A noise-induced transition in the Lorenz system

We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.


Introduction
The classical Lorenz system [Lor ] is a very popular prototypical toy model for chaos / turbulence [Rue ]. The traditional way of writing this system is given bẏ with parameter values σ = 10 and β = 8/3. Changing the value ̺ allows to explore a variety of different behaviours [Spa , BPV ]. In particular, for ̺ < 1, ( . ) admits a unique fixed point at the origin which eventually attracts every single solution, while for ̺ > 1 it admits two further 'non-trivial' fixed points. These fixed points become unstable at ̺ = σ 3+β+σ σ−β−1 ≈ 24.74, after which the system I exhibits either a chaotic attractor or stable limit cycles, see for example [Tuc ].
For very large values of ̺ (̺ 313), the system admits a stable limit cycle which undergoes a cascade of period-doubling bifurcations as one decreases ̺.
The vertical axis H = {(X, Y, Z) : X = Y = 0} is invariant for all parameter values and, for β > 0, it is contained in the stable manifold of the origin. The aim of this article is to explore how ( . ) behaves in the stable regime ̺ < 1 under the addition of noise to the Z-component. More precisely, we consider the modified systeṁ ( . ) whereα > 0 and ξ denotes white noise. For all values ofα, this system admits as invariant measure the measure ν 0 with ν 0 (H) = 1 under which Z ∼ N (0,α 2 /2β). The question we consider is whether it admits other invariant measures supported in R 3 \ H. Our main result then states that while ν 0 is the unique invariant measure for ( . ) for small values ofα, it necessarily admits a second invariant measure for large values.
For ̺ ≥ 1, there exists α ⋆ ≥ 0 such that the second statement still holds. Proof. The fact that forα > 0 ( . ) admits at most one ergodic invariant measure besides ν 0 is the content of Theorem . . Theorem . links the existence of the additional invariant measure ν ⋆ to the sign of the quantity λ α whose asymptotic behaviour for both small and large values ofα is obtained in Theorem . . It remains to note that one always has λ α > 0 (and therefore existence of ν ⋆ ) forα large enough, while its sign asα → 0 is negative when ̺ < 1 and positive when ̺ > 1.

Remark . .
One would naturally expect to have α ⋆ = α ⋆ , but we cannot guarantee this at the moment. It does however follow from our analysis that, for all values of α, ( . ) admits at most one ergodic invariant measure besides ν 0 and that the set of valuesα for which ν 0 is the unique invariant measure consists of finitely many intervals. One would also expect to have α ⋆ = 0 when ̺ > 1 since then H is already linearly unstable for the deterministic system, but although our results do indeed guarantee that in this case the system admits a unique additional invariant measure ν ⋆ for allα ∈ (0, δ) for some δ > 0, we cannot rule out the existence of an intermediate range of values for which ν 0 would be the unique invariant measure. The blue trajectory shows a simulation of ( . ) in the (x, y) plane with σ = 10, β = 8/3, ̺ = 1/2, andα = 30. The red trajectory is obtained by using the same parameters, initial condition, and realisation of the noise, except thatα = 10.

Remark . .
One motivation for the study of ( . ) is that this provides a toy model for the following situation. Consider the D Navier-Stokes equations on a torus with additive translation-invariant Gaussian forcing. It was shown in [HM , HM ] (see also the earlier works [EMS , BKL , KS ] showing similar results under stronger non-degeneracy assumptions) that this system admits a unique invariant measure under a very weak non-degeneracy assumption on the noise. This result however fails to apply to the situation where the noise is itself periodic with a period strictly smaller (say half) than that of the torus on which the system is posed. In this case, we know by [HM ] that it admits a unique invariant measure concentrated on functions with the same period as the noise and, by [Mat ], that this measure is unique at small enough Reynolds number. A natural question then is whether a bifurcation appears at high Reynolds number, as one would expect from heuristic considerations. Our result can be viewed as a first mathematically rigorous result pointing in the direction of a positive answer.

Remark . .
The behaviour observed here can be contrasted with that observed in [HM a, HM b], where the authors exhibit a system with a quadratic nonlinearity which can explode in finite time in the absence of noise but admits global solutions (and a unique invariant measure) in the presence of noise.
Figure shows a simulation of ( . ) for ̺ = 1/2 and two different values ofα. Forα > α ⋆ , a typical trajectory consists of relatively long stretches of time spent in the vicinity of H, interspersed with excursions away from H. These excursions all escape in roughly the same direction and the Z-coordinate (not depicted in the figure) is always quite negative when this happens. When Z becomes positive, they so that one can reduce oneself to Y = 0 when ̺ > 1 and X = 0 when ̺ < 1. We deduce from this simple observation a slight strengthening of [BKP , Thm . ], namely that if κ > β(1 ∧ ̺), then ( . ) admits exactly 3 ergodic invariant measures when ̺ = 1 and uncountably many ergodic invariant measures when ̺ = 1. Note that this does not depend onα, it is in particular also true forα = 0 in which case the invariant measures are concentrated on fixed points. A version of the Lorenz system in which the parameter ̺ randomly switches between two "unstable" values ( and close to ) was analysed in [BH , Str ]. In these works, the existence of exactly two invariant measures was proved by combining an extension of Hörmander condition to piecewise deterministic systems and the precise knowledge of the Lyapunov exponents of the system.

Remark . .
A general framework for verifying the stability / instability for an invariant subset of a Markov process was given in [Ben ] in terms of existence of suitable Lyapunov-type functions. From this point of view, our approach is much less sophisticated, as we simply look for a function V : R 3 \H → R + with compact sublevel sets and such that LV ≤ K − cV . The main difficulty in our case is to be able to actually build a Lyapunov function when α is large and to show that this construction must break down at some sufficiently low value of α.
The structure of the remainder of this article is as follows. First, in Section we perform a simple change of variables that brings ( . ) in a slightly more canonical form and we introduce some notation. In Section , we then provide a preliminary analysis of the equation which shows that it is irreducible and strong Feller on R 3 \ H, so that in particular it can have at most one additional ergodic invariant measure ν ⋆ besides ν 0 . The core of our analysis is contained in the last two sections. First, in Section , we construct a Lyapunov function which allows to reduce the N existence / non-existence of ν ⋆ to the behaviour of the invariant measure µ α for an auxiliary problem describing the behaviour of a "linearised" version of ( . ) around H. The construction of the Lyapunov function uses a philosophy similar to that used in [HM , Hai ], namely to exhibit a "fast" dynamic in the regime of interest and to use this to build a "corrector" which then allows to turn a "naïve" Lyapunov function for the effective "slow" dynamic into a proper Lyapunov function for the full system. Finally, Section analyses the behaviour of µ α as α → ∞, which allows us to conclude that ( . ) is indeed destabilised for any value of its parameters provided that α is sufficiently large. Since µ α describes a non-equilibrium system, it is not explicit and moreover has a quite complicated structure. The study of its asymptotic behaviour as α → ∞ is therefore highly non-trivial and constitutes one of the main points of this article.

Acknowledgements
MCZ gratefully acknowledges support by the Royal Society through a university research fellowship. MH gratefully acknowledges support by the Royal Society through a research professorship.

Notations
It will be convenient to write ( . ) in such a way that all of the arbitrary constants appear in the equation for Z. This will be convenient since we will be mostly interested in the regime where x 2 + y 2 ≪ 1, so that Z is close to a simple Ornstein-Uhlenbeck process. To this end, we define the constants as well as z ⋆ = 2 + χ 2 σ(̺ − 1) .
If we then perform the change of variables the system ( . ) can be rewritten aṡ We also introduce "polar coordinates" x = e r sin θ , y = e r (cos θ − sin θ) , so that one can alternatively write the equations of motion aṡ H (These coordinates are chosen in such a way that ( . ) is as simple as possible when z = 0 and can be "guessed" by looking at the explicit solution of the damped harmonic oscillator describing the (x, y) system with z = 0.) An important role will be played by the "linearisation" obtained by replacing the last equation in ( . ) by the Ornstein-Uhlenbeck procesṡ We will use the notation L for the generator of ( . ) and L 1 for the generator of the "linearised" system, namely We will also use L 0 for the generator of the (θ, z)-component of the linearised system, namely We henceforth fix the values of the constants η, γ and z ⋆ appearing in our dynamic, but we will keep track on the dependence on α.
In particular, we write µ α for the invariant measure on S 1 × R for the diffusion with generator L 0 . Such an invariant measure clearly exists by Krylov-Bogoliubov. It is also quite easy to see that it is unique as a consequence of the controllability result shown in Proposition . below and the regularity result given by Proposition . . (See Theorem . for a reference.) Remark . . A "naïve" heuristic for the stability of H goes as follows. Writing λ + (z) for the largest real part of the eigenvalues of the system (x, y) given in ( . ) (with z frozen), one can verify that λ + (z) = −1 + √ z − 11 z>1 . This then suggests that α ⋆ is the smallest value such that where the expectation is taken over the invariant measure N (z ⋆ , α 2 /(2γ)) for ( . ). While this heuristic is incorrect, it is quite accurate in practice. For example, for the parameters used in Figure , it suggests α ⋆ ≈ 27.04 while numerical simulations suggest α ⋆ ≈ 27.7.

Hypoellipticity and control
The goal of this section is to analyse irreducibility and regularity properties on R 3 \ H of our stochastic Lorenz system. We will prove the following result. Proof of Theorem . . Since the restriction of our system to H is simply an Ornstein-Uhlenbeck process, the Gaussian with variance α 2 /(2γ) centred at z ⋆ is the unique ergodic invariant measure there. On R 3 \ H, one has at most one invariant probability measure since Proposition . implies that the transition probabilities are strong Feller, while Proposition . , combined with the support theorem [SV ] implies that every point of R 3 \ H is accessible. We conclude by for example [Hai , Cor. . ].
The fact that the transition probabilities are strong Feller is contained in the following proposition. ( . ) satisfies Hörmander's condition on R 3 \ H.

Proof.
We want to show that the C ∞ -module M generated by the iterated Lie brackets of the two vector fields so that L = X 0 + 1 2 X 2 1 , is of maximal rank at every point of R 3 \ H. A simple calculation shows that their Lie bracket is given by and that furthermore, Note that span{X 1 , X 2 , X 3 } = R 3 whenever x = 0, while span{X 0 , X 1 , X 3 } = R 3 when x = 0 and y = 0, and therefore Hörmander's condition is satisfied everywhere on R 3 \ H.
The next result concerns controllability properties of the Lorenz system, and, in particular, accessibility to any point in R 3 \ H.
Proof. Since z in ( . ) can be completely controlled by h, the proof relies on finding a smooth curve z so that x 0 = (x 0 , y 0 ) andx = (x,ȳ) can be connected by a solution of the first two equations in ( . ). The proof is divided in three steps.

H
Step . We first identify a non-smooth trajectory that connects x 0 withx, as depicted in Figure . Notice that for ζ ∈ R given, the first two equations of ( . ) constitute a two-dimensional linear system, that can be written as The eigenvalues of A are The idea is now to alternate between the globally stable dynamics (when ζ < 1) for which (0, 0) is attractive, and the case in which there are a stable and an unstable manifold (when ζ > 2). We therefore choose ζ = 0, with corresponding matrix A 0 , to implement the first scenario, and ζ = 5, with corresponding matrix A 1 , to implement the second scenario. Notice that A 1 has eigenvalues with corresponding eigenvectors so that the diagonal ℓ u = {(x, y) : x = y} is the unstable manifold of the system.
x y Figure : The non-smooth trajectory x s constructed in ( . ). A few orbits of the dynamic with ζ = 0 are drawn in light blue and a few orbits with ζ = 5 are drawn in light red.
H Now, given x 0 = (x 0 , y 0 ), consider the forward solution toẋ = A 0 x with initial condition x(0) = x 0 . It is clear from ( . ) that e A 0 t x 0 → 0 as t → ∞ and that e A 0 t x 0 intersects ℓ u infinitely many times. We define The definition of the time t 0 makes sure that not only e A 0 t 0 x 0 ∈ ℓ u , but also that it is on the same ray as e −A 0tx and that enough time as passed so that e A 0 t 0 x 0 is closer to the origin than e −A 0tx . Finally, we define t 1 ≥ 0 to be the time such that Therefore, defining T := t 0 + t 1 +t, we find that the piecewise continuous curve is such that x s (0) = x 0 and x s (T ) =x.
Step . We proceed with a suitable smoothing of the trajectory constructed in step . For a small δ ∈ (0, 1), define a smooth function ζ δ such that ζ δ (0) = z 0 , ζ δ (T ) =z and We now compare the dynamics of the ODEs generated by the matrices and Moreover, the underlying dynamics is the same for the two systems, up to a time-interval of size 4δ. Since for possibly a different C, independent of δ.
Step . To conclude the proof, fix ε > 0 and choose δ in ( . ) It is then clear that (x c (t), y c (t), z c (t)) is the unique solution to ( . ) with initial datum (x 0 , y 0 , z 0 ). Moreover, z c (T ) =z and by ( . ) and our choice of δ, the proof is over.

Remark . .
A suitable modification of the above proof implies that the time T can be made arbitrarily small thanks to the fact that the eigenvalues of the 2 × 2 system can be completely controlled through the z variable. However, we will not need this enhancement for our purposes.

Construction of a Lyapunov function
We now proceed with the analysis of the "linearised" model obtained from ( . )- ( . ). Here, we are taking advantage of the fact that r does not appear in the right-hand side of ( . )- ( . ), and it is therefore completely determined once (θ, z) are. Note that by Propositions . and . , ( . ) is strong Feller and topologically irreducible. Since the state space of θ is compact and z is a simple Ornstein-Uhlenbeck process, it admits Lyapunov functions with compact sublevel sets (the map (θ, z) → |z| 2 will do) and therefore has a unique invariant measure µ α . The averaged behaviour of the right-hand side of the r equation in ( . ) turns out to characterise the invariant measure of the full Lorenz system ( . ), as the following theorem shows.
The proof of this theorem, which is given in Section . below, is based on the construction of a Lyapunov functional for the Lorenz system ( . ) that blows up as |x| 2 + |y| 2 → 0 (in the unstable case λ α > 0) and at infinity. This follows by a perturbative analysis from a similar analysis of the linearised system, which we carry out in the next section.

. Lyapunov functional for the linearised system
To show the existence of a Lyapunov functional for the linearised system ( . ) we need to analyse the regularity properties of the solution g α to the problem Notice that g α does not depend on the r variable. The following result contains all the properties needed later.

Proof.
Smoothness of g α (once we know that it exists) follows from Hörmander's theorem [Hör ], noting that Proposition . guarantees that its assumptions are verified (Hörmander's condition for L is equivalent to that of L 1 , which in turn implies it for L 0 ). Note first that we can assume without loss of generality that ε is small enough since ( . ) then holds automatically for larger values of ε. Setting we see that ( . ) is equivalent to the equation and This operator belongs to the class K 0 as defined in [EH , Def. . ], on the space integration by parts shows that, for any ψ ∈ C ∞ 0 , In particular, for any positive ε such that ε ≤ γ/α 2 , and using the bound |∂ θ b| |z|, we infer by Cauchy-Schwarz that C L which can be extended to all ψ ∈ D(K) by a simple approximation argument. By [EH , Cor . ], it follows that K has compact resolvent. Let now λ > 0 be in the resolvent set for K (such a λ exists since the spectrum is discrete) and write R λ for the resolvent so that, since R λ is a bijection between L 2 and D(K), ( . ) is equivalent to Note now that if f ∈ ker K * , thenf := e −εz 2 /2 f satisfies L * 0f = 0. Sincef ∈ L 1 , we conclude as in the proof of [EPRB , Prop. . ] that |f | is the density of an invariant measure for ( . ). Since the latter is unique and since e −εz 2 /2 ∈ ker K (so that dim ker K = dim ker K * ≥ 1), we conclude that ker K * is one-dimensional and spanned by the element f ∈ L 2 such that Since µ α (G) = 0 by definition of λ α and therefore f, Φ = 0 so that Φ ∈ (ker K * ) ⊥ , we deduce from Fredholm's alternative [Rud , Thm . ] that ( . ) admits a unique solution ϕ with the additional property that the corresponding function g α is centred with respect to µ α .
Note that it immediately follows from ( . ) that To derive similar bounds on higher derivatives, we simply take derivatives of ( . ), making sure that the corresponding commutators are well-behaved. We have In light of ( . ), we have that [∂ θ , K]ϕ is in L 2 , hence ( . ) holds also for ∂ θ ϕ.
Thus [∂ z , K]ϕ ∈ L 2 as well and the same conclusion for ∂ z ϕ follows. Proceeding iteratively, we deduce that ϕ ∈ H 3 (S 1 × R), and the conclusion follows from the embedding H 3 ⊂ W 1,∞ .
The construction and properties of the Lyapunov functional for the full linearised system ( . )- ( . ) is contained in the following proposition.
and let V 0 be given by for some constants κ ∈ R and δ > 0. Then, for every α ∈ R such that λ α = 0 there exists a choice of δ, κ with sgn κ = sgn λ α such that and such that furthermore Proof. From ( . ) and the definition of the generators ( . b)- ( . c), it follows that A simple calculation then shows that where we used the fact that ε α ≤ γ 2α 2 in order to obtain the last inequality. Note now that since the function u → e u 2 (3−u 2 ) is bounded from above by 8, we deduce that so that We now use the fact that, by Lemma . , there exists a constant c α such that We now make the choices (as well as sgn κ = sgn λ α as in the statement) so that we obtain the bound If we furthermore impose |κ| ≤ 1/c 4 α , then it follows from ( . ) that |κ||g α | ≤ |κ|c α e εαz 2 /2 ≤ 1 2 1 + δe εαz 2 , so that we do indeed have the bound ( . ). In particular, since δ ≤ 1 this implies This finally leads to the bound as required.

. Lyapunov functional for the stochastic Lorenz system
We also need the following standard result which immediately follows from the Dambis-Dubins-Schwarz representation of continuous martingales as a time-changed Brownian motion [RY , Thm V. . ].

Lemma . . Let
Then, provided that lim t→∞ M t = ∞ almost surely, one has lim t→∞ X t = −∞ almost surely.

C L
We are now ready to give the proof of Theorem . .
Proof of Theorem . . We treat the cases λ α > 0 and λ α < 0 separately. In the case λ α > 0, it suffices to find a function V : (R 2 \ {(0, 0)}) × R → R + with compact level sets and such that LV ≤ K − cV for some positive constants c and K.
For this, we first go back to the original formulation ( . ), write U = (X, Y, Z) and define the norm For anyc > 0 to be fixed, we define the functional Applying the generator L of ( . ), we see that V 1 satisfies the identity where dα ∼α −2 and K > 0 is independent ofα. In the variables (x, y, z) of system ( . ), it then follows from ( . ) that the functional satisfies for different constants K and d α , explicitly computable from ( . ). The bound ( . ) suggests the natural choice leading to the identity C L Combining ( . ) with Proposition . and the bound ( . ) yields the estimate for possibly different constants d α and K.
We then note that, as a consequence of Lemma . , there exists a constant c such that It is then immediate that in the region 3x 2 + η 2 y 2 ≤ d α /(2c), we have the bound On the other hand, in the region 3x 2 + η 2 y 2 ≥ d α /(2c), since λ α > 0 we have that V 0 e εαz 2 e 2εα(z−z⋆) 2 .
Now, comparing the above upper bound with the definitions of V 1 in ( . ) and of V 1 in ( . ) via the change of variables ( . ), it is not hard to see that if (compare with ( . )) In particular, we can find K > 0 such that Combining these bounds does indeed yield as required.
Regarding the case λ α < 0, it suffices to show that, for any realisation of the process, lim t→∞ |x 2 t + y 2 t | = 0 almost surely, which follows in particular if we can show that lim t→∞ V 0 (t) = 0. (Note that in this case κ < 0 in the definition of V 0 !) By Proposition . , we have the bound Note furthermore that since κ < 0, it follows by ( . ) that V 0 is bounded below by 1. Hence, ( . ) entails It follows that there exists a constant δ such that, as long as x 2 t + y 2 t ≤ δ, one has for some Wiener process W t . It then follows from Lemma . that there exists some ε 0 > 0 such that, uniformly over all initial conditions with x 2 0 + y 2 0 = δ/2, one has lim t→∞ log V 0 (t) = −∞ and sup t>0 (x 2 t + y 2 t ) ≤ δ with probability at least ε 0 . It furthermore follows from Proposition . and the fact that V 1 is a global Lyapunov function for our system that, for any fixed initial condition with x 2 +y 2 > δ/2, the stopping time τ = inf{t > 0 : x 2 + y 2 = δ/2} is almost surely finite and admits exponential moments. If we now split the trajectory into excursions between the cylinders {x 2 + y 2 = δ} and {x 2 + y 2 = δ/2}, a simple renewal argument shows that one has indeed lim t→∞ log V 0 (t) = −∞ almost surely, as required.

Behaviour of the angular motion for small and large α
Recall that the system under consideration is given bẏ and write as before µ α for its invariant measure (we consider γ and z ⋆ as fixed). Then, one can write λ α as Remark . . Since both ( . ) and ( . ) are invariant under θ → θ + π, we can view θ as an element of the real projective line RP 1 ≡ [− π 2 , π 2 ]/{ π 2 , − π 2 }. For fixed z > 1, the equation for θ in ( . ) then admits exactly two fixed points θ ± with The fixed point θ + is stable, while θ − is unstable with a saddle-node bifurcation at z = 1 when θ ± = π 2 . In this section, we exhibit the precise asymptotic behaviour of λ α for small and for large values of α. Since the map α → λ α is smooth, this immediately yields Theorem . when combined with Theorems . and . .

Remark . . Our proof actually shows that
B α (for any κ > 0) while we have Heuristics suggest that ( . ) should hold with exponents 1 and 2 respectively, which we expect to be optimal. It is however not clear to us what the optimal exponent in ( . ) should be.

Proof.
We start with the case α → ∞ since this is the harder one. Setting the claim follows if we can show that where µ α denotes the invariant measure for the system ( . ). This is because z ∼ N (z ⋆ , α 2 /(2γ)) under µ α and, for X ∼ N (0, 1), one has In order to show that ( . ) holds, we consider the process given by ( . ) and we define the following increasing sequence of stopping times. We set τ 0 = 0 and then inductively τ n+1 = inf{t > τ n : z(t) ∈ Z(z(τ n ))} , ( . ) where the regions Z(z) ⊂ R are defined by Our definitions guarantee that |z(τ n+1 ) − z(τ n )| ≥ 1/4, so that since z is a simple Ornstein-Uhlenbeck process (and therefore has continuous sample paths), the stopping times τ n necessarily increase to +∞. Using the explicit representation of the Ornstein-Uhlenbeck process z(t), it is furthermore straightforward to see that for every p ≥ 1 there exists a constant C such that the bound holds almost surely, uniformly over n (see Lemma . below for a much more precise upper and lower bound). Writing X = RP 1 × R for the state space of the Markov process ( . ), we define a space of excursionsX = R × C(R + , X )/ ∼, where we set (τ, u) ∼ (τ ,ū) iffτ = τ and u(t) =ū(t) ∀t ≤ τ , B α and we define a sequence of excursions E n ∈X by E n = (τ n+1 − τ n , u(τ n + • )) , ( . ) where this time u( • ) = (θ( • ), z( • )) is the Markov process defined by ( . ). Given any continuous function F : X → R, we can lift it to a functionF :X → R byF Given an excursion E, we also write τ (E) for the time τ such that E = (τ, u). Note that the sequence E n given by ( . ) is Markovian as a consequence of the Markov property of u. Writingμ α for its invariant measure onX (whose existence is shown in Lemma B. below and whose uniqueness will not be used), it is then shown in Lemma B. that, for every F : X → R with at most polynomial growth in the z direction, one hasF ∈ L 1 (μ α ) and Write nowP for the transition probabilities of the Markov chain E n onX and assume that we can find a function G α : X → R + such that, uniformly over E and over α ≥ 1, with F and F ∞ as in ( . ). This then immediately implies that If we can choose G α in such a way that furthermore then our claim follows. We claim that for every κ > 0 one can find a constant C > 0 such that, setting if |z| ≤ α 2/3 , α|z| −3/4 + α 1/3 + |z| 1/3 (|z|/α) 16 if |z| ≥ α 4/5 , α 8/15 + α 2 |z| −2 otherwise, B α the bounds ( . ) and ( . ) are satisfied. Since a simple calculation shows that G α dµ α α κ+1/3 , it only remains to show ( . ). Note that ( . ) holds provided we can show that, setting τ = inf{t > 0 : z(t) ∈ Z(z 0 )}, one has the bound where we write u = (θ, z) and set In fact, since G α has the property that G α (z) G α (z) uniformly over all z, ( . ) is implied by the same bound with G α replaced by G α . It remains to show that such a bound is indeed satisfied for all initial conditions (θ 0 , z 0 ). In order to show this, we consider a number of different regimes separately.
The case |z 0 | ≤ α 2/3 . This case is trivial since, provided that C is large enough, one has |F ∞ | + |F | ≤ G α in this region.
The case z 0 ≤ −α 4/5 . We first note that by Fernique's theorem, combined with the upper bound in Lemma . and Corollary . , we can restrict ourselves to the event where δ > 0 is any (fixed, small) exponent and • α denotes the α-Hölder seminorm. This is because the event on which ( . ) fails has probability bounded by Cα −p for any p (and a fortiori by α −2/3 ). Let N > 0 be such that |z 0 |2 −N ∈ (1/2, 1] and, for n ∈ {0, . . . , N − 1}, write D n ⊂ RP 1 for the region defined by We also set Note that one has |D n | ≈ 2 −n in the sense that the ratio between these quantities is bounded from above and below by strictly positive constants. Note that for n ∈ {1, . . . , N − 1}, D n = D + n ∪ D − n consists of a pair of intervals placed symmetrically around θ = 0, while D 0 is a single interval centred around π 2 and D N is a single interval centred around 0. In this regime, the system spins around RP 1 many times: on each full spin, we exploit cancellations on every pair {D + n , D − n }, via Lemma A. ; however, depending on the starting point and the point at which the system lies at the stopping time τ , some intervals cannot be paired and contribute of a remainder bounded by at most two full spins. This contribution will be taken care of at the end of the proof, in ( . ).

B
α Using Lemma A. , we now have everything in place to bound the contribution in this regime (here, 'of order' means bounded above and below by a fixed multiple independent of z 0 , α, n). Before time τ , the right hand side for ( . ) in D n is of order 2 −2n |z 0 | while the integrand is at most of order 2 −n |z 0 |. (Recall that F ∞ = 0 in this regime and that 2 −n |z 0 | ≥ 1/4 by our choice of N .) As a consequence, the time t n it takes to cross one of the intervals is of order t n ∼ 2 −n /F n ∼ 2 n /|z 0 |. (This also includes the intervals D 0 and D N .) Summing up these bounds shows that the time it takes to go around RP 1 once is of order |z 0 | −1/2 . We now apply Lemma A. for each of the pairs of intervals {D + n , D − n }, except that that on D − n we replace θ by −θ and run time backwards. We then have right hand sides F n , F ′ n and integrands G n , G ′ n with, as a consequence of ( . ), The total contribution over the two intervals is therefore bounded by Summing this and using the fact that N ∼ log |z 0 |, we get a total contribution per round of By ( . ), the total number of rounds in this regime is bounded by |z 0 |(|z 0 |/α) 2 α δ , so that the from the above bound we obtain a total contribution of We now distinguish two cases: if |z 0 | ≤ α, then from Lemma . the total contribution is at most of order which is consistent with ( . ) provided we choose 3δ < κ. Otherwise, if |z 0 | > α, Lemma . implies that 1 Eτ . As a consequence, from ( . ) the total contribution is ( . ) provided δ ≤ 1/12. The additional contribution coming from the fact that some intervals may not be paired up is, for each of the D n 's, of order at most G n t n 1, so the total contribution coming from this is at most which is again of the desired order.
The case −α 4/5 ≤ z 0 ≤ −α 2/3 . In this case, we only have a contribution of log |z 0 | for each round, and the expected number of rounds is bounded by |z 0 | Eτ + 1. Hence, the total contribution in this case is B α A straightforward calculation shows that one can take the constant K appearing in Proposition A. to be some constant that is independent of both α and z 0 . Since, by ( . ), one furthermore has the bound |z sin(2θ + (z)) − 2 √ z| 1 √ z in the region under consideration, we conclude that

This yields the bound
as required, it remains to bound the first term. For this, write The contribution of τ⋆ τ 1 |F ∞ (u(t)) − F (u(t))| dt is then bounded by O(log z 0 ) in exactly the same way as the contribution ( . ) considered in the regime z 0 ≤ −α 4/5 . Writing similarly τ 0 for the first time such that |θ(t) − θ − (z(t))| ≥ z −2/3 0 , we can bound the contribution from τ 0 to τ 1 by writing similarly to beforė and making use of the lower bound in Corollary A. . Since K ≈ 1 and a 0 ≈ √ z 0 as before, one has z −2/3 0 ≫ K/a 2 0 so that the assumptions of the corollary are satisfied, which shows that |τ 1 − τ 0 | log z 0 √ z 0 . The contribution of this regime is therefore bounded by log z 0 as before.
To bound the contribution up to time τ 0 , we make use of Lemma A. . Our aim is to use it in order to show that so that, as a consequence of Corollary . , we obtain α as desired. As before, we can furthermore assume that we are on the event ( . ).
In particular, as long as t < α κ z −1/2 0 , we have the bound provided that both κ and δ are sufficiently small. In order to place ourselves in the framework of Lemma A. , we setθ = θ + 1/ √ z, so that Note that there existsε sufficiently small such that if we consider times such that |z − z 0 | ≤ε|z 0 |, we are in the setting of Lemma A. with a = 2 √ z 0 and b = α/(2z Applying Lemma A. with K ≈ z 13/12 0 /α, we conclude that there exists a constant C such that which indeed implies the required bound ( . ).
The case α small and z ⋆ < 1. We now proceed to the proof of the second limit in in our statement. The methodology is the same as above, but with F ∞ = −1, different choices of G α and τ n , and the different terms are less delicate to estimate. This time, we set and we write τ n+1 = (τ n + α −2 ) ∧ inf{t > τ n : z(t) ∈ Z(z(τ n ))} .
We also write t ⋆ for the period of the ODĖ which is finite since z ⋆ < 1. By symmetry, we have independently of the initial conditionθ(0). We then break the interval [0, τ ] into chunks [t k , t k+1 ] of length t ⋆ and note that, by settingθ n to be the solution to ( . ) withθ n (t n ) = θ(t n ), one has except possibly for the last chunk which yields at most a contribution of order 1. It follows that we have τ 0 (F (θ(t), z(t)) + 1) dt 1 + α 3 4 −2 α 3/4 Eτ G α (z 0 )Eτ , as required.
The case α small and z ⋆ > 1. We use the same definitions as in the previous case and note again that we only need to consider the case |z 0 − z ⋆ | ≤ α 3/4 and τ = α −2 . This time we note that as before the reference dynamic ( . ) admits two fixed points θ ⋆ ± such that sin The claim therefore follows if we can show that, for τ = α −2 , To understand the behaviour of λ α in this case, one could heuristically think as follows: in the limit α → 0, one expects the sequence of invariant measures {µ α } α to accumulate more and more on the stable fixed point represented by the point-mass δ (θ ⋆ + ,z⋆) . From ( . ), we see that λ 0 can then be computed explicitly, leading to ( . ). Estimate ( . ) makes this heuristic precise by asserting that the system indeed spends most of the time near the stable fixed point.
The proof of this bound follows the same lines as the proof of the case α 4/5+κ ≤ z 0 ≤ α 24/23 given above, except that the various regimes are much easier to treat. Since |θ + (z(t)) − θ ⋆ + | α 3/4 and since this is a stable fixed point (at least for α sufficiently small depending on z ⋆ ), it follows immediately that once |θ(t) − θ ⋆ + | ≤ Cα 3/4 for a suitable C, this bounds holds for all subsequent times. Set τ 1 = inf{t > 0 : |θ(t) − θ ⋆ + | ≤ Cα 3/4 } and break the integral in ( . ) into a contribution up to τ 1 and a remainder. Since the integrand in the remainder is bounded by Cα 3/4 and since we are restricted to the event τ ≤ α −2 , this term is indeed bounded by O(α 3 4 −2 ) as required.
Setting as before τ 0 = inf{t > 0 : |θ(t) − θ ⋆ − | ≥ Cα 3/4 }, a very brutal bound shows that τ 1 − τ 0 α −3/4 . This is simply because the right hand side of the equation for θ is of size at least α 3/4 during that time so that it takes time at most O(α −3/4 ) to move by an order 1 distance. It therefore remains to show that a bound of the form ( . ) holds for τ 0 , for which we would like to apply Lemma A. again.
If we set similarly to beforeθ(t) = θ(t) − θ − (z(t)), then we see that it satisfies an equation of the form and where the nonlinearities G 1,2 satisfy the bounds Unfortunately, the second term appearing in the right hand side is of order α 3/4 , which is much larger than b √ a = α/z ⋆ so it appears that Lemma A. doesn't apply.
The trick is to look for a constant c such that if we set this term cancels out. A simple calculation shows that this is the case if we choose With this choice,θ then satisfies andĜ 1,2 satisfying the same bounds ( . ) as G 1,2 . We are now in the setting of Lemma A. . Since there exists a constantĈ such that |θ| >Ĉα 3/4 implies that |θ − θ ⋆ − | > Cα 3/4 , the required bound on τ 0 follows at once. The case α small and z ⋆ = 1. This is the case where the two fixed points θ ⋆ ± are merged into one. We proceed again in the same way as above with G α as in ( . ), but this time equal to Cα 1/4 for |z − z ⋆ | ≤ α 3/4 /2. Again, it suffices to consider the case τ = α −2 and |z 0 − z ⋆ | ≤ α 3/4 . Define the two intervals and write D c 1 for the complement of D 1 . The trajectory t → θ(t) can then be partitioned into intervals according to whether θ(t) ∈ D 1 (called 'slow intervals') or θ(t) ∈ D c 1 ('fast intervals'). A simple comparison with scaled translates of the ODEu = u 2 shows that it takes at most a time of order α − 1 8 to traverse a fast interval and at least a time of order α − 3 8 to go from the boundary of D 1 to that of D 0 (and therefore a fortiori also to traverse a slow interval). Since one has |F (θ, z)| 1 in D c 1 and |F (θ, z)| |θ − θ ⋆ ± | 2 α 1 4 in D 1 , we conclude that the average of F over any time interval of size at least α − 3 8 is bounded by Cα We complete this section with a bound on the stopping time τ given as in ( . ) by where z denotes the solution to ( . ) with initial condition z 0 .

B α
If |z 0 | ≥ α, we then have for some c > 0 as required. In the regime α ≥ |z 0 |, we have for some (possibly different) strictly positive constant c as required.
To show the upper bound, note first that we can assume without loss of generality that z 0 > 0. For every stopping time t ≤ τ we then have, similarly to before, It follows from the standard small ball estimates for Brownian motion [Chu , LS ] that, for all t ≤ 1, one has the bound P(τ ≥ t) ≤ C exp −c t ε 2 , for some constants c, C > 0. On the other hand, for every t ≤ τ , we also have which, by [Hal , Eq. ], implies that so that in particular, for all t ≥ 2, P(τ ≥ t) (1 ∧ ε 2 ) exp(−ε 2 t/8) .
For ε ≤ 1/2 it follows that, for a suitable c > 0, as claimed. For ε ≥ 1/2, we have where we used Lemma . to get the last bound. The claim now follows at once by choosing k sufficiently large.

Appendix A Useful bounds
The following can be verified without encountering any surprises.
Lemma A. . There exists a universal constant ε > 0 such that the following holds. Let W be a standard Wiener process, let E and C be continuous processes adapted to the filtration generated by W , and let x solve for some constants a, b > 0. Assume that, almost surely, one has |C(t) − 1| ≤ ε and |E(t)| ≤ εb √ a for all t > 0 and write τ K for the first time when |x| ≥ Kb/ √ a.
Then, the bound holds uniformly over x 0 ∈ R, K > 0 and every integer N > 0.

Proof. By consideringx(t) =
√ a b x(t/a) we can reduce ourselves to the case a = b = 1, which we assume from now on. We write It follows that for any deterministic time t > (log 2)/2 we can write ) where η is a Gaussian random variable with variance in [ 1 4 , 1 2 ] and, by Bernstein's inequality [RY , Ex. IV. . ], ζ is a random variable (correlated with η in general) with the property that P(|ζ| > 1 + M ) ≤ 2 exp(−M 2 ).
Using the same argument, combined with the fact that E and C are adapted, we obtain the almost sure bound Iterating this bound then yields the claim. Proposition A. . Let x and y be solutions tȯ for a function a such that a(t) ≥ a 0 > 0 for all t > 0 and such that |a(t)−a(s)| ≤ K whenever |t − s| ≤ 1/a 0 . There exists a universal constant C such that, setting x ⋆ (t) = 1/a(t), one has ) for all t ≥ 0. Setting y ⋆ (t) = −1/a(t), there exists a constant C such that, provided that |y(0) − y ⋆ (0)| ≥ CK/a 2 0 and that a(t) ∈ [a 0 , 2a 0 ] for all t > 0, one has |y(t) − y ⋆ (t)| ≥ |y(0) − y ⋆ (0)|e a 0 t/2 /2 .
On the other hand, one has Since |x ⋆ (0)| ≤ 1/a 0 , we conclude that (A. ) holds for t ≤ a 0 with C = 2. It then suffices to iterate this bound.
Regarding the lower bound on y, a similar calculation shows that, for t ≤ 1/a 0 , |ỹ(t)| ≥ e a 0 t |ỹ (0) Choosing C sufficiently large and iterating this bound proves the claim.
Corollary A. . Let x and y be the solutions tȯ with a as in Proposition A. and f ≥ f 0 . Then, the conclusions still holds, but with e −a 0 t and e a 0 t/2 replaced by e −a 0 f 0 t and e a 0 f 0 t/2 respectively. Proof. It suffices to perform a time change to eliminate f .

Appendix B Path decomposition
In this appendix, we justify the relation between the Markov process determined by the diffusion ( . ) and the Markov chain E n on path segments defined in ( . ). The natural topology onX (which turns it into a Polish space) is the one given by the metric d(τ, u;τ ,ū) = |τ −τ | + sup t≥0 |u(t ∧ τ ) −ū(t ∧τ )| .
. For every α > 0, the function V (E) = exp(αN (E)) is a Lyapunov function forP in the sense that there exists c < 1 and C > 0 such that . For every ε ≥ 0 and every n ≥ −1 there exists a compact set K ⊂X such thatP(E, K) ≥ 1 − ε, uniformly over all E ∈X n .
Proof. The first property follows immediately from the fact that solutions to ( . ) depend continuously on their initial condition, combined with [Bog , Cor. . . ] and the fact that their law does not charge the points of discontinuity of the map u → inf{t > 0 : |z(t)| = a} for any a = z(0). For the second property, a simple comparison with Brownian motion with drift shows that, for any fixed ε > 0 and any n large enough, one hasP(E, {N = n − 1}) ≥ 1 − ε for all E ∈X n , whence the claim follows.
The last property follows from the upper bound of Lemma . and Arzelà-Ascoli, using the fact that the 1/3-Hölder norm (say) of a Wiener process admits exponential moments over any fixed time interval.
Combining the second and third part, we conclude that the sequence {P k (E, · )} k≥1 is tight for every E ∈X , so that the existence of an invariant measureμ α forP follows from the Krylov-Bogolyubov theorem. The integrability of V is an immediate consequence of part .
Note that the invariant measure exhibited in this lemma is actually unique, but we will not use this and therefore do not give a proof. We then have the following result.
(B. ) holds almost surely. (The reason why we need to invoke the Harris chain property is that the initial condition u(0) in (B. ) is distributed according to π * 0μ α , which is singular with respect to µ α .) In particular, since lim n→∞ τ n = ∞ almost surely (take F (τ, u) = τ in (B. )), one has F (u) µ α (du) = lim