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A Multiscale-Analysis of Stochastic Bistable Reaction-Diffusion Equations

A multiscale analysis of 1D stochastic bistable reaction-diffusion equations with additive noise is carried out w.r.t. travelling waves within the variational approach to stochastic partial differential equations. It is shown with explicit error estimates on appropriate function spaces that up to lower order w.r.t. the noise amplitude, the solution can be decomposed into the orthogonal sum of a travelling wave moving with random speed and into Gaussian fluctuations. A stochastic differential equation describing the speed of the travelling wave and a linear stochastic partial differential equation describing the fluctuations are derived in terms of the coefficients. Our results extend corresponding results obtained for stochastic neural field equations to the present class of stochastic dynamics.


Introduction
Consider the reaction-diffusion equation x)) , t > 0 , x ∈ R for strictly positive constants ν, b > 0 and bistable reaction term (A1) f (0) = f (a) = f (1) = 0 for some a ∈ (0, 1) f (x) < 0 for x ∈ (0, a) , f (x) > 0 for x ∈ (a, 1) It is well-known that under the above assumptions (1.1) admits a travelling wave solution, i.e. a monotone increasing C 2 -functionv, connecting the stable fixed points 0 and 1 of the reaction term, satisfying cv x = νv xx + bf (v) (1.2) for some wavespeed c ∈ R and boundary conditionsv(−∞) = 0,v(+∞) = 1, see, e.g. Theorem 12 in [6]. It is easy to see thatv(· + ct) and all spatial translatesv(· + x 0 + ct) are solutions of (1.1). In the particular case of the cubic nonlinearity f (v) = v(1 − v)(v − a), equation (1.1) reduces to the well-known Nagumo equation (cf. [14]), for which the travelling wave is explicitly given bŷ propagating along the axis with speed c = √ 2νb ( 1 2 − a). It can be observed in numerical simulations that the travelling wave solution persists up to apparently lower order fluctuations also under the impact of noisy perturbations to (1.1). The term "lower-order" here is to be understood w.r.t. the order of the noise amplitude and will be made precise in our subsequent analysis. Even more holds for small noise amplitude: similar to the deterministic case, the solution of (1.1) will converge quickly to some profile of the type (1.3) v(t, x) =v(x + γ(t)) + u(t, x) , where γ(t) is the random position of the wave front and u denotes lower order fluctuations. For this reason we will call the solution v also a stochastic travelling wave. It is the main purpose of this paper to rigorously derive a decomposition of type (1.3) in the case of noisy perturbations induced by function-space valued additive Wiener noise (see e.g. [2]), together with an identification of γ (resp. u) as solution of a stochastic differential equation (resp. linear stochastic partial differential equation) accompanied with explicit error estimates in terms of u in appropriate function spaces. More specifically, we consider the stochastic reaction-diffusion equation dv(t, x) = ν∂ 2 xx v(t, x) dt + bf (v(t, x)) dt + ε dW (t, x), t > 0, x ∈ R (1. 4) where W (t), t ∈ [0, T ], denotes a Q-Wiener process, taking values on a suitable Hilbert space H and ε > 0 will be considered to be a small parameter.
The problem is obviously connected with the stability properties of the travelling wave solutionv and the latter problem has been intensively studied in the deterministic setting for a long time, mainly based on maximum principle and comparison techniques (cf. in particular [5]), and on spectral considerations (cf. [4,7] and the more recent monograph [3]). Both approaches however are not easy to carry over to the stochastic case, or even worse, require unnatural monotonicity conditions on the noise terms. Instead, we rather apply a pathwise stability analysis in the spirit of the classical Lyapunov approach to dynamical systems, first developed in [17,18].
Note that decomposition (1.3) is not well-posed without further assumptions, since it involves two unknowns: the position γ(t) of the wave front and the remainder u(t, x). In addition, there are various possibilities to define the position of the wave front. Since pointwise criteria like, e.g. level sets of v(t, x) do not make much sense in the stochastic case, because of (spatially non-monotone) fluctuations, we take a minimiser γ(t) := argmin γ∈R v(t, ·) −v(· + γ) of the distance between v(t, ·) and the spatial translates ofv as the (in general non-unique) definition of the position of the wave-front.
Since global minima are difficult to characterise and difficult to handle in the context of dynamical equations, we will require only the necessary condition v(t, ·) −v(· + γ(t)),v x (· + γ(t)) = 0 for local minima in the decomposition (1.3). Here, ·, · denotes the scalar product of the underlying Hilbert space H. In [8], the authors used this approach and obtained results on the corresponding decomposition up to the first random time τ , when the local minimum becomes a saddle point. Apart from the necessity of introducing the above mentioned stopping time, the position of the wave front will be a semi-martingale only, in particular non-differentiable. In addition, the decomposition obtained in this way will be non-explicit w.r.t. the small parameter ε.
To avoid non-differentiability we will consider in this paper the following (differentiable) approximation γ m (t), defined as the solution of the following pathwise ordinary differential equatioṅ for suitable initial condition and a priori chosen relaxation parameter m > 0. We will then show in Theorem 3.3 as our first main result that γ m (t) admits the decomposition and c m 0 is the unique strong solution of the stochastic ordinary differential equation where u m 0 is the unique variational solution of the (affine) linear stochastic partial differential equation To compare our results to the analysis in [8], based on the variational characterisation of local minima of v(t, ·) −v(· + γ) , we also consider the asymptotics of our decomposition for m → ∞, i.e. in the limit of immediate relaxation. We will show in Lemma 3.4 below that both processes γ m (t) and u m 0 (t) converge as m → ∞ and we will identify the limiting processes again as solutions of stochastic (partial) differential equation together with explicit error estimates in the associated decomposition (1.3) (see Theorem 3.5).

Stochastic Reaction-Diffusion Equations in Weighted Sobolev Spaces
2.1. Realisation as stochastic evolution equation. Thinking of a typical trajectory of a stochastically perturbed travelling wave one cannot expect the solution v of (1.4) to take values in L 2 (R), which raises the question in which sense and especially on which function space to model the above reaction-diffusion equation. If we assume the noise to be on L 2 (R), what is true however, is that the difference between the solution v and a deterministic reference profilev takes values in L 2 (R). This motivates us to derive a decomposition of v into v = u +v and investigate properties of the difference u under a smallness condition on u 0 := v 0 −v. This approach to study the dynamics of stochastic travelling wave solutions has also been used in [1], [17], [10] and [11]. In the stability analysis below it turns out, that controlling u in L 2 (R) with respect to the Lebesgue measure is not sufficient, but that we need an additional control with respect to the measure with a positive constant Z > 0. This measure is naturally associated to the equation and will be derived and motivated in Section 2.3. Assume (W (t)) t∈[0,T ] to be a Q-Wiener process on the space This SPDE can now be formulated as a stochastic evolution equation on L 2 (1 + ρ). Following the approach of [16] we choose the Gelfand triple with the norms on H 1 (1 + ρ).
Define the operator ∆ : In this terminology (2.1) corresponds to the stochastic evolution equation on L 2 (1 + ρ). To prove well-posedness of this equation we impose the following additional assumptions on the global behaviour of the reaction term f . We assume that the derivative f ′ is bounded from above by that there exists a finite positive constant L such that which is typically satisfied for polynomials of third degree with leading negative coefficient and that there exists η 2 such that Furthermore, to ensure even higher regularity of solutions of (2.2), namely weak spatial differentiability, we demand the following growth condition of the derivative f ′ : There exists a constant η 3 > 0 with For simplicity of the following analysis we also assume This condition on the input parameter f or, in other words, on the associated potential F = f dv implies that the wave speed c, as part of the travelling wave solutionv, is nonnegative. To briefly illustrate this relation we multiply (1.2) byv x and integrate over the real line to obtain due to the boundary conditions ofv at ±∞. This shows that the wave speed c and the integral on the right-hand side have the same sign. Assumption (A1) is required in our analysis only to ensure that ̺ is monotone decreasing. Dropping this assumption would require to consider both cases, monotone decreasing (resp. increasing).
Under the above conditions the following existence and uniqueness result can be stated Here the Bochner integrals are defined in H 1, * (1 + ρ). Moreover, there exists a constant and C(T, ω, u 0 2 1+ρ ) < ∞ for almost all ω ∈ Ω.
Proof. We show that the coefficients in (2.3) satisfy the conditions of Theorem 1.1 in [12]. Let u, w ∈ H 1 (1 + ρ). G can be realised as a continuous mapping H 1 (1 + ρ) → H 1, * (1 + ρ): Using property (B2) and the elementary estimate u ∞ ≤ u H 1 (1+ρ) one obtains The drift and diffusion operators are clearly hemicontinuous. For showing coercivity we estimate 2 for all x, y ∈ R using (B1). Weak monotonicity of the linear part of the drift operator is covered by the coercivity condition. For the nonlinear part with the one-sided Lipschitz estimate from above one obtains for u 1 , u 2 ∈ H 1 (1 + ρ) Furthermore, G is of admissible growth: Let w ∈ H 1 (1 + ρ) with w H 1 (1+ρ) ≤ 1. Applying condition (B2) yields ) For the second part of the statement an application of Itô 's formula as stated in [16,Theorem 4.2.5] together with the coercivity of the drift yields for all t ∈ [0, T ] where M (t), t ∈ [0, T ], denotes the martingale part and · L2(L 2 (1+ρ)) the Hilbert-Schmidt norm on L 2 (1 + ρ). In particular, setting θ := 2 ν + c 2 2ν + bη 1 this implies such that Gronwall's lemma allows us to bound the expectation of the left-hand side by Taking first the supremum and then the expectation of all terms in (2.5) and inserting (2.6) we obtain the estimate A control of the last expectation in terms of the martingale's quadratic variation [M ] t , t ∈ [0, T ], is provided by the Burkholder-Davis-Gundy inequality Summarizing the above estimates, there exists a positive constant C(T, u 0 2 1+ρ ) such that and C(T, ω, u 0 2 1+ρ ) < ∞ for P -almost every ω ∈ Ω. From the above decomposition it follows that v := u +v is a solution of the original reactiondiffusion equation (1.4).

Continuity of the solution in
We will now show that, assuming higher spatial regularity of the noise as well as an additional growth condition of f ′ , the unique solution u of (2.3) is even continuous with values in H 1 (1 + ρ).
Remark 2.3. In the following analysis we will frequently consider translations of the measure ρ w.r.t. deterministic as well as stochastic shifts, for which reason the following two properties of ρ will become important: (i) ρ is decreasing, i.e. ρ(x + y) ≤ ρ(x) for all y > 0.
(ii) ρ is of (at most) exponential growth, i.e., ρ(x − ξ) ≤ e M|ξ| ρ(x) for M = c ν and all ξ ∈ R . Lemma 2.4. Assume (B4). Let p t = e tν∆ , t ≥ 0, denote the semigroup generated by ν∆. Then there exists a finite constant C = C(T, v x ∞ ) such that for all u ∈ H 1 (1 + ρ) and for all s ≤ t the following bound holds: It is well-known that ν∆ generates the Gaussian semigroup Applying it to the measure 1 + ρ we obtain With this at hand we are able to formulate the following regularity statement: from Theorem 2.1. Using the semigroup p t = e tν∆ , t ≥ 0, u allows for the mild solution representation We will show that indeed u is differentiable in space and the gradient takes values in L 2 (1 + ρ): =: I + II + III, say. Clearly, For uniformly bounding III note that for a Q-Wiener process W on and thus The bound (2.4) is now applied to reduce the second term to for constants C 2,1 = C 2,1 (T, ω, u 0 1+ρ )) and C 2,2 = C 2,2 (T, ω, u 0 1+ρ ). Altogether, this leads to for a positive constantC. The previous P -nullsets on whichC is possibly infinite do not depend on t such that we even have To prove continuity it remains to show that u Considering clearly I and III have the required regularity. For the second term note that with Lemma 2.4 for all s ≤ t the integrand takes values in L 2 (1 + ρ) such that II is a continuous process on this space.
The objective of the following sections is to more thoroughly study the behaviour of u, providing information about the deviation of the stochastic solution from the deterministic wave. We will identify a suitable stochastic phase of the chosen reference profilev and subsequently quantify the remaining fluctuations around this profile in terms of the noise strength ε. This first-order stability analysis with respect to ε follows closely the methods and ideas developed in [11] (and partially also in [10]) for the study of stochastic neural field equations. In the case of bistable reactiondiffusion equations the involved diffusion and reaction operators however are no longer bounded, such that an extended notion of solution as well as refined estimates have to be introduced. To have the flexibility of also working in the larger spaces L 2 (ρ) and L 2 (R) we introduce a second Gelfand triple 2.3. The frozen-wave setting. Since our aim is to study the dynamics of solutions whose initial profile is already close to a travelling wave solution, we linearise the nonlinearity f around the travelling wavev and obtain the representation and the nonlinear remainder To identify the dynamical equations for the wave-speed for the stochastic travelling wave, as well as for the remaining fluctuations, it will be useful to introduce the frozen-wave operator L # : V → V * given by Note that by (1.2) we have L #v x = 0. Likewise, one can easily check that an eigenvector Ψ of the adjoint operator corresponding to the eigenvalue 0 is given by Ψ(x) = e − c ν xv x (x). Ψ is strictly positive and we assume the following regularity property: Indeed, this regularity can be shown under the following additional assumption on the reaction function f , namely saying that f is strictly convex in a small neighbourhood around 0 and strictly concave in a small neighbourhood around 1. The main implication of this additional assumption is then formulated in the next Lemma.
Proof. We will first show that lim |x|→∞ e − c ν xv2 In particular, 2ν xv x and note that is strictly positive (resp. negative) for x ↓ −∞ (resp. x ↑ +∞). According to the previous part of the proof, lim |x|→∞ w 2 (x) = 0, hence vx is strictly increasing at ±∞. To prove the remaining identities for γ ± observe that by l'Hospital's rule we obtain that . Since γ − < 0 we obtain the assertion. γ + can be computed similarly. Proof. We have The previous Lemma implies that x < ∞ which implies that Combining (2.12) and (2.13) we obtain that e −2 c ν xv2 x dx < ∞. Finally, The above supremum can be bounded by which is finite again by the previous Lemma 2.7.
We normalise Ψ such that Ψ,v x = 1. As mentioned earlier, in the following stability analysis it will turn out to be of advantage to work in weighted measure spaces instead of the unweighted choices of V and H. A natural choice for such a measure is where Z denotes the normalising constant. One reason for ρ to be a natural choice is that in the space L 2 (ρ) the frozen-wave operator L # separatesv x , which due to the identity ∂ tv (x + ct) = cv x (x + ct) is the direction of movement of the wave, from its orthogonal complement v ⊥ x in the following sense: Note that for u ∈ H 2 (R) this is the usual orthogonality in L 2 (ρ), i.e. L # (H 2 (R)) ⊂v ⊥ x in L 2 (ρ). Due to the fact thatv x is a zero-eigenvector of L # perturbations in that direction lead to a random phase shift in the dynamics. To also bound the spread of perturbations in the shape of the wave profile (see Theorem 3.3) we need to control the behaviour of the dynamics in directions orthogonal tov x , more precisely we assume the following contraction property i.e. the flow generated by the frozen-wave operator is contracting on the orthogonal complement ofv x in L 2 (ρ).
Alternatively, this abstract assumption can again be replaced by the additional assumption (A2) on the reaction term f . We will prove the contraction property (C1) in analogy to [18,Theorem 1.5], where a similar spectral gap inequality for the (unfrozen) operator in the unweighted space L 2 (R) has been shown. Proof. For u ∈ C 2 c (R) we write u = hv x . Due to the identity L #v x = 0 we obtain and the associated quadratic form Rewriting also (C1) in terms of u = hv x and setting w(x) =v x (x) e − c 2ν x our aim is to prove an inequality of the type Recall from the proof of Lemma 2.7 that x dy e −(2 c ν −γ−)xv2 x dy < ∞ and e − c ν xv2 x is locally bounded from below we can find a finite constant K − such that x dy e −(2 c ν −γ+)xv2 x dy < ∞ and e − c ν xv2 x is locally bounded from below, we can also find a finite constant K + such that x for all x ≥ 0 .

Estimate (2.14) now implies for
and similarly, using (2.15), Combining both estimates we obtain the weighted Hardy type inequality for any h ∈ C 1 b (R) with h(0) = 0. For a general h ∈ C 1 b (R) centering allows us to derive the inequality We introduce the normalising constant W = w 2 dx and the normalised measurew 2 = W −1 w 2 to further estimate the above right hand side by Altogether, this yields the desired Poincare inequality Lemma 2.11. Under Assumption 2.9 L # generates a contraction semigroup (P # t ) t≥0 onv ⊥ x ⊂ L 2 (ρ) satisfying Proof. Note that (L # , C 2 c (R)) is symmetric on L 2 (ρ) and the subspacev ⊥ x is invariant under L # , i.e. for u ∈v ⊥ x we have L # u,v x ρ = u, L #, * Ψ = 0 and thus L # u ∈v ⊥ x . By Assumption 2.9, the operator L # is bounded from above by L # u, u ρ ≤ −κ u 2 ρ on the orthogonal complementv ⊥ x . It follows that L # is essentially self-adjoint and its Friedrichs extension (L # , D(L # )) generates a symmetric C 0 -semigroup (P # t ) t∈[0,T ] onv ⊥ x (see [9]). It is easy to see that Assumption 2.9 extends to all u ∈ D(L # ). In particular, for all u ∈ D(L # ) and subsequently for all u ∈v ⊥ x . We would like to highlight that this contraction property is only needed for the asymptotic second moment estimate in Section 3.4 but not for the main multiscale decompositions in Theorem 3.3 and Theorem 3.5. A similar contraction property can in general not be expected to hold true for the non-autonomous linearisation (L t ) given in (2.9). But yet we will show that this family of linear operators generates an evolution family on H 1 (1 + ρ) facilitating a mild solution representation of u on this space. The family of linear operators (L t ), t ∈ [0, T ], can be seen as operators from for a constant L * > 0.
Proof. We first show that the Gaussian semigroup p t = e tν∆ , t ≥ 0, acts on H 1 (1 + ρ): For u ∈ H 1 (1 + ρ) using the symmetry of the semigroup. Together with (2.7) this enables us to obtain the bounds as well as   Conditions for variational solutions to satisfy a mild solution representation are stated in [16].

Dynamical equation for phase-adaptation.
To establish a description of the noiseinduced phase shift of the wave profile arising due to the nonlinearity of the system the idea is to determine the stochastic phase by dynamically matching the deterministic profilev with the stochastic solution v. This matching is achieved by minimising the L 2 -distance over all possible phases C. Again, it turns out to be of advantage to work in the weighted measure space L 2 (ρ), where the measure will now be moved with the wave. The following dynamics are designed to point in the direction of the negative gradient of (3.1). Let m > 0 and consider the (pathwise) ODEĊ 2) can be regarded as an alternative approach to the phase conditions specified by certain algebraic constraints in the classical stability analysis (refer to [7]) and has also been introduced in [10], [11], [13] and [18]. Proof. With analogous arguments as in [18] and [10] It then suffices to show that B(t, C) is Lipschitz continuous in C with a Lipschitz constant independent of t: For C 1 , C 2 ∈ R, t > 0 Therefore, since u is adapted and in L 0 (Ω; C([0, T ]; H)), there exists a unique adapted process C ∈ L 0 (Ω; C 1 ([0, T ])) that solves (3.2).
We now derive an SDE for the stochastic perturbation c m (t) :=Ċ m (t) of the wave speed. Note that while the phase adaptation C m (t) is a process of bounded variation, the resulting adapted wave speed is of unbounded variation.   In order to investigate the dynamics on different scales of the noise strength we first formally identify the highest order terms in (3.5) as well as (3.4). Expecting that both C m and u m are of order ε leads us to define c m 0 to be the unique strong solution of dc m ). This regularity holds true since u m 0 has the mild solution representation We define the first order phase adaptation by C m 0 (t) = t 0 c m 0 (s) ds and the first order phase by γ m 0 (t) = ct + εC m 0 (t). For ε > 0 and q ∈ [0, 1] set where u is the solution from Proposition 2.5 and τ m q,ε = inf{t ∈ [0, T ] : |C m 0 (t)| ≥ ε −q } Theorem 3.3. Let q < 1 2 . On {τ q,ε ∧ τ m q,ε = T } the stochastic travelling wave v can be decomposed into v(t) =v(· + ct + εC m 0 (t)) + εu m 0 (t) + εr m (t) with r m T ≤ α(T )ε 1−2q where the constant α(T ) is independent of m and ε. Moreover, Proof. Letũ m 0 (t) := v(t) −v(· + γ m 0 (t)) = u(t) +v(· + ct) −v(· + γ m 0 (t)). By Taylor's formula there exists is a variational solution of the following pathwise evolution equation: dt with r m (0) = 0. Thus, it can be represented as a mild solution Using (2.17) and Remark 2.3 (i) we estimate The first part is bounded by applying condition (B3) as follows: Note that even though f ′′ and f ′′′ are not assumed to be globally bounded, we can control the above expression using local bounds on {τ q,ε ∧ τ m q,ε = T }. Sincev ∈ [0, 1] and |ξ(t, x)| ≤ |ũ m 0 (t, x)| ≤ ũ m 0 (t) H 1 (1+ρt) ≤ C 1 ε 1−q < 1 for ε small enough, we know thatv(x) +ũ m 0 (t, y) ∈ [−1, 2] for all t > 0, x, y ∈ R, and therefore The second part can be controlled by Thus, we obtain By Taylor's theorem there exists ξ(t, x) with |ξ| ≤ ε|C m 0 | such that the order of the first summand can be estimated by For estimating the order of the second summand one needs to control also higher derivatives of R m 3 : for a constant C 4 > 0 and for a constant C > 0 independent of ε. It remains to show that in the small-noise limit the above order estimate holds for P -almost all paths ω ∈ Ω. If τ q,ε < T then, due to continuity, for t 0 =: τ q,ε (ω) we obtain )v x (· + ct 0 + ξ(t 0 )) + u m 0 (t 0 ) H 1 (1+ρt 0 ) + α(T )ε 2−2q and therefore, using Markov's inequality Likewise, the second stopping time converges as follows 3.3. Immediate relaxation. From the definition of the stochastic phase adaptation process C m it is clear that the initial goal of minimising the L 2 (ρ)-distance between v andv for every time t ∈ [0, T ] can only approximately be achieved when choosing a finite relaxation rate m. Although Theorem 3.3 shows that already for finite m a multiscale decomposition into processes of the expected order can be installed, we are interested in investigating the case of so-called immediate relaxation, i.e. the limit m → ∞, and show that in that case indeed a rigorous minimisation (on relevant orders of the noise strength) is achieved. As an alternative to this description of a stochastic phase one could try to adapt the analysis developed in [8] for a generalised framework of neural field equations, where the dynamics of the local minimum of (3.1) are explicitly described by an SDE up to a certain stopping time. This approach does not incorporate a gradient-descent procedure. As pointed out by the authors, studying instead the behaviour of the global minimum of (3.1) would be much more complicated, since its dynamics will be highly discontinuous. To our knowledge, this has not been investigated for bistable reaction-diffusion equations. Below we will adapt the methods from [11]. It will turn out that the limit phase adaptation is a process of unbounded variation behaving almost like a Brownian motion, which is in accordance with the phenomenological description of stochastic travelling waves in nonlinear systems developed in [1] for stochastic neural field equations.
Lemma 3.4. Define the processes C 0 and u 0 as Then for any δ > 0 almost surely With this, the difference between the approximate relaxation and the immediate relaxation process is given by Thus, where Γ denotes the Gamma function Note that since L tvx (· + ct) = 0 we have P t,svx (· + cs) =v x (· + ct). Thus, the mild solution representation of the first-order fluctuations u m 0 is given by Indeed, in contrast to [8] where the existence of a phase-adaptation process has been shown up to a stopping time, Lemma 3.4 provides us with effective formulae for the first-order stochastic phase-adaptation and fluctuations for all times t ∈ [0, T ]. We now show that passing over to the limit from finite to immediate phase relaxation preserves the previous multiscale decomposition (Theorem 3.3).
with analogous arguments as in Theorem 3.3.

3.4.
Statistical and geometrical properties of first-order phase adaptation and of fluctuations. We would like to compare our stochastic phase adaptation C 0 to the phase description obtained in [1], where the phenomenon of stochastic wave propagation has been (formally) investigated for (nonlocal) stochastic neural field equations. They concluded that to first order of the noise strength the stochastic perturbation of the phase is a Brownian motion. Taking a look at the variance of the immediate phase adaptation we see that for t > 0 Var(C 0 (t)) = Var showing that C 0 is roughly diffusive if Q is "almost" translation invariant. Note that strict translation invariance is excluded since Q is of finite trace. Thus, the above statement can only be an heuristic approximative description. Our first-order fluctuations are indeed orthogonal to the direction of movement of the wave: For t > 0 u 0 (t),v x (· + ct) ρt = P t,0 Π 0 η, Ψ(· + ct) + t 0 P t,s Π s dW (s), Ψ(· + ct) = Π 0 η, P * t,0 Ψ(· + ct) + t 0 P * t,s Ψ(· + ct), Π s dW (s) (3.10) = Π 0 η, Ψ + t 0 Ψ(· + cs), Π s dW (s) = 0 .

Minimisation.
It is still open to verify that the above choice of C 0 indeed realises the declared objective of minimising the distance between the stochastic wave v and all possible translations of the deterministic profilev, thus offering an apt description for a stochastic phase. Since all relevant dynamics have been considered on a scale of order ε, it is natural to also investigate the minimisation property on this scale.