A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear Sensitivity

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We show that for any suitable initial data there exists a pathwise unique, global solution to the SPDE. Furthermore we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a H\"older-Besov space of positive regularity, which the solution law converges to exponentially fast. We also establish tail bounds on the invariant measure that are heavier than Gaussian when measured using any $L^p$ norm.


Introduction
We study the Cauchy problem and establish exponential ergodicity for the SPDE on T, u| t=0 = ζ, on T, where ξ is a space-time white noise, T := R/Z is the one dimensional torus with unit volume and χ > 0 is a positive constant. The spatial average,ū := u, 1 L 2 (T) , is equal to the 0 th Fourier mode and we will take ζ to be a Hölder-Besov distribution of possibly negative regularity. Without loss of generality we assume that the solution to the second equation of (1.1) is mean free.
Without noise, (1.1) is a Keller-Segel model of chemorepulsion with nonlinear sensitivity and is an example from a wide family of parabolic PDE models of chemotaxis written, with some generality on the d-dimensional torus, T d , for a ≥ 0, as on R + × T d , u| t=0 = u 0 ≥ 0, ρ| t=0 = ρ 0 , on T d .

(1.2)
In this paper we study (1.1), a one-dimensional additive noise, stochastic version of (1.2) in the chemorepulsive and parabolic-elliptic (a = 0) regime, with nonlinear sensitivity function, f (u, ρ) = χu 2 . Deterministic models including nonlinear sensitivity have been studied since the early work of Keller and Segel, see for example [7,16,18,26]. These works focused on modelling nonlinear sensitivity with respect to the chemical signal, i.e. f (u, ρ) = χuf (ρ). More recently there has been interest in modelling nonlinear sensitivity with respect to the cell density, i.e. f (u, ρ) =f (u). This interest has been motivated, in part, by desires to include volume filling, saturation and density dependent (quorum sensing) regularisation effects into chemotaxis models, [9,10,13,17,31,32]. The nonlinearity that we treat in (1.1) was already considered in the deterministic context in [17,31]; in [17] it was shown that in the chemorepulsive setting the parabolic-elliptic model exhibits global existence and convergence to equilibrium whenf (u) ∼ u m for all m > 0. In this case, the nonlinear sensitivity models an increased response to the chemorepulsant when cell density is high.
While it is common to study chemotaxis using PDE models, SPDE approaches can be used to model exogenous processes, study meta-stability or replicate physically relevant fluctuations around the continuum limit. In this work we focus on an additive noise model for two reasons. Firstly, it is a relatively simple model for which we can accommodate space-time white noise and establish a full ergodic analysis. Secondly, it allows us to make meaningful comparisons with additive noise stochastic reaction-diffusion equations which have been extensively studied; we refer to [4,20,23,27,34] for an incomplete list of works concerning these models. In particular, we highlight a comparison between our stochastic model, (1.1), and those considered in [20]. As in the above works, the main technical step we require is to obtain sufficiently strong a priori bounds on the pathwise solution which are independent of the initial data. In our context we divide the solution into a regular and an irregular part and the key step is a careful analysis of the more regular, remainder equation exploiting the repulsive nature of our nonlinearity to compensate for lower order terms without definitive sign. This is the content of Theorem 4.4. Furthermore, we keep careful track of the exponents and constants through the proof, which in particular allows us to obtain a tail bound, which is heavier than Gaussian, on the L p norm of the invariant measure. We give a detailed discussion on the implications of this bound and comparisons to the literature on stochastic reaction-diffusion equations in Remarks 4.5, 4.6 and 4.7.
There are many natural questions that arise from our work. For this model it would be interesting to understand if the heavier than Gaussian tails that we establish are in fact optimal. Furthermore, applying our current methods, it appears that the low integrability of the solution, as compared with stochastic reaction-diffusion equations, imposes a significant barrier to considering m > 2 in (1.1). Understanding whether this is a genuine issue or simply one of methodology would be interesting. Going further, one could naturally ask if it is possible to extend the well-known global well-posedness of attractive chemotaxis models in one dimension, [12,25], to the stochastic case with white noise forcing. In addition, since chemotaxis is typically observed in two or three dimensions, extending the above study to higher dimensional versions would certainly be of interest. Using simple power counting one would expect (1.1) to be sub-critical in the sense of regularity structures, [8], on R + × T d for d < 4. While local analysis is tractable in the presence of whitenoise forcing, establishing global well-posedness for the repulsive model in dimension two seems challenging by our current methods, see [19,Ch. 7]. Finally, beyond the additive noise case, both multiplicative and conservative noise models are highly relevant in the context of chemotaxis. Extending our results to those cases, especially in a two or three dimensional setting, is a natural goal.
In the remainder of the introduction we summarise some notation in Section 1.1, and present our main results in Section 1.2. In Section 2 we recap the definitions and properties of the linear stochastic heat equation, which plays a key role in our analysis. In Section 3 we obtain local well-posedness of (1.1) and some ancillary properties of the solution. Section 4 contains the main contribution of this paper, establishing an a priori bound that is independent of the initial data and constitutes a coming down from infinity property. This a priori bound enables us to establish global well-posedness, existence of invariant measures and the tail bound, (1.5), below. In Section 5 we establish the existence of invariant measures to (1.1) and closely following the approach of [34], we establish the strong Feller property, irreducibility, and exponential ergodicity of the associated semi-group. The appendices, A and B respectively contain summaries and useful properties of the inhomogeneous Hölder-Besov spaces and the stochastic heat equation on T.

Notation
Let N = {0, 1, . . . , } denote the set of non-negative integers. For k ∈ N, we denote by C k (T) the space of k times continuously differentiable, 1-periodic, real functions. We denote by C ∞ (T) the space of smooth, periodic functions with values in R and S ′ (T) for its dual. For k ∈ N, p ∈ [1, ∞) (resp. p = ∞) we write W k,p (T) for the spaces of periodic functions with p-integrable (resp. essentially bounded) weak derivatives up to k th order, and write L p (T) = W 0,p (T). We write B α p,q (T) for the Besov space associated to α ∈ R, p, q ∈ [1, ∞] and use the shorthand C α (T) = B α ∞,∞ (T). Note that for α ∈ N the spaces C α and C α are not equal. See Appendix A for a full definition.
For f ∈ S ′ (T) we denote its spatial mean byf := 1, f L 2 (T) . For m ∈ R we write, for example, S ′ m (T), C k m (T), C α m (T), L p m (T), for the corresponding spaces with the additional constraint thatf = m. When the context is clear we drop dependence on the domain in order to lighten notation.
Given a Banach space E, a subset I ⊆ [0, ∞) and κ ∈ (0, 1), we write C I E := C(I; E) (resp. C κ I E := C κ (I; E)) for the space of continuous (resp. κ-Hölder continuous) maps f : For a Banach space, E, equipped with its Borel σ-algebra, E, we use the notation B b (E), C b (E) and C 1 b (E) respectively for the sets of bounded Borel measurable, continuous and continuously Fréchet differentiable maps Φ : E → R. We write P(E) for the set of probability measures on E which we equip with the topology of weak convergence. For a sequence (µ n ) n≥1 ⊂ P(E) we write µ n ⇀ µ to indicate the weak convergence of (µ n ) n≥1 to µ ∈ P(E). Using Π µ,ν to denote the set of all couplings between µ, ν ∈ P(E), we write the total variation distance as, We write to indicate that an inequality holds up to a constant depending on quantities that we do not keep track of or are fixed throughout. When we do wish to emphasise the dependence on certain quantities, we either write K,v or define C := C(α, p, d) > 0 and write ≤ C.

Main Results
We fix a filtered probability space (Ω, F , (F t ) t≥0 , P) carrying a mean-free, space-time white noise, ξ, defined in Section 2 below. We fix α 0 ∈ − 1 2 , 0 , α ∈ 0, α 0 + 1 2 and η > 0 such that, . Then there exists a unique, probabilistically strong, mild solution, u(ζ), to (1.1) such that P-a.s. u(ζ) ∈ C η;T C α m (T). Remark 1.2. While we frame Theorem 1.1 as a probabilistic statement, the proof is mainly deterministic and based on PDE techniques. In the course of the proof we additionally establish local Lipschitz continuity of the solution in both initial data and the noise, see Proposition 4.8. This will be important in Section 5.3 when we establish full support of the law on C α m (T). Remark 1.3. The statement of Theorem 1.1 remains valid for η = α−α0 2 . However, we take η > α−α0 2 to simplify some statements and proofs below. In general we may think of η as being arbitrarily close to α−α0 2 . Let L(u t (ζ)) := u t (ζ)#P ∈ P(C α m (T)) denote the law of the solution, started from ζ ∈ C α0 m (T), at time t > 0. Theorem 1.4 (Exponential Ergodicity). Let m ∈ R and δ ∈ (0, 1/2). Then there exists a unique measure ν ∈ P(C 1/2−δ m (T)) which is invariant for the semi-group associated to (1.1). Furthermore, ν has full support in C 1/2−δ m and there exists c > 0 such that for all t > 1 and ζ ∈ C α0 m , (1.4) Finally, for any p ∈ [1, ∞) there exists Λ := Λ(p, δ) > 0 such that The proof of Theorem 1.1 is completed at the end of Section 4 and the proof of Theorem 1.4 is completed at the end of Section 5.

Stochastic Heat Equation
In what follows we will decompose the solution u to (1.1) into a regular and irregular part, with the irregular part being the solution to a linear stochastic heat equation (SHE). We briefly recap some necessary definitions of space-time white noise, the solution to the SHE and some of its properties.
Definition 2.1 (White Noise). Given an abstract probability space (Ω, F , P) we say that an Rvalued, stochastic process, indexed by For ϕ ∈ L 2 (R + × T) we write ξ(ϕ) in the convenient form of a stochastic integral, even though ξ is almost surely not a measure. We refer to [6,Ch. 4] for further details and a proper construction of (2.1).
We define the filtration and let (F t ) t≥0 denote its usual augmentation.
Definition 2.2. Let 0 ≤ t 0 < T and H be the periodic heat kernel on T defined in (A.9). We say that the S ′ (T) valued process, is a mild solution to the SHE, with zero initial condition at t = t 0 .
Proof. See Appendix B.

Local Well-Posedness
Throughout this section, we fix T > 0 and χ ∈ R (not necessarily positive). From Theorem 2.3 we see that P-a.s. the map t → v 0,t is finite only in C T C α 0 , for α < 1/2. We therefore cannot expect to find solutions to (1.1) of any higher regularity. Obtaining the a priori estimates in Section 4 requires at least one degree of spatial regularity. Therefore, although the equation is not singular itself, we solve for a remainder process with higher regularity using a Da Prato-Debussche trick, [4]. Concretely we decompose the solution as u t := w t + Z t , where t → Z t is a deterministic function taking values in the Hölder-Besov space C α 0 with zero spatial mean. Subsequently we will take Z t to be a P-a.s. realisation of the stochastic heat equation t → v 0,t . The unknown, w, solves, on T, w| t=0 = ζ, on T. For the rest of this section, we fix ζ ∈ C α0 (T) and Z ∈ C T C α 0 . We first show that under suitable regularity assumptions on w, the right hand side of the first equation in (3.1) is a well-defined element of C η;T C α (T). Lemma 3.1. Let w ∈ C η;T C α . Then the map w → Ψw, defined for any t ∈ (0, T ], by is well-defined from C η;T C α to itself. Proof. Applying (A.10), for any t > 0, we have that Concerning the integral term, expanding the square and applying Theorem A.5, along with (A.12) we obtain the bounds Combining these yields that Therefore, applying (A.10), (A.5) for any s < t ∈ (0, T ∧ 1] we have Since η < 1 4 we may integrate s −3η near 0 and so for t ∈ (0, T ∧ 1] we have where both exponents are positive due to (1.3) and the norms on the right hand side are finite by assumption. For t > 1 one may argue in almost exactly the same way, only splitting the time integral at t = 1 and replacing the multiplication by t η with (t ∧ 1) η .
Definition 3.2 (Mild Solutions to (3.1)). We say that w ∈ C η;T C α is a mild solution to (3.1) on [0, T ] (started from ζ and driven by Z) if for every t ∈ (0, T ], Remark 3.3. Lemma 3.1 demonstrates that for any solution, the right hand side of (3.6) is welldefined.
Theorem 3.4 (Local Well-Posedness of (3.1)). Let R ≥ 1 be such that Z 3 CT C α + ζ C α 0 (T) < R. Then there exists C > 0, independent of R, ζ, and Z, such that (3.1) has a unique mild solution w ∈ C η;T * C α where Furthermore sup and lim Proof. Denoting B T * := w ∈ C((0, T * ]; C α (T)) : sup is a contraction on B T * for T * defined by (3.7) for C > 0 sufficiently large. By (3.5), for w ∈ B T * and t ∈ (0, T * ], there exists C > 0 such that and so Ψ maps B T * into itself for T * defined by (3.7). To show that Ψ is a contraction we let w,w ∈ B T * . For any s ∈ (0, T * ], using similar steps to those in the proof of (3.5), we have that So then, for any t ∈ (0, T * ], we have It follows that there exists C > 0 such that, for T * given by (3.7), Hence Ψ is a contraction on B T * , and therefore there exists a unique fixed point w ∈ B T * of Ψ which, by construction, is a mild solution to (3.1) in the sense of Definition 3.2 and satisfies (3.8).
To show that w is the unique solution in all of C η;T * C α , letw be another mild solutions of (3.1). Then, by a similar argument to the above, there exists aT ( w Cη;T * C α , w Cη;T * C α ) =:T ∈ (0, T * ] such that w,w ∈ BT . Since both must be fixed points of Ψ on BT we have that w =w on [0,T ]. Iterating the argument, using the sameT at each step, shows that w =w on [0, T * ]. To show (3.9), observe first that lim t→0 e t∆ ζ − ζ C α 0 = 0 by Remark A.7 and so it only remains to show that the integral term converges to zero in C α0 . Considerη ∈ α−α0 2 , η . Since Cη ;T * C α ֒→ C η;T * C α , applying what we have proved so far to (α 0 , α,η) in place of (α 0 , α, η), we see that, for S > 0 sufficiently small, w is also the unique mild solution to (3.1) in Cη ;S C α , and that sup t∈(0,S] tη w t C α ≤ 1. Applying (A.10) and (3.4), where we used the fact that α0−α+1 2 ∨ 3η < 1 to evaluate the integral. We now chooseη sufficiently close to α−α0 2 so that 1 2 + α−α0 2 − 3η > 0, from which (3.9) follows.
Proof. From the mild form of the equation started from data w t0 at t = t 0 , it follows that for Integrating by parts in the first term on the right hand side gives Using the same argument for the second term on the right hand side we have that Changing the order of integration gives Putting this together with (3.16) gives that Rearranging and integrating the left hand side by parts once proves (3.15) for any φ ∈ C ∞ (T). For It follows from the proof of Lemma 3.6 that any solution to (3.1) has constant spatial mean.

A Priori Estimate and Global Well-Posedness
In this section we make use of the specific sign choice χ > 0 in (1.1) to obtain an a priori estimate on the solution in Theorem 4.4. Throughout this section we fix T > 0, and a zero spatial mean Proof. Let β ∈ (1, α + 1) and κ ∈ (1/2, β/2). By Lemma 3.5, Now, let ϕ ∈ C ∞ (R) and observe that for any t 0 ≤ r < s ≤ T we have the identity So for any t ∈ (t 0 , T ], and n ≥ 2, defining a family of partitions, by setting t i = t 0 + i(t − t 0 )/n, for i = 0, . . . , n and applying Lemma 3.6 with the test function ϕ(w ti ), we have By continuity of s → ∂ x ϕ(w s ) we see that For R n (t), Taylor's formula gives We show that R n 2 (t) converges to zero. Using (4.2), we have the bound where we used that ϕ ′′ is continuous. Regarding R n;1 (t), we may apply Lemma 3.6 once again to each bracket, this time with the test function w ti+1 ϕ ′ (w ti ), to give that So again, by regularity of the map s → ∂ x w s and ϕ, we have that So combining (4.3) and (4.4) gives, Observe that one of the terms appearing in the second term on the right hand side of (4.1) is We will use this term in combination with the first term, − w p−2 t |∂ x w t | 2 L 1 , coming from the Laplacian, to obtain an a priori estimate in Theorem 4.4 on w t p L p that is independent of the initial data. To do so we make use of an ODE comparison lemma which can be found with proof as [34,Lem. 3.8].
In light of Remark 4.3, we phrase the following a priori estimate only for the caseζ ∈ {0, 1}.
Proof. Since t → ∂ x w t and t → (w t + Z t ) 2 ∂ x ρ wt+Zt are both continuous mappings from (0, T ] into L ∞ (T) by Lemma 3.5, we may differentiate (4.1) with respect to t in order to obtain where we used that p is even in the first term on the right hand side. Regarding the second term, Integrating the first term by parts, using that −∂ xx ρ wt = w t −ζ by Corollary 3.7 and recalling that we have setζ = m, Therefore, applying the chain rule in the remaining terms gives We demonstrate that all terms without a definite sign can be bounded by a constant multiple of the two negative terms, From Theorems A.2 and A.8, for any α ∈ R and p, q ∈ [1, ∞], we have that and For any 0 < q ≤ p + 2, by Jensen's inequality we have (4.8) Furthermore, by applying Cauchy-Schwarz followed by (4.8), for 2 p < q ≤ p + 1 we have To keep track of dependence on p, we also make use of the following inequality which readily follows from Young's inequality for products: for γ 1 , γ 2 ∈ (1, ∞) such that 1 γ1 . From now on we let C > 0, c ≥ 1 be constants, that are independent of ζ, Z, p, and χ. Later, we will fix c ≥ 1 sufficiently large at the end of the proof. If we write in an inequality below, the implied proportionality constant is equal to C > 0 which we take sufficiently large so that the inequality holds.
Remark 4.5. Observe that the bound (4.5) is trivial in the limit p → ∞, since γ ≥ 1 and so (p ∨ χ) γ → ∞. This is in contrast to [20], where an a priori bound of the same form in L ∞ (T) is shown directly for stochastic reaction-diffusions.
Remark 4.6. In [20], the equivalent of (4.5), in the context of stochastic reaction-diffusion equations, CT C α . This leads to lighter than Gaussian tail bounds in the case of the reaction-diffusion equation, again contrasted with the heavier than Gaussian tails that we are able to establish for (1.1), see Theorem 1.4.
Remark 4.7. While a mild generalisation of this analysis to f (u), smooth and asymptotically quadratic (c 1 u 2 < f (u) < c 2 u 2 ), would likely be possible, more natural generalisations such as f (u) = |u| or f (u) = u m−1 for m ≥ 5 and odd, seem to present significant challenges. In the case of f (u) = |u|, since the testing argument we follow only applies to the remainder w = u − Z, the nonlinearity becomes When testing with w p−1 t this no longer produces the necessary damping term − w p+2 t L 1 and so it is not directly clear how one could proceed in this case. Regarding higher polynomial powers, it appears that controlling the transport term causes a problem for m > 3. In particular estimating the equivalent of (4.15) in the same way, leads us to control | ∂ x Z, w p+m−2 ∂ x ρ w |. Repeating similar estimates as in (4.16) and (4.17) lead to the condition m < 3−α 1−α . For α ∈ (0, 1/2) this restricts us to m < 5. Letting α → 1 would allow m → ∞. Informally speaking, integrability of the solution resulting from the transport term appears to be linked to regularity of the noise -higher regularity leads to higher integrability. This was also observed in the case of stochastic reaction-diffusion equations in [20]. Note also that with α ≥ 1 one could apply the testing argument directly to u, rather than working with the remainder. In [17] it was in fact shown that for the same model but with ξ ≡ 0, global existence and convergence to equilibrium holds for all m > 0. In the case of more regular noise v ∈ C T C 1 (T), we would therefore expect at least an analogue of Theorem 4.4 to hold directly for u also for all m > 0.
Proof. We recall from Theorem 3.4 that there exists a T * ∈ (0, 1), depending only on ζ C α 0 and Z CT C α , such that a mild solution w ∈ C η;T * C α exists to (3.1). Without loss of generality let us assume T > T * and that we fix an even p ∈ (− 1 α0 , ∞) so that L p (T) ֒→ C α0 (T). In this case, it is clear that we can extend the solution for a positive time of existence so long as w t L p remains finite. However, by Theorem 4.4, w t L p is bounded above by a function of t independent of the initial data and so we may continue the solution to all of [0, T ]. From Corollary 3.7, for u t := w t +Z t , we haveū t =ζ +Z t , for all t ∈ (0, T ]. Similarly, for all t > 0, u t C α ≤ w t C α + Z t C α . Hence the solution map (4.22) is well-defined and u Cη,T C α depends only on Z CT C α .
To continue the proof, we state the following Lemma 4.9. Consider R > 0 and define the set Then for T * = T * (R) > 0 sufficiently small, S : Proof. Let (ζ, Z), (ζ,Z) ∈ D R and consider the corresponding solutions From Theorem 3.4, there exists T * (R) > 0 such that u Cη;2T * C α ∨ ũ Cη;2T ⋆ C α ≤ 2. For t ∈ (0, 2T * ], using Theorem A.6 and similar bounds as in the proof of Theorem 3.4 we see that where the proportionality constant does not depend on ζ,ζ, Z,Z. So multiplying through by t η and taking the supremum over t ∈ (0, 2T * ] we have that By lowering T * further, we obtain We now prove that S is jointly locally Lipschitz. Consider R > 1, D R , and T * (R) > 0 as in Lemma 4.9. Then S is K(R)-Lipschitz from D R to C η ((0, 2T * ], C α ). In particular, the map If T ≤ 2T * , then we are done. Hence, suppose T > 2T * . We will prove that, forT * (R) ∈ (0, T * ] sufficiently small, u| [T * +T * ,T ] ∈ C([T * , T ], C α ) is a Lipschitz function of (ζ, Z), from which the conclusion will follow.
We conclude this section with the proof of Theorem 1.1.

Invariant Measure
Having established global well-posedness of the SPDE (1.1) we turn to study its long time behaviour. In this section we prove Theorem 1.4 which implies existence and uniqueness of the invariant measure and demonstrates exponential convergence of the adjoint semi-group to the invariant measure in the topology of total variation. By Remark 4.3, appropriately adjusting the spatial mean m of the initial condition ζ, it suffices to prove Theorem 1.4 in the case χ = 1. We thus assume χ = 1 throughout this section. We define the semi-group associated to (1.1) by setting, for all t ≥ 0, for all Φ ∈ B b (C α0 (T)).
We first show that (u t ) t>0 is a Markov process with Feller semi-group (P t ) t>0 and that for each ζ ∈ C α0 m (T), there exists a measure ν ζ ∈ P(C α0 m ), invariant for (P t ) t>0 . Lemma 5.2. Let u be a mild solution to (1.1). Then for every t ∈ [0, T ] and h ∈ (0, T − t) we have the identity u t+h =w t,t+h + v t,t+h , (5.1) wherew t,t+h solves Proof. A simple consequence of the heat semi-group property.

Existence of Invariant Measures
We use the decomposition (5.1) to show that t → u t defines a Markov process.
Theorem 5.4. Let ζ ∈ C α0 (T) and u t (ζ) as in Definition 5.1. Then for any In particular, t → u t is a Markov process with associated Markov semi-group (P t ) t>0 .
The following lemma shows that (P t ) t>0 is a Feller semi-group in the sense of [5, Sec. 3.1].
We are now in a position to show the existence of an invariant measure, ν ζ ∈ P(C α0 m ) for every ζ ∈ C α0 m (T). Theorem 5.6. Let m ∈ R. Then for every ζ ∈ C α0 m (T), there exists a measure ν ζ ∈ P(C α0 m (T)) and an increasing sequence of times t k ր ∞ such that, Furthermore ν ζ is invariant for (P t ) t>0 .
Proof. By the compact embedding L p ֒→ C α0 for sufficiently large p ≥ 1, it follows from Theorem 5.3 that the family of measures {P * t δ ζ } t≥1,ζ∈C α 0 (T) is tight. In particular, for every sequence of times (t k ) k>0 , R * t k δ ζ k>0 := 1 is a tight family of measures. Applying Prokhorov's theorem gives the existence of a sequence (t k ) k>0 for which the necessary weak convergence holds. From [5, Thm. 3.1.1] we see that the limit measure is necessarily invariant for (P t ) t>0 .

Strong Feller Property
We prove that (P t ) t>0 possesses the strong Feller property closely following the method employed in [34]. The main step is to establish a Bismut-Elworthy-Li (BEL) type formula, similar to that given in [34,Thm. 5.5].
For technical reasons in this section we replace (1.3) with the assumption that α 0 ∈ − 1 3 , 0 , α ∈ 0, α 0 + 1 3 and η > 0 such that, We note that all previous results also hold for this more restrictive parameter range and this restriction does not affect the main result as stated, see the proof of Theorem 1.4 at the conclusion of Section 5.4. Let (e m ) m∈Z , e m (x) = e i2πmx , be the usual Fourier basis elements of L 2 (T) and (∆ k ) k≥−1 be the Littlewood-Paley projection operators, see Appendix A for more details. For ε ∈ (0, 1), we define Π ε (L 2 (T)) as the space of real functions spanned by (e m ) |m|< 1 ε and Observe that there exists ℓ ε : where F is the Fourier transform. Furthermore (see e.g. [34, p. 1213 Π ε u C β ≤ 1 for all ε ∈ (0, 1) and β ∈ R, ii) for every β ∈ R and δ > 0, there exists C > 0 such that for all ε ∈ (0, 1) We fix for the rest of the subsection δ > 0 Such a δ exists due to (5.6).
Remark 5.7. We will later set, right before Theorem 5.13, Z ε =Π ε v, where v is the SHE.
To continue, we state the following Lemma 5.9. Consider R > 0 and define D R as in Lemma 4.9. There exists whereũ = S (ζ,Z) with S as in Proposition 4.8.
Remark 5.10. We will henceforth identifyC T C Nε with a subspace of C T C ∞ 0 ⊂ C T C α+δ 0 by mapping (B m ) m∈Nε to Z ε via (5.11). Observe that the integral in (5.11) is well-defined as a Riemann-Stieltjes integral for anyB ∈C T C Nε . For fixed ζ ∈ C α0 , we will treat in this way u ε (ζ, Z) := S ε (ζ, Z) as a function ofB ε whenever it is well-defined.
We now prove the differentiability of S ε with respect to both arguments separately, using D for derivatives with respect toB ε and D for derivatives with respect to ζ. For R ≥ 1, we recall the definition of T * (R) given by (3.7).

Lemma 5.11 (Derivative in Noise). There exists an open neighbourhood OB
ε ⊂C T C Nε containinĝ B ε such that u ε (ζ, ·) is Fréchet differentiable as a mapping from OB ε to C T Π ε L 2 (T). Furthermore, for any f ∈C T C Nε , such that f | t=0 = 0, the directional derivative D f u ε satisfies the equation (5.13) Finally, for ε ∈ (0, ε 0 ), there exists a C(T, ζ C α 0 , Z ε CT C α ) > 0 such that, Proof. Integration by parts implies that, for any m ∈ Z and f ∈ C T C with f 0 = 0 It follows that Z ε is a bounded, linear function ofB ε with values in C T Π ε L 2 0 (T), and so is continuously Fréchet differentiable. Furthermore, for any f ∈C T C Nε with f 0 = 0 Regarding the approximate solution, u ε;t , the mappings h → h 2 and h → ∂ x ρ h are Fréchet differentiable on Π ε (L 2 (T)), so the map is Fréchet differentiable as a mapping Φ T : (C T Π ε L 2 (T),C T C Nε ) → C T Π ε L 2 (T) and is such that Φ T (u ε ,B ε ) = 0. Moreover, writing D for the Fréchet derivative, is a Banach space isomorphism for T * ( u ε CT C α ) > 0 sufficiently small. Applying the implicit function theorem for Banach spaces, [1, Thm. 2.3], we obtain that u ε (ζ, ·)| [0,T * ] is Fréchet differentiable in a neighbourhood ofB ε . Patching together a sufficient (but finite) number of intervals of length T * to cover [0, T ], we obtain the first claim.
Due to the global existence of u ε ∈ C η;T C α (shown in Theorem 5.8 for ε ∈ (0, ε 0 ) where ε 0 depends on ζ C α 0 , Z ε CT C α ), for a new constant C(T, ζ C α 0 , Z CT C α ) > 0, Regarding the derivative of u ε (ζ) with respect to the initial condition, for g ∈ C α0 , we set g ε :=Π ε g and then for any 0 ≤ s ≤ t ≤ T we let J ε s,t g solve the equation We show below that for any ζ ∈ C α0 (T) and t ∈ [0, T ], J ε 0,t g = D g u ε (ζ), the directional derivative of u ε (ζ) in ζ. Note that J ε satisfies the flow property, that is for 0 ≤ s ≤ t ≤ T one has J ε 0,t = J ε s,t J ε 0,s . In particular J ε s,s = id. Lemma 5.12. There exists an open neighbourhood O ζε ⊂ Π ε (L 2 (T)) containing ζ ε such that u ε ( · ,B ε ) is Fréchet differentiable as a mapping from O ζε to C T Π ε (L 2 (T)). For any g ∈ C α0 , the derivative is given by D g u ε;t (ζ) = J ε 0,t g. Furthermore, setting R = Z CT C α + ζ C α 0 , there exists C(R) > 0 and T * (R) > 0 such that for all t ∈ (0, Proof. The same argument as in the proof of Lemma 5.11 shows that the map Π ε L 2 m (T) ∋ ζ ε → u ε (ζ) ∈ C T Π ε (L 2 m (T)) is Fréchet differentiable in a neighbourhood of ζ ε . It is then readily checked that on O ζε , for any g ∈ C α0 the Fréchet derivative is equal to the map t → J ε 0,t g. To prove (5.16), observe that Therefore, for any t ∈ (0, T ], Since u ε − u Cη;T C α ≤ 1, by Theorem 3.4 there exists a T * (R) ∈ (0, 1) such that u ε Cη;T * C α ≤ 2.
Hence, for all t ∈ (0, so then choosing a sufficiently small time t 1 (R) ∈ (0, t], Repeating this procedure, we find a constant C := C(R) > 0 such that We finally reintroduce probability and consider in the remainder of the section a finite dimensional approximation B ε;t of the white noise defined by Note that (B m ) m∈Nε is a family of complex valued Brownian motions which satisfy the reality condition (5.12). Our approximation of the SHE corresponding to (5.11) is then v ε;s,t :=Π ε v s,t = m∈Nε t s ℓ ε (m)e −4π 2 |m| 2 (t−r) dB m;r e m , (5.17) By Remark 5.10, sinceB ε ∈C T C Nε , (s, t) → v ε;s,t is well defined pathwise. Furthermore, by Property ii), there exists a P-null set N ⊂ Ω such that, fixing any realisation ξ(ω) for ω ∈ Ω \ N gives realisations of v ε (ω), v(ω) and for which v ε;0, · (ω) → v 0, · (ω) in C κ T C α−2κ for every κ ∈ [0, 1/2). In the rest of the subsection, we will let v ε denote the random path t → v ε;0,t .
For each ε > 0, the Cameron-Martin space ofB ε is By the Sobolev embedding, W 1,2 (R) ֒→ C 1/2 (R), we see CM ε ⊂C T C Nε . Therefore, Lemma 5.11 applies with f ∈ CM ε . We also choose a smooth, compactly supported, cut-off function Θ : R + → [0, 1] such that Θ(z) = 1 for z < 1 2 and Θ(z) = 0 for z ≥ 1. We introduce the notion of right sided derivatives, D + , of Z CT C α , which, for f ∈C T C α , is defined by Finally, for the third term, Therefore there exists a constant C(f, Θ, t, Φ C 1 b ) > 0 such that for all λ ∈ (0, 1] and p ≥ 1 where we applied Jensen's inequality to obtain ( where A pλ is also a strictly positive martingale with expectation 1 and exp p 2 −p 2 λ 2 t 0 |∂ t f ε s | 2 ds is uniformly bounded in L ∞ (Ω; R) across λ ∈ (0, 1] due to the assumptions on f ε . Regarding the second term, using the explicit form, and Cauchy-Schwarz, we see that The first factor is uniformly bounded for allλ ∈ (0, 1] by the same arguments as we applied for (A λ t ) p . The second factor can be controlled using the Burkholder-Davis-Gundy inequality and again our assumption of uniform boundedness on ∂ t f ε L 2 ([0,t]) . Hence and we have uniform integrability. Therefore, by Vitali's convergence theorem, we may exchange differentiation and expectation to give that In order to conclude the proof it only remains to show that the identities The first follows directly from Lemma 5.11, since one sided derivatives and full derivatives agree for any Fréchet differentiable map. The second identity follows after setting λ = 0 in (5.19) and observing that A 0 t ≡ 1. The final identity follows in the same way from our assumptions on Θ and the chain rule.
We can apply the identity (5.18) to obtain an initial bound on the difference of the semi-group acting on elements of C 1 b (C α0 ), from two initial points. The following result and its proof is close to that of [34,Prop. 5.8].
Proposition 5.14. Let ζ,ζ ∈ C α0 m withζ ∈ B 1 (ζ), set R := 1 + 2 ζ C α 0 and let T * (R) > 0 be defined according to (3.7). Then there exists a constant C := C(R, α, α 0 , η) > 0 and exponent θ := θ(η) > 0 such that for any Φ ∈ C 1 b (C α0 m ) and t ∈ (0, T * ], Proof. First, defining the semi-group for the approximate equation and applying the triangle inequality, for every Φ ∈ C 1 b (C α0 ) one obtains For the second term, and using the fact that v ε → v in C t C α in probability we obtain the second term on the right hand side of (5.20). Regarding the first term, we use the fundamental theorem of calculus along with Fubini to write, We now note that for any f ε ∈ CM ε , it follows from the mild forms of D f ε u ε;t and J ε s,t ∂ t f ε m;s given in (5.13) and (5.15) respectively, that, and for notational ease, we have suppressed the dependence on z λ in J ε 0,s (ζ −ζ). Then Furthermore, for f ε defined in this way, ∂ t f ε ∈ L ∞ (Ω; L 2 ([0, t]; C Nε )). Note also that z λ C α 0 ≤ R for all λ ∈ (0, 1), so the local bounds of Lemmas 5.11 and 5.12 both hold uniformly in z λ . Using (5.23) and (5.18) we obtain the bound For the first term, one can apply Cauchy-Schwarz, Itô's isometry and the results of Lemma 5.12 to obtain For the second term, using the explicit form of the directional derivative and Lemma 5.12 again applied to our choice of f ε , we have where in the last line we also used that Θ ′ is bounded.
We conclude this section by obtaining a local Hölder bound for the dual semi-group. It follows from this bound that, for all t > 0, P t : , which concludes our proof of the strong Feller property for (P t ) t>0 .
Theorem 5.15. Let ζ,ζ ∈ C α0 m withζ ∈ B 1 (ζ). Then there exists C > 0, θ ∈ (0, 1) and σ > 0 such that, for every t ≥ 1, Sketch of Proof. It follows from [5,Lem. 7.1.5] that (5.20) is equivalent to the statement, From this bound, one may follow exactly the steps of [34,Thm. 5.10]. The key idea is to use the Gaussian nature of v to obtain the control P( v CtC α > 1) t θ2 for some θ 2 ∈ (0, 1). Then we use the explicit form of T * (R) defined in (3.7) and monotonicity of the map t → P * t δ ζ − P * t δζ TV to obtain the final result.

Full Support
We demonstrate that u T (ζ), and thus any invariant measure ν ζ for (P t ) t>0 as in Theorem 5.6, has full support in C 1/2−δ ζ (T) for any δ ∈ (0, 1/2). In this subsection we are not concerned with the behaviour of the solution near zero, so until the start of Section 5.4 we just consider α ∈ (0, 1/2) and m ∈ R.
Let L 2 0 (R + × T) be the space of square integrable functions on R + × T such that for any t ≥ 0, f t = 0. Then for any T ≥ 0 we define the Cameron-Martin space of v := v 0, · , Note that by Theorems A.2 and A.6, H T is continuously and densely embedded in {h ∈ C T C α 0 : h(0) = 0}. The following is a direct consequence of the Cameron-Martin theorem.
Proof. We first show Recall that the map S T (ζ, · ): C T C α 0 (T) → C α m is continuous and H T ⊂ C T C α 0 (T). Consider now h ∈ H T . Then for any ε > 0, there exists a δ > 0 such that where the last inequality follows from Lemma 5.16 and this shows (5.26). It now suffices to show that, and Since ζ ∈ C α m , we have u y ∈ C T C α m and it also follows that h y ∈ C T C α 0 with h y (0) = 0. Furthermore, by construction, S T (ζ, h y ) = u y T = y. (5.28) Approximating h y by functions in C ∞ 0 ([0, T ] × T) ∩ H T and using the density of C ∞ (T) in C α (T) concludes the proof.

Exponential Mixing
In Theorem 5.18 and Corollary 5.19 below, we keep (α 0 , α) satisfying (5.6), the more restrictive parameter set from the start of Subsection 5.2.
We now complete the proof of Theorem 1.4.
We denote by B α p, q (T) the completion of C ∞ (T) with respect to (A.1), which ensures these spaces are separable.

A.1 Parabolic and Elliptic Regularity Estimates
We define the action of the heat semi-group on f ∈ L 1 (T) by, defines the heat kernel on (0, ∞) × T. We refer to [22,Prop. 5 & 6] for a proof of the following theorem.
The equivalent elliptic regularity estimate is an easy consequence of [2, Lem. 2.2] concerning Fourier multipliers.

B Regularity of the Stochastic Heat Equation
We outline the main steps necessary to prove Theorem 2.3 using the characterisation of Hölder-Besov spaces. We note that a similar result can be proved using a version of the Garsia-Rodemich-Rumsey lemma, [14]. A more detailed presentation of similar results in our context can be found in [22,34]. We include a proof directly in our setting for the reader's convenience. A central tool is the following Kolmogorov regularity result which can be found in a similar form as [22,Lem. 9 & 10]. We recall the kernels h k ∈ C ∞ (T) from the beginning of Appendix A.
Proof of Theorem 2.3. Below we show that L(I t0,t ) depends only on |t − t 0 | so without loss of generality we set t 0 = 0. It is also clear, by translation invariance of H in space and the law of the white noise, that I t0,t is translation invariant, i.e I t0,t (φ( · + x)) ∼ I t0,t (φ( · )) for any φ ∈ C ∞ (T) and x ∈ T.
For k ≥ −1, t ∈ (0, T ] and setting x = 0 in (B.1), we have v 0,t (h k ) = I 0,t (h k ). So splitting the increment using the semi-group property, Parseval's theorem, the covariance formula for the spacetime white noise and the action of the heat kernel in Fourier space we have for all 0 ≤ s < t ≤ T where F h k denotes the Fourier transform of h k and we may choose any γ ∈ [0, 1]. So then using Nelson's estimate [24,Sec. 1.4.3], for any p ≥ 2, there exists a constant C := C(p, T ) > 0 such that for all k ≥ −1 and γ ∈ [0, 1), Then applying Theorem B.1 and the Besov embedding (A.3), we see that there exists a modification of t → v 0,t (which we do not relabel) such that for any p ≥ 1, α ∈ (0, 1/2), κ ∈ [0, 1/2) and T > 0, To see stationarity of the processes t 0 → v t0,t0+h for fixed h > 0, we use the fact that v t0,t0+h (φ) is Gaussian, and has zero mean, so that its law is entirely determined by its second moment. Therefore, letting φ ∈ C ∞ (T), and by similar computations as above, we see that E v t0,t0+h (φ) 2 = m 1 8π 2 |m| 2 1 − e −8π 2 |m| 2 h |φ m | 2 , and hence L(v t0,t0+h ) depends only on h ∈ [t 0 , T −t 0 ]. Finally, by the assumption that ξ is spatially mean free, for any t 0 ∈ [0, T ) and h ∈ (T − t 0 , T ], L(v t0,t0+h ) ∈ C α 0 (T).