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Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt

We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in the production of non-woven textiles. Following a micro-macro decomposition, we use hypocoercivity techniques to show exponential convergence to equilibrium with an explicit rate assuming the conveyor belt moves slow enough. This work is an extension of (Dolbeault et al., 2013), where the authors consider the case of a stationary conveyor belt. Adding the movement of the belt, the global Gibbs state is not known explicitly. We thus derive a more general hypocoercivity estimate from which existence, uniqueness and exponential convergence can be derived. To treat the same class of potentials as in (Dolbeault et al., 2013), we make use of an additional weight function following the Lyapunov functional approach in (Kolb et al., 2013).


Introduction
The mathematical analysis of the fibre lay-down process in the production of non-woven textiles has seen a lot of interest in recent years [14,15,6,10,11,4,12]. Non-woven materials are produced in melt-spinning operations: hundreds of individual endless fibres are obtained by continuous extrusion through nozzles of a melted polymer. The nozzles are densely and equidistantly placed in a row at a spinning beam. The visco-elastic, slender and in-extensible fibres lay down on a moving conveyor belt to form a web, where they solidify due to cooling air streams. Before touching the conveyor belt, the fibres become entangled and form loops due to highly turbulent air flow. In [14] a general mathematical model for the fibre dynamics is presented which enables the full simulation of the process. Due to the huge amount of physical details, these simulations of the fibre spinning and lay-down usually require an extremely large computational effort and high memory storage, see [15]. Thus, a simplified two-dimensional stochastic model for the fibre lay-down process, together with its kinetic limit, is introduced in [6]. Generalisations of the two-dimensional stochastic model [6] to three dimensions have been developed by Klar et al. in [10] and to any dimension d ≥ 2 by Grothaus et al. in [7].
We now describe the model we are interested in, which comes from [6]. We track the position x(t) ∈ R 2 and the angle α(t) ∈ S 1 of the fibre at the lay-down point where it touches the conveyor belt. Interactions of neighbouring fibres are neglected. If x 0 (t) is the lay-down point in the coordinate system following the conveyor belt, then the tangent vector of the fibre is denoted by τ (α(t)) with τ (α) = (cos α, sin α). Since the extrusion of fibres happens at a constant speed, and the fibres are in-extensible, the lay-down process can be assumed to happen at constant normalised speed x 0 (t) = 1. If the conveyor belt moves with constant speed κ in direction e 1 = (1, 0), then dx dt = τ (α) + κe 1 .
Note that the speed of the conveyor belt cannot exceed the lay-down speed: 0 ≤ κ ≤ 1. The fibre dynamics in the deposition region close to the conveyor belt are dominated by the turbulent air flow. Applying this concept, the dynamics of the angle α(t) can be described by a deterministic force moving the lay-down point towards the equilibrium x = 0 and by a Brownian motion modelling the effect of the turbulent air flow. We obtain the following stochastic differential equation for the random variable X t = (x t , α t ) on R 2 × S 1 , where W t denotes a one-dimensional Wiener process, A > 0 measures its strength relative to the deterministic forcing, τ ⊥ = (− sin α, cos α), and V : R 2 −→ R is an external potential carrying information on the coiling properties of the fibre. More precisely, since a curved fibre tends back to its starting point, the change of the angle α is assumed to be proportional to τ ⊥ (α)·∇ x V (x). It has been shown in [12] that under suitable assumptions on the external potential V , the fibre lay down process (1) has a unique invariant distribution and is even geometrically ergodic (see Remark 1.2). The stochastic approach yields exponential convergence in total variation norm, however without explicit rate. We will show here that a stronger result can be obtained with a functional analysis approach. Our argument uses crucially the construction of an additional weight functional for the fibre lay-down process in the case of unbounded potential gradients inspired by [12,Proposition 3.7].
The probability density function f (t, x, α) corresponding to the stochastic process (1) is governed by the Fokker-Planck equation with diffusivity D = A 2 /2. We state below assumptions on the external potential V that will be used regularly throughout the paper: (H1) Regularity and symmetry: V ∈ C 2 (R 2 ) and V is spherically symmetric outside some ball B(0, R V ).
(H2) Normalisation: (H4) Pointwise regularity condition on the potential : there exists c 1 > 0 such that for any (H5) Behaviour at infinity: Remark 1.1. Assumptions (H2-3-4) are as stated in [4]. Assumption (H1) assumes regularity of the potential that is stronger and included in that discussed in [4] since (H1) implies V ∈ W 2,∞ loc (R 2 ). Assumption (H5) is only necessary if the potential gradient |∇ x V | is unbounded. Both bounded and unbounded potential gradients may appear depending on the physical context, and we will treat these two cases separately where necessary. A typical example for an external potential satisfying assumptions (H1-2-3-4-5) is given by for some constants K > 0 and s ≥ 1 [5,12]. The potential (3) satisfies (H3) since see for instance [17, A.19. Some criteria for Poincaré inequalities, page 135]. The other assumptions are trivially satisfied as can be checked by direct inspection. In this family of potentials, the gradient ∇ x V is bounded for s = 1 and unbounded for s > 1.
Remark 1.2. The proof of ergodicity in [12] assumes that the potential satisfies Under these assumptions, there exists an invariant distribution ν to the fibre lay-down process (1), and some constants C(x 0 ) > 0, λ > 0 such that where · T V denotes the total variation norm. The stochastic Lyapunov technique applied in [12] however does not give any information on how the constant C(x 0 ) depends on the initial position x 0 , or how the rate of convergence λ depends on the conveyor belt speed κ, the potential V and the noise strength A. This can be achieved using hypocoercivity techniques, proving convergence in a weighted L 2 -norm, which is slightly stronger than the convergence in total variation norm shown in [12]. Conceptually, the conditions (4) ensure that the potential V is driving the process back inside a compact set where the noise can be controlled. Our framework (H1-2-3-4-5) is more general than conditions (4) in some aspects (including bounded potential gradient) and more restrictive in others (assuming a Poincaré inequality). The proof in [12] relies on the strong Feller property which can be translated in some cases into a spectral gap; it also uses hypoellipticity to deduce the existence of a transition density, and concludes via an explicit Lyapunov function argument. With our framework (H1-2-3-4-5), and adapting the Lyapunov function argument presented in [12] to control the effect of κ∂ x 1 , we derive an explicit rate of convergence in terms of κ, D and V .
To set up a functional framework, rewrite (2) as where the collision operator Q := D∂ αα acts as a multiplicator in the space variable x, P κ is the perturbation introduced by the moving belt with respect to [4]: and the transport operator T is given by We consider solutions to (5) We denote by ·, · κ the corresponding scalar product and by · κ the associated norm. Here, ζ > 0 is a free parameter to be chosen later. The construction of the weight g depends on the boundedness of ∇ x V . When it is bounded, no additional weight is needed to control the perturbation, and so we simply set g ≡ 0 in that case. When the gradient is unbounded, the weight is constructed thanks to the following proposition: Proposition 1.3. Assume that V satisfies (H1) and (H5) and that lim |x|→∞ |∇ x V | = +∞.
If κ < 1/3 holds true, then there exists a function g(x, α), a constant c = c(κ, D) > 0 and a finite radius R = R(k, D, V ) > 0 such that where L κ is defined by The weight g is of the form where the parameter β > 1 and the function Γ ∈ C 1 ([−1, 1]), Γ > 0 are determined along the proof and only depend on κ.
We show in Section 3 the existence of such a weight function g under appropriate conditions following ideas from [12].
We denote C := C ∞ c R 2 × S 1 , and define the orthogonal projection Π on the set of local equilibria Ker Q and the mass M f of a given distribution f ∈ L 2 (dµ κ ), Integrating (2) over R 2 × S 1 shows that the mass of solutions of (2) is conserved over time, and standard maximum principle arguments show that it remains non-negative for non-negative initial data. The collision operator Q is symmetric and satisfies In the case κ = 0, if the entropy dissipation − Qf, f 0 was coercive with respect to the norm · 0 , exponential decay to zero would follow as t → ∞. However, such a coercivity property cannot hold since Q vanishes on the set of local equilibria. Instead, Dolbeault et al. [5] applied a strategy called hypocoercivity (as theorised in [17]) and developed by several groups in the 2000s, see for instance [9,8,13,2,3]. The full hypocoercivity analysis of the long time behaviour of solutions to this kinetic model in the case of a stationary conveyor belt, κ = 0, is completed in [4]. For technical applications in the production process of non-wovens, one is interested in a model including the movement of the conveyor belt, and our aim is to extend the results in [4] to small κ > 0.
We follow the approach of hypocoercivity for linear kinetic equations conserving mass developed in [5], with several new difficulties. Considering the case κ = 0, Q and T are closed operators on L 2 (dµ 0 ) such that Q − T generates the C 0 -semigroup e (Q−T)t on L 2 (dµ 0 ). When κ > 0, we use the additional weight function g > 0 to control the perturbative term P κ in the case of unbounded potential gradients; and show the existence of a C 0 -semigroup for L κ = Q − T + P κ (see Section 4.1). Unless otherwise specified, all computations are performed on the operator core C, and can be extended to L 2 (dµ κ ) by density arguments.
When κ = 0, the hypocoercivity result in [5,4] is based on: microscopic coercivity, which assumes that the restriction of Q to (Ker Q) ⊥ is coercive, and macroscopic coercivity, which is a spectral gap-like inequality for the operator obtained when taking a parabolic drift-diffusion limit, in other words, the restriction of T to Ker Q is coercive. The two properties are satisfied in the case of a stationary conveyor belt: • The operator Q is symmetric and the Poincaré inequality on S 1 , In the case κ = 0, the unique global normalised equilibrium distribution F 0 = e −V lies in the intersection of the null spaces of T and Q. When κ > 0, F 0 is not in the kernel of P κ and we are not able to find the global Gibbs state of (5) explicitly. However, the hypocoercivity theory is based on a priori estimates [5] that are, as we shall prove, to some extent stable under perturbation. Our main result reads: hold. For 0 < κ < 1 small enough (with a quantitative estimate) and ζ > 0 large enough (with a quantitative estimate), there exists a unique non-negative stationary state F κ ∈ L 2 (dµ κ ) with unit mass M Fκ = 1. In addition, for any solution f of (2) in L 2 (dµ κ ) with mass M f and subject to the initial condition f (t = 0) = f in , we have where the rate of convergence λ κ > 0 depends only on κ, D and V , and the constant C > 0 depends only on D and V .
In the case of a stationary conveyor belt κ = 0 considered in [4], the stationary state is characterised by the eigenpair (Λ 0 , F 0 ) with Λ 0 = 0, F 0 = e −V , and so Ker L 0 = F 0 . This means that there is an isolated eigenvalue Λ 0 = 0 and a spectral gap of size at least [−λ 0 , 0] with the rest of the spectrum Σ(L 0 ) to the left of −λ 0 in the complex plane. Adding the movement of the conveyor belt, Theorem 1.4 shows that Ker L κ = F κ and the exponential decay to equilibrium with rate λ κ corresponds to a spectral gap of size at least [−λ κ , 0]. Further, it allows to recover an explicit expression for the rate of convergence λ 0 for κ = 0 (see Step 5 in Section 2.1). In general, we are not able to compute the stationary state F κ for κ > 0 explicitly, but F κ converges to F 0 = e −V weakly as κ → 0 (see the discussion in Section 5). Let us finally emphasize that a specific contribution of our paper is to introduce two (and not one as in [5,4]) modifications of the entropy: 1) we first modify the space itself with the coercivity weight g, then 2) we change the norm with an auxiliary operator following the hypocoercivity approach.
The rest of the paper deals with the case κ > 0 and is organised as follows. In Section 2, we prove the main hypocoercivity estimate. This allows us to establish the existence of solutions to (2) using semigroup theory and to deduce the existence and uniqueness of a steady state in Section 4 by a contraction argument. In Section 3, we give a detailed definition of the weight function g that is needed for the hypocoercivity estimate in Section 2.
Note that the estimate (10) is stronger than what is required for the uniqueness of a global Gibbs state, and represents an extension of the estimate given in [4]. When applied to the difference of two solutions with the same mass, (10) gives an estimate on the exponential decay rate towards equilibrium.

Proof of Proposition 2.1
where the entropy dissipation functionals D 0 , D 1 , D 2 and D 3 are given by Note that the term LAf, f 0 vanishes since it has been shown in [5] that A = ΠA and hence Af ∈ Ker Q. Further, Tf, f = 0 since T is skew-symmetric. We estimate the entropy dissipation of the case κ = 0 as in [4]: We will show the boundedness of D 0 , which is in fact the dissipation functional for a stationary conveyor belt. We thus recall without proof in the following lemma some results from [4]. Lemma 2.2 (Dolbeault et al. [4]). The following estimates hold: In order to control the contribution ATΠf, Πf 0 in D 0 , we note that shares its spectral decomposition with (TΠ) * TΠ, and by macroscopic coercivity Hence,

Now, recalling Lemma 2.2 and using
We now turn to the entropy dissipation functional D 1 , which we will estimate using elliptic regularity. Instead of bounding AP κ , we apply an elliptic regularity strategy to its adjoint, as for AT(1 − Π) in [4]. Let f ∈ L 2 (dµ 0 ) and define h := (1 + (TΠ) * TΠ) −1 f so that u h = e V Πh satisfies We have used here the fact that in the space L 2 (dµ 0 ): Then and since the adjoint for ·, · 0 of the perturbation operator P κ is given by where in the last inequality we have used an elliptic regularity estimate. This estimate turns out to be a particular case of [4, Proposition 5 and Sections 2-3], where the positive constant C V is the same as in Lemma 2.2 reproduced from [4]. This concludes the boundedness of AP κ , Using a similar approach for the operator P * κ A, we rewrite its adjoint as where we defineh := (1 + (TΠ) * TΠ) −1 P κ f for a given f ∈ L 2 (dµ 0 ), or equivalently Multiplying by uh and integrating over R 2 , we have This inequality is a H 1 (e −V dx) → H −1 (e −V dx) elliptic regularity result. Hence, and so we conclude Combining (11) and (12), the entropy dissipation functional D 1 is bounded by Using integration by parts, we have The estimation of this term goes differently depending on the boundedness of ∇ x V .
If ∇ x V is bounded, we write where we have used f κ = f 0 , since g ≡ 0.
Assume now that |∇ x V | → ∞ as |x| → ∞. Thanks to the choice of g, we have the estimate with which is finite by (H5).

# Step 4: Estimation of D 3 [f ].
We start by recalling that this estimate is only relevant when ∇ x V is unbounded. Indeed, in the opposite case, D 3 [f ] = 0 since g ≡ 0 by definition. By the identity with L κ as defined in (7), we have Proposition 1.3 allows us to control the g-weighted L 2 -norm outside some fixed ball. More precisely, take R > 0 in (6) large enough s.t. |∇ x V | ≥ 1 for all |x| > R, then where C 4 (R) := sup |x|≤R |L κ (g) + cg|e −V .

Remark 2.3.
Observe here that one could take advantage of the growth of ∇ x V by playing with the cut-off parameter R and keeping track of min |x|≥R |∇ x V | in the negative term. It could lead to more optimal constants but we chose instead to vary the parameter ζ in front of the coercivity weight g in the measure µ κ for simplicity.

#
Step 5: Putting the four previous steps together.
Combine the previous steps into by Young's inequality with the choice b = λ 2 /γ 2 , and where we used the fact that Here, ξ(κ) is explicit, and given by since the minimum in the first term is realised when the two arguments are equal, fixing ε 1 = 2Dγ 2 / γ 2 2 + 2γ 2 + λ 2 2 . Note that this choice of ε 1 satisfies ε 1 < D and ε 1 < 1. Choosing κ small enough ensures ξ(κ) > 0. From this analysis we conclude Let us now add the control of by the norm equivalence (9). Here, we defined When ∇ x V is unbounded, (13)-(14)-(15)- (16) imply again by norm equivalence (9), and where we defined This requires ζ > 0 to be large enough, and the upper bound for κ should be chosen accordingly: In order to maximise the rate of convergence to equilibrium given κ, D and V , one can optimise γ 1 over ζ whilst respecting the above constraints.
Remark 2.4. The condition γ 1 > 0 translates into an explicit upper bound on κ. More precisely, we require ξ(κ) > κu/4 where u := ∇ x V ∞ in the case of a bounded potential gradient, and u := ζC 4 (R) otherwise. This condition is satisfied for small enough κ: which also implies ξ(κ) > 0. Recall that Proposition 1.3 requires κ < 1/3 in the case of unbounded potential gradients. These conditions provide a range of κ for which Proposition 2.1 holds.

The coercivity weight g
In this section, we define the function g in such a way that it allows us to control the loss of weight in the perturbation operator P κ . When ∇ x V is bounded, we do not need any extra weight since then we may control the perturbation thanks to the stationary weight e V , and so we set g ≡ 0 in that case. When it is not, Proposition 1.3 provides a suitable weight function g by constructive methods.

Proof of Proposition 1.3
The proof is strongly inspired from [12], however our weight is different since we work in an L 2framework rather than in an L 1 one. Assuming ∇ x V is unbounded, we seek a weight g of the form where the parameter β > 1 and the function Γ ∈ C 1 ([−1, 1]), Γ > 0 are to be determined. We define and split the proof into four steps: 1) we rewrite statement (6) using the explicit expression of the weight g, 2) we simplify the obtained expression using assumption (H5), 3) we prove the equivalent statement obtained in Step 2 by defining a suitable choice of Γ(·) and β, and 4) we demonstrate that it is indeed possible to choose suitable parameters for the calculations in Step 3 to hold, fixing explicit expressions where possible.
Applying the operator L κ defined in (7) to g, we can compute explicitly Since In order to see which Γ to choose, let us divide by |∇ x V | and denote the diffusion and transport part by Now, we can rewrite the statement of Proposition 1.3: we seek a positive constant c > 0 and a radius R > 0 such that for any α ∈ S 1 and |x| > R, To achieve this bound, note that |Y | ≤ 1 and |tran| ≤ 1 for all (x, α) ∈ R 2 × S 1 .
Further, the diffusion term diff(·) can be made arbitrarily small outside a sufficiently large ball. Indeed, and both |∇ x Y | and |∇ x (|∇ x V |)|/|∇ x V | converge to zero as |x| → ∞ by assumption (H5), and Γ is bounded. In other words, using the fact that the potential gradient is unbounded, it remains to show that we can find constants γ > κβ > 0 and a radius r 1 > 0 such that Then we can choose r 2 > 0 such that and we conclude for the statement of Proposition 1.3 with R := max{r 1 , r 2 } and c := (γ−κβ)/2 > 0.
Putting these estimates together, we obtain for |x| > r 1 : Step 4: Choice of parameters.

Existence and uniqueness of a steady state 4.1 Existence of a C 0 -semigroup
The proof of the next theorem relies on the a priori estimates from Section 2.
Theorem 4.1. The linear operator L κ : Proof. Let us denote by L * κ the adjoint of L κ in L 2 (dµ κ ). Both domains D (L κ ) and D (L * κ ) contain the core C and are dense. The operator L κ is closable in L 2 (dµ κ ). To see this, take a sequence (f n ) n∈N ∈ D(L κ ) converging to zero in L 2 (dµ κ ) such that the sequence (L κ f n ) n∈N converges to some limit h ∈ L 2 (dµ κ ). Then for any test function ϕ ∈ C, ϕ , L κ f n κ = L * κ ϕ , f n κ → 0 as n → ∞ .
Since the left-hand side converges to ϕ , h κ for all ϕ ∈ C, we conclude h ≡ 0 a.e., and so L κ is closable. Similarly, L * κ is closable. We denote byL κ andL * κ some closed extensions of L κ and L * κ , respectively. Lumer-Phillips Theorem in the form [16,Corollary 4.4] states that an operatorL generates a C 0 -semigroup ifL is closed and bothL andL * are dissipative. Since the core C is dense in both D(Q κ ) and D(Q * κ ), which in turn are both dense in L 2 (dµ κ ), then for any constant C > 0, L κ − CId is dissipative if and only ifL * κ − CId is dissipative. Therefore, it remains to show that L κ − CId is dissipative for some C > 0. Since the restriction ofL κ to C is L κ , it is enough to prove that L κ − CId is dissipative on C for some constant C > 0. The estimates in Section 2 show that there exists C > 0 s.t.
∀f ∈ C, L κ f , f κ ≤ C f 2 κ for some explicit constant C > 0, which concludes the proof.

Proof of Theorem 1.4
Proposition 2.1 is the key ingredient to deduce existence of a unique steady state. The set is convex and bounded in L 2 (dµ κ ) by the norm equivalence (9). By Theorem 4.1, the operator L κ generates a C 0 -semigroup (S t ) t≥0 . Then let us show that B is invariant under the action of (S t ) t≥0 . Integrating in time the hypocoercivity estimate (10) in Proposition 2.1 for any f in ∈ L 2 (dµ κ ) with mass 1, we obtain the bound and thus Since in addition, (S t ) t≥0 conserves mass and positivity, we conclude S t (B) ⊂ B for all times.
Integrate again the hypocoercivity estimate (10) in Proposition 2.1, now for the difference of two solutions with same mass, to get for any t > 0 and f, h ∈ B. It follows by Banach's fixed-point theorem that there exists a unique u t ∈ B such that S t (u t ) = u t for each t > 0. Let t n := 2 −n , n ∈ N, and u n := u tn . Then S 2 −n (u n ) = u n , and by repeatedly applying the semigroup property, Let us prove that B is weakly compact in L 2 (dµ κ ). Consider a sequence (f n ) n∈N ∈ B. It has a cluster point f for the weak convergence since B is bounded in L 2 (dµ κ ), and the corresponding subsequence is still denoted f n for simplicity. By lower semi-continuity of the equivalent norm G: Further, since f n ≥ 0 for all n ∈ N, it follows that f ≥ 0 (the set of non-negative functions is a strongly closed convex set, hence weakly closed). It remains to show that the limit f has mass 1 by preventing loss of mass at infinity. Use Cauchy-Schwarz's inequality and the norm equivalence (9) to get for r > 0 This shows that sup n∈N |x|>r Together with M fn = 1 for all n ∈ N, it follows that M f = 1. Hence f ∈ B. The weak compactness of B implies the existence of a subsequence u n j of u n and a function u ∈ B such that u n j converges weakly to u in L 2 (dµ κ ). Letting n j → ∞ in (21) implies that (since S t is a continuous operator) ∀ m ∈ N, ∀ k ∈ N, S k2 −m (u) = u .
Finally the density of the dyadic rationals {k2 −m : k ∈ N, m ∈ N} in (0, +∞) and continuity of S t (u) in t for all u ∈ B imply that ∀ t ≥ 0, S t (u) = u .
This shows the existence and uniqueness of a global stationary state F κ := u ∈ B.
To complete the proof of Theorem 1.4, we apply the hypocoercivity estimate Proposition 2.1 to the difference between a solution f ∈ L 2 (dµ κ ) and the unique stationary state of the same mass, M f F κ , to show exponential convergence to equilibrium in · κ : first of all, we deduce from the contraction estimate (10) that which allows then to estimate the difference to equilibrium in the L 2 (dµ κ )-norm. Indeed, by norm equivalence, we obtain Hence, we obtain (8) with rate of convergence λ κ := γ 1 /2.

Concluding remarks
From our previous estimates, we have that G(F k ) is uniformly bounded in κ for κ sufficiently small. As a consequence, (F κ ) κ>0 is a relatively weakly compact family in L 2 (dµ κ ), and by uniqueness of the stationary state in the case κ = 0, we deduce that F κ → F 0 as κ → 0. It could also be proved with further work that the optimal (spectral gap) relaxation rate is continuous as κ → 0. Working in L 2 (dµ κ ) ⊂ L 2 (dµ 0 ) we are treating the operator L κ as a small perturbation of the case κ = 0 with stationary conveyor belt. The natural space to investigate the convergence to F κ in the case κ > 0 however is L 2 F −1 κ dx dα . In this L 2 -space the transport operator T − P κ is not skew-symmetric and the collision operator Q is not self-adjoint, so the hypocoercivity method [5] cannot be applied. To get around this, one can split the operator L κ differently into a transport and a collision part following the approach in [1]. More precisely, we can write L κ =Q −T where Then in L 2 F −1 κ dx dα the operatorQ is symmetric and negative semi-definite, and the operatorT is skew-symmetric. Furthermore, the stationary state F κ lies in the intersection of the kernels of the collision and transport operators, i.e. F κ ∈ KerQ ∩ KerT. The hypocoercivity approach requires microscopic and macroscopic coercivity ofQ andT. Which then requires as in [1] to control the behaviour of the stationary state at infinity, i.e. for large enough |x|, for some constants σ 1 , σ 2 > 0. If true, this would be an important physical information on the stationary state, but we do not know how to prove it at now. Even with this information at hand, this approach requires that the existence of the stationary state is known a priori. The rate of convergence one obtains in this case may be different from the rate obtained here, and it is not clear which method yields the better rate as both are most likely not optimal.