Section 3. In Section 5, we obtain a local version of this result: namely that there are no constraints on the Lévy area under small perturbations of the fast dynamics.Finally, in Section 6, we verify that, in reasonable situations, nonvanishing Lévy area leads as anticipated to nontrivial corrections to limiting stochastic integrals.

The setup
We consider fast-slow ordinary differential equations (ODEs) on where Λ ⊂ R m is a compact subset and µ is a Borel probability measure on Λ.The functions are assumed to be C r (for some r ≥ 1) with Λ v dµ = 0. Let g t : Λ → Λ denote the flow on Λ generated by the ODE ẏ = g(y).We assume that µ is g t -invariant, ergodic and mixing.
The aim of homogenisation is to establish, as ǫ → 0 in (2.1), a limiting stochastic differential equation (SDE) of the form such that x = x (ǫ) converges weakly to X. Here, W is d-dimensional Brownian motion, the stochastic integral b(X) • dW has the Stratonovich interpretation, and ã is a modified drift term incorporating the correction (if any) to the Stratonovich integral.
In the deterministic setting of (2.1), convergence to an SDE of the form (2.2) was obtained in [13] under suitable chaoticity assumptions (subsequently optimised in [5,8,14]) on the fast dynamics ẏ = g(y).Under these assumptions, we have convergent series of Green-Kubo-type: These series define a positive semi-definite symmetric matrix Σ ∈ R d×d and a skew-symmetric matrix E ∈ R d×d .Moreover (under smoothness assumptions on a and b which play no further role in this paper), the solutions x (ǫ) converge weakly to solutions of the SDE (2.2) where the Brownian motion W has covariance matrix Σ and the modified drift term is given by ã (2.3) Here, Z ij denotes the (i, j)'th entry of a matrix Z and b β denotes the β'th column of b.Generally, Σ is positive definite in the setting of [13].Indeed, the case det Σ = 0 is infinitely unlikely in a sense that can be made precise, see for example [4,Section 2.3].Given its antisymmetry, a natural question is to ask whether the Lévy area E may be forced to vanish in certain circumstances.Clearly, if where R is a time-reversal symmetry for the fast dynamics, then again it is easily verified (see Remark 3.2) that E = 0.This situation can occur in simple physical situations where v represents position or velocity, such as in the dispersing billiards examples in [3].
In this paper we show that the cases v•R = v and v•R = −v are the only situations where time-reversal symmetry forces the Lévy area to vanish, and typically E = 0 for the remaining time-reversible systems.In particular, we show that v transforms as and that E = 0 where E 0 is a general d + × d − matrix.

Time-reversal symmetry
In this section, we introduce time-reversal symmetry into the fast-slow ODE (2.1) and derive simplified formulas for the covariance Σ and Lévy area E. Our only assumption in this section is that the Green-Kubo-type formulas for Σ and E converge.We assume that there is a time-reversal symmetry (x, y) → (Sx, Ry) where S ∈ R d×d and R : Λ → Λ satisfy S 2 = I and R • R = I.As usual, this means that (Sx(−t), Ry(−t)) is a solution of (2.1) whenever (x(t), y(t)) is a solution.We suppose also that µ is R-invariant.
For the fast dynamics, time-reversibility means that Equivalently, g(Ry) = −Rg(y) for all y ∈ Λ.
Proof By invariance of µ under the fast flow g t and R, and (3.1), Substituting this into the covariance and Lévy area formulas from Section 2 yields the result.
Remark 3.2 It can be seen already from Proposition 3.
(This was the case in [3].)Our main result, Theorem 4.3 below, implies that typically E = 0 outside of these cases.
For the slow dynamics, time-reversibility means that This simplifies to the requirement that To avoid pathologies, from now on we suppose that b : R d → R d×d defines a nonsingular matrix in R d×d on a dense subset of R d and that {v(y) : y ∈ Λ} spans R d in the sense that {v(y) : y ∈ Λ} is not contained in a proper affine subspace of R d .Then the second condition in (3.2) simplifies further:

2) holds if and only if there exists
Proof It is immediate that if a, b and v satisfy these restrictions for some A with Then Hence v(y) = A(x 1 )A(x 2 )v(y) for all y and it follows from the spanning assumption on v that A(x 1 )A(x 2 ) = I for all x 1 , x 2 ∈ X. Taking , where The result follows.
4 Generality of E 0 In this section, we show that there are no further restrictions on the Lévy area E beyond those in Corollary 3.4.Recall that the vector field g in (2.1) is assumed to be C r for some r ≥ 1.We suppose that the fast dynamics defined by g is mixing sufficiently quickly, so that the series for E in Section 2 and hence E 0 in Corollary 3.4 converge for all C r functions v : Λ → R d with Λ v dµ = 0. We exclude the uninteresting case where Λ is a fixed point.(For then v| Λ ≡ 0, so Σ = E = 0 and there is no stochasticity in the limit.) Remark 4.1 In the setting of [5,8,13,14], the matrices Σ and E depend continuously on v ∈ C r .
We say that a function Proof Recall that g t denotes the fast flow generated by the vector field g in (2.1). We Clearly, v + •R = v + = Av + .By the chain rule, (∇h) Ry = (∇h) y R.This combined with the identity g Since the flow is mixing and By the definition of E 0 in Corollary 3.4 (up to a factor of 2 which can be incorporated into f ) we have proved the result.Proof We claim that it is possible to choose v so that E 0 has full rank, namely min{d + , d − }.Assuming this is the case, let . By standard linear algebra, this results in any desired matrix in R d + ×d − .It remains to prove the claim.The first step is to construct suitable R-invariant has full rank.The second step is to approximate f 1 and h 1 by smooth Rinvariant functions f : R m → R d + and h : R m → R d − with Λ f dµ = 0 so that Λ f ⊗ h dµ has full rank.The third step is to apply Proposition 4.2.
Step 1 Since Λ is not a fixed point, we can choose infinitely many orthonormal Rinvariant functions φ j ∈ L 2 (Λ) with Λ φ j dµ = 0 and Let F 1 = Λ f 1 ⊗ h 1 dµ.Taking β j to be the j'th canonical unit vector, F 1 is the d + × d − matrix with columns α 1 , . . ., α d − .In particular, F 1 is arbitrary and we can choose the α i so that F 1 is a matrix of full rank.
Step 2 Now choose R-invariant functions φj ∈ C ∞ (R m ) with Λ φj dµ = 0 and Λ |φ j − φj | 2 dµ small 1 and define f , h using φj in place of φ j with α i , β j unchanged.This results in a matrix F = Λ f ⊗ h dµ close to F 1 .In particular, taking the approximation close enough ensures that F is still of full rank.
Step 3 It follows from Proposition 4.2 that we can choose In particular, such E 0 has full rank, proving the claim.
1 For instance, to approximate φ 1 by a C ∞ function, first approximate φ 1 in L 2 by a simple function small.By Urysohn's Lemma, there exists a continuous function In this way we obtain a continuous function

Local surjectivity for perturbations of E
In Section 4, we showed that E 0 was a general matrix in the sense that χ is surjective.As mentioned in Remark 4.1, χ is continuous in reasonable situations.In this section, we complete the picture by establishing a local surjectivity result for χ; namely that if χ(v 0 ) = E 0 and E is close to E 0 of full rank, then there exists v 1 close to v 0 with χ(v 1 ) = E.
The first step is to show that if E 0 = χ(v 0 ) does not have full rank, then the rank can be increased under small perturbations.
Proof Suppose without loss of generality that ) so p(t) = 0 for large t.Since p is a polynomial, it follows that there exists ǫ > 0 such that p(t) = 0 for all t ∈ (0, ǫ).Hence A t is invertible, and so rank E t = d + , for all t ∈ (0, ǫ).
Next, we require a basic result from linear algebra.Proposition 5.2 Let E 0 , E ∈ R m×n be matrices of full rank and suppose that E is close to E 0 .Then there exist near identity matrices Proof Suppose without loss of generality that m ≤ n.Since rank E 0 = m, there exist invertible matrices P 0 ∈ R m×m , Q 0 ∈ R n×n such that P 0 E 0 Q T 0 = I m 0 .Then P 0 EQ T 0 is close to I m 0 and it is easily seen that there exist near identity matrices P 1 ∈ R m×m , Q 1 ∈ R n×n corresponding to near identity row and column operations such that (P We can now state and prove the main result of this section.Theorem 5.3 Suppose that E 0 = χ(v 0 ) and that E is of full rank and close to E 0 .Then there exists v that is C r -close to v 0 such that χ(v) = E.
Proof Suppose without loss of generality that d + ≤ d − .By Lemma 5.1, we can suppose that rank E 0 = d + .By Proposition 5.2, there exist near identity matrices Hence we can take v = (P ⊕Q)v 0 .

Corrections to limiting stochastic integrals
In the previous sections, we gave a complete description of the Lévy area, namely the skew symmetric matrix E whose entries determine the correction to the drift term a(X) as given in (2.3).This is not the complete story since even when E is nonzero, it might be the case that the correction is forced to vanish due to the structure of b.Indeed, this happens when b : R d → R d×d satisfies an exactness condition, namely that b −1 = dh for some h : R d → R d , see for example [11].
In general, b satisfies the time-reversibility constraint b(Sx) = −Sb(x)A, (6.2) from Lemma 3.3 which places restrictions on the correction (6.1).
In this section, we consider some simple time-reversible situations with E nonvanishing and show that in these cases the correction (6.1) is typically nonzero.Given the bilinearity of this term, it suffices to exhibit a single b for which the term is nonzero.
Recall that A 2 = S 2 = I and that we can choose coordinates such that A = I d + ⊕ I d − .Recall also that E γβ = 0 if 1 ≤ β, γ ≤ d + and if d + + 1 ≤ β, γ ≤ d, while the remaining entries of E are general.Since E is assumed to be nonvanishing, we have d ± ≥ 1.
Our simplifying assumption, which is natural in term-reversible situations, is that S is also diagonal in these coordinates, so S = diag{S 1 , S 2 , . . ., S d } where S β ∈ {±1}, and that S = ±I.Define B ⊂ {1, . . ., d} so that S β = 1 if and only if β ∈ B. Then the constraint (6. and setting the remaining entries to zero.Substituting into the sum in (6.1), we see immediately that nonzero terms require α = i.Then the factor b αγ (X) is nonzero only for γ = d.The column vector b β (X) has nonzero entries only for β = 1, d.Since E dd = 0, the correction (6.1) reduces to a single term 1 2 E d1 ∂ i b 1 (X)b id (X) = 1 2 E d1 X i e j where e j is the canonical unit vector with 1 in the j'th entry.We know that typically E d1 = 0, so this yields a nontrivial correction to the drift term a as required.
We immediately obtain that v • R = Av.Also b • S = −SbA on the dense set X ⊂ R d and hence on the whole of R d by continuity of b.Let π ± : R d → R d be the projections onto the ±1 eigenspaces of A. Then we can write v in (2.1) uniquely as

Theorem 4 . 3
For any d + × d − matrix F , there exists a C r function v : R m → R d with v • R = Av and Λ v dµ = 0 such that E 0 = F .