The KPZ Equation on the Real Line

We prove existence and uniqueness of distributional solutions to the KPZ equation globally in space and time, with techniques from paracontrolled analysis. Our main tool for extending the analysis on the torus to the full space is a comparison result that gives quantitative upper and lower bounds for the solution. We then extend our analysis to provide a path-by-path construction of the random directed polymer measure on the real line and we derive a variational characterisation of the solution to the KPZ equation.


Introduction
In this work we solve the Kardar-Parisi-Zhang (KPZ) equation on the real line, i.e. we construct a unique h : R ≥0 × R → R such that where ξ is a Gaussian space-time white noise, the generalized Gaussian process on R ≥0 × R with singular covariance structure E[ξ(t, x)ξ(s, y)] = δ(t−s)δ(x−y).
The KPZ equation was introduced in [32] as a model for the growth of a onedimensional interface that separates two two-dimensional phases of which one invades the other. The conjecture of [32], now called strong KPZ universality conjecture, was that any (1 + 1)-dimensional (one time and one space dimension) interface growth model that is subject to random influences, surface tension, and lateral growth, shows the same large scale behavior under the now famous 1−2−3 scaling, and that the KPZ equation provides a prototypical example of such a model. Since then it became apparent that there is a second, weaker universality in the class of (1 + 1)-dimensional interface growth models: If the lateral growth or the random influence is very weak, then according to and Quastel [1] gave a different, probabilistic construction of Q, based on Kolmogorov's extension theorem, but they did not establish the link to the SDE beyond formal calculations. Here we combine the approaches of [15] and [1], which have existed independently so far, and give a rigorous explanation of the above SDE. We give a path-by-path construction of the random directed polymer measure that does not depend on the statistical properties of the white noise, but only on the "model" associated to it. We also show that the KPZ equation can be interpreted as the value function of the stochastic control problem where under P x for a Brownian motion W that is independent of ξ and where the supremum is taken over all processes v on [0, T ] adapted to W , but possibly dependent on the realization of ξ. This representation was previously derived in [24] for the KPZ equation on the torus.

Structure
In the first section we introduce techniques from paracontrolled calculus for SPDEs in a weighted setting, cf. [21,24,36]. Among them are the commutation and product estimates from Lemmata 2.10 and 2.8, as well as tailor-made Schauder estimates for the weighted setting, e.g. Lemma 2.14. The existence and uniqueness of solutions to the KPZ equation follows from a comparison result, Lemma 3.10. The lower estimate guarantees that the Cole-Hopf solution is the unique paracontrolled solution to the KPZ equation and that the latter depends continuously on the parameters of the equation (Theorem 3.19). When considering uniqueness of solutions to PDEs on the whole space it is important to add some weight assumptions on the initial conditions. In this work we assume roughly linear growth of the initial condition (for the precise statement see Assumption 3.7 and Table 2). This is imposed upon us by the Cole-Hopf transform and the weighted Schauder estimates. These conditions suffice to start the equation in the invariant measure, the two sided Brownian motion [31,7]. Section 4 addresses the random directed polymer measure Q. We prove subexponential moment estimates (Lemma 4.5) and we show that the polymer measure is absolutely continuous with respect to a reference measure P U which we refer to as the partial Girsanov transform (cf. [24,Section 7]) and which is in turn singular with respect to the Wiener measure. We conclude with two characterizations of the solution h to the KPZ equation. The first, via the Feynman-Kac formula (Remark 4.13), states that the solution h is the free energy associated to the measure Q. The second is a variational representation à la Boué-Dupuis (Theorem 4.16), cf. [8,43]. The rest of this work is dedicated to technical, yet crucial, results. In particular (Section 5) we prove a solution theorem for linear SPDEs, which applies to all linear equations studied in [29,24,36]. Remark 1.1. Our approach uses the Cole-Hopf transform in several crucial steps. But we expect that the transform can be entirely avoided by making stronger use of the variational formulation of the KPZ equation, as soon as we can prove the following conjecture: Let, with the notation of Section 2 below, X ∈ CC −1/2−ε p(ε) for all ε > 0, let (∂ t − 1 2 ∆ x )Q = ∂ x X, and let Q X ∈ CC −ε p(ε) for all ε > 0. Then we conjecture that the paracontrolled solution u to KPZ on R grows sublinearly in the L ∞ norm in space, provided that f does. That is, for some δ < 1 sup t∈[0,T ] sup x∈R |f (t, x)| 1 + |x| δ < ∞ ⇒ sup t∈[0,T ] sup x∈R |u(t, x)| 1 + |x| δ < ∞.
While this conjecture seems very plausible, we are at the moment not able to prove or disprove it, and we leave it for future work.

Notations
Define N = {1, 2, 3, . . . } and N 0 = N {0}. For inequalities we use the following convention. For a set X and two functions f, g : X → R we write f g if there exists a constant C > 0 such that for all x ∈ X : f (x) ≤ Cg(x). When the constant C depends on some parameter κ and we want to underline this fact we shall write: f κ g. For vector spaces X ⊆ V and f, g ∈ V we shall write equations of the kind f = g + X if f −g ∈ X.

Fourier transform
We review basic knowledge and notations regarding the Fourier transform. We define the space of Schwartz functions S (R) as the space of smooth and rapidly decaying functions. The dual space S (R) is the space of tempered distributions. Let ϕ ∈ S (R), then we define for all ξ ∈ R:φ (ξ) = F ϕ (ξ) = R ϕ(x)e −ixξ dx and for ϕ ∈ S (R) we define the Fourier transform in the sense of distributions: φ, ψ = F ϕ, ψ = ϕ, F ψ , ∀ψ ∈ S (R) .
Since we will consider functions that have more than just polynomial growth at infinity, it is necessary that we go beyond the setting of tempered distributions and consider tempered ultra-distributions. This theory is presented in [42] or [6]. For a simple and hands-on introduction to all the tools we need we refer to [36]. Consider the function ω(x) = |x| δ , δ ∈ (0, 1) with δ fixed once and for all. Using this weight, we define spaces of exponentially decaying Schwartz functions and their duals as follows. S ω (R) = f ∈ S (R) p α,λ (f ) < ∞, π α,λ (f ) < ∞ ∀ λ > 0, α ∈ N d 0 , and we denote by S ω (R)its dual, which we call the space of tempered ultra-distributions.
Finally, we can define the Fourier transform on the space of tempered ultra-distributions just as before: For f ∈ S ω (R) we set We have introduced the Fourier Transform for exponentially decaying functions so that we can extend the Littlewood -Paley theory also to weighted functions. We now fix the weights that we are allowed to use.
If this bound holds true we say that z is ω-moderate.
The need for considering ω-moderate weights can be explained by the following calculation: to estimate the convolution ϕ * f we can compute |ϕ(x − ·)e λω(x−·) |, |f (·)/z(·)| . Now we can intuitively bound the last term by some weighted norm of f assuming that ϕ is fixed and rapidly decaying.
Note that we distinguish the parameters t and l because later we will consider time dependent weights.

Littlewood-Paley theory
In this section we review the construction of weighted Hölder-Besov spaces. For a comprehensive introduction to Littlewood-Paley theory we refer to [2]. For a treatment of weighted spaces we also refer to [36,35]. Following their constructions we fix a dyadic partition of unity generated by two smooth functions ρ −1 and ρ, that belong to S ω (R) and are supported in a ball around the origin B and an annulus around the origin A , respectively. We then define ρ j (x) = ρ(2 −j x), j ≥ 0. Now we define the Littlewood-Paley blocks: for ϕ ∈ S ω (R) and j ≥ −1 let ∆ j ϕ = F −1 (ρ j ϕ). We will use the following notation for paraproducts: Definition 2.4 (Hölder-Besov spaces). For any α ∈ R and weight function z ∈ ρ(ω) we define the space: We denote with C α for the space C α z with weight z = 1 and use the norm where in the last step we used that ω(x) = |x| δ , λ ≥ λ 0 and c = ν/λ δ 0 . So eventually we can estimate: where in the last step we changed variables and used the assumption that ψ is in S ω (R) and the growth assumptions on z to conclude that the second norm is finite uniformly over all λ ≥ λ 0 .
Proof. The first three estimates are shown in [36,Lemma 4.2]. The estimate for the commutator C is from [36,Lemma 4.4].

Time dependence
Throughout this work we mostly use an arbitrary but finite time horizon T > 0 which will be fixed from now on. When we change the time horizon we will explicitly state it.
We define the heat operator L and the associated semigroup P t : x . In this section the aim is to encode the following information: 1. Time dependent weights.
Here as before we follow the notation of [36]. For an arbitrary horizon T r ≥ 0 we denote by X = (X (s)) s∈ [0,Tr] an increasing sequence of Banach spaces. A typical example could be the sequence X (s) = C α e(l+s) . In fact we will use only two kinds of time dependent weights, which we will refer to through the following abuse of notation: e(l + t)p(a) = (e(l + t)(·)p(a)(·)) t∈[0,T ] , e(l + t) = (e(l + t)(·)) t∈[0,T ] , where the t on the left-hand side is only a formal way of representing time dependence. In applications it will always be clear whether t is fixed or whether we are considering a time-dependent weight. Now we define the following space of functions for given β ≥ 0: . Similarly we can define Hölder continuity through the following norm for α ∈ (0, 1): we can easily control the spatial regularity. Let us concentrate on the time regularity. We estimate for t ≥ s: Now let us fix a j 0 such that 2 −j0 ≤ |t − s| 1/2 < 2 −j0+1 . We will use different estimates on small scales and on large scales. Indeed, an application of Bernstein's inequality (Proposition 2.5) gives for the large scales We conclude the preliminaries by stating some important estimates regarding the heat semigroup, commonly referred to as Schauder estimates. We write where P t is the semigroup defined by Equation (2.1). Proposition 2.12. Fix α ∈ (0, 2) and z ∈ ρ(ω).
In the previous result the role of e(l + t) and p(a) only comes into play with the parameter a/δ. Although this seems a minor detail in the statement, it is actually the key point that allows us to solve linear singular SPDEs on the whole real line with exponential weights. This approach has been developed by Hairer and Labbé in [28,29] and it is also present in [15,30].
In the next result we show our last product estimate. Since our definition of the L α spaces does not allow for negative α we state in the following theorem the classical result of Young integration with parabolic scaling. Recall the definition of L from Equation (2.1). Lemma 2.13 (Young Integration). As before let z i , i = 2 be pointwise increasing, timedependent weights. Conisder f ∈ L β,α z1 and g ∈ L γ z2 with β ∈ [0, 1) and α, γ ∈ (0, 2). If α+γ−2 > 0, we have f · ∂ t g ∈ LL β,α+γ z1·z2 and the following two estimates hold true: for any ε > 0 and 0 ≤ T ≤ T r ≤ T h .
Proof. The proof of this result is the content of Lemma D.3 and the preceding results.
The next result shows how to interpolate between different L β,α z spaces.
Note that here we allow also for ε = 2β. This is possible because, using the same arguments as in the proof of [24, Lemma 6.8] we obtain the following bound (uniformly However, it is a priori not clear that f : [0, T ] → C ζ z(T ) is a continuous function. Since we have Hölder continuity in L ∞ z(T ) of f and a uniform bound in C α−2β z we can conclude by interpolation, at the price of an arbitrarily small loss of regularity, which explains the strict inequality ζ < α−ε.
With these results we end our brief introduction to the theory of paracontrolled analysis and Schauder estimates.

The paracontrolled KPZ equation
Here we briefly review the notion of paracontrolled solutions to the KPZ equation first introduced in [24] and [21]. For counting regularity we will use the index α. We will use the index a for counting the polynomial growth of the noise at infinity and we recall that δ ∈ (0, 1) is used in our definition of ultra-distributions. We will work under the following standing assumptions on the parameters. Assumption 3.1.
Here α counts the regularity of the solution to the heat equation with additive space-time white noise and a counts its polynomial growth at infinity. In general α ∈ (1/3, 1/2) will be sufficient, we will need some tighter control only in Section 4 (see e.g. Proposition 4.2), while a has to be sufficiently small to apply the results of Theorem 5.5.
We can compile the following rule-of-thumb table. Table 1: Rule-of-thumb for the Parameters α δ a 1/2− 1− 0+ Let us introduce the extended data for the KPZ equation. We collect in the next table all the terms involved. Here and throughout this work we will use the notation Here we assume that θ ∈ LC α/2 (R; C ∞ b (R)) is a (spatially) smooth noise and C ∞ b (R) is the space of bounded and infinitely differentiable functions with all derivatives bounded.
The reason for assuming only distributional regularity of θ in the time variable is that we do not want to exclude spatial mollifications of the space-time white noise, which are convenient from a probabilistic point of view because they preserve the Markovian structure of the equation. We solve the equations for the elements in Y(θ) by taking all initial conditions equal to zero, except Y (0) = Y 0 is assumed to be non-trivial. We are interested in starting the KPZ equation at its invariant measure, and for that purpose it is convenient to let Y 0 be of the form where B is a two sided Brownian motion and C ∈ R (cf. [41,Section 1.4]). Note that we have added X = ∂ x Y to the table because we assume that it has a better behavior at infinity than Y . Indeed, while Y may have superlinear growth at infinity, its derivative X is started in the invariant measure for the rough Burgers equation, which has the growth of white noise on R, i.e. it grows less than any polynomial.
We now rigorously define the spaces of functions we will work with. For a finite collection I of Banach spaces X i we call product norm on × i∈I X i the norm: · ×Xi = max i∈I π i (·) Xi with π i being the projection on the i-th coordinate.

Definition 3.2.
We shall call Y ∞ kpz be the image of the map Y(θ, Y 0 , c , c ) in the space We define Y kpz as the closure of the image of Y(θ, Y 0 , c , c ) in the above space endowed with the product norm, which we will refer hereafter as · Y kpz . For any Y ∈ Y kpz we define the distribution ξ by ξ = LY .
These tools are sufficient to define paracontrolled solutions to the KPZ equation. Definition 3.3. We say that h is a paracontrolled solution to the KPZ equation (1.1) with initial condition h ∈ C(R, R) and with external data Y ∈ Y kpz if there exists an κ ∈ R and β ∈ (0, 1), β ∈ (0, α+1 2 ) such that h is of the form: , and if the following conditions are satisfied:

1)
and h = X + ∂ x h P . purpose are the paracontrolled nature of h P and the fact that the resonant product ∂ x Y ∂ x Y is given a priori in Y kpz . For smooth noises θ the definition amounts to h satisfying the equation In this sense a paracontrolled solution h to the KPZ equation with white noise forcing with w ξ ="w(ξ − ∞)" with ∞ = lim c . Definition 3.5. We say that w is a paracontrolled solution to the RHE equation (3.3) with initial condition w 0 of the form w 0 = w 0 e Y (0)+Y (0)+Y (0) with w 0 ∈ C β e(l) , for some β ∈ (0, 2α+1], and external data Y ∈ Y kpz if there exists a κ ∈ R such that w is of the form: and w ∈ Lβ ,2α+1−ε e(κ) (3.4) and such that w P solves the equation The existence of global in space solutions to the RHE is already established in [29]. In Section 5 we review their approach and prove an existence result for the paracontrolled setting (Proposition 5.6). Now we briefly discuss how white noise can be lifted to extended data for the KPZ equation.
Theorem 3.6 (Renormalisation). Let ξ be a white noise on [0, T ] × R, let B be an independent two-sided Brownian motion on R, and let C ∈ R. Then for any α < 1/2 and a > 0 (see Table 2), (ξ, B + Cx) is almost surely associated to a Y(ξ, B + Cx) ∈ Y kpz : There exists a sequence (ξ n , Y n 0 , c n , c n ) in LC α/2 (R; C ∞ b (R)) × C ∞ b × R × R such that almost surely (ξ n , Y n 0 ) → (ξ, B + Cx) in the sense of distributions and such that where the convergence is in L p (Ω; Y kpz ), for all p ∈ [1, +∞). Moreover, while ξ n and Y n 0 are of course random processes, the constants c n , c n can be chosen deterministic.
Finally also the following asymmetric product converges: Proof. Letξ n (ψ) = ξ| [−n,n] ( k∈Z ψ(·, 2kn + ·)) and be the (spatial) 2n-periodization of ξ and B + Cx, respectively. Let ϕ ∈ C ∞ c (R) be even and such that ϕ(0) = 1 and define as the spatial regularization ofξ n respectivelyỸ n 0 through the Fourier multiplier ϕ(n −1 ·). It is not hard to show that (ξ n , Y n . In Section 9 of [24] the construction of Y(ξ, B) is performed in the periodic case, and slightly adapting the arguments of that paper we also obtain the convergence in our setting (for C = 0): it suffices to change the definition of E = Z \ {0} to E = R and to replace L p (T) by L p (R, p(a)) for a > 1/p in the computations following equation (84). Since p can be arbitrarily large, a > 0 can be as small as we need.
It remains to treat the case C = 0. But adding Cx to Y 0 only results in changing Y (t) → Y (t) + Cx (here we used that spatial convolution with the heat kernel leaves Cx invariant, because it is a harmonic function). Also, the approximations to Cx are smooth uniformly in n, and therefore adding these additional terms does not change the regularities or divergences. The result regarding the asymmetric resonant product follows along the same lines: for clarity it is postponed to Lemma C.1. Assumption 3.7. We will work under the following assumptions: for some β ∈ (0, 2α+1] and assume that there exists a sequence h 3. For α satisfying Assumption 3.1 we write, as in Equation (3.4): In addition, write M > 0 for a constant such that: Remark 3.8. In particular, we can choose any initial conditionh in the space C β p(δ) .
Indeed, in Theorem 3.6 we can set C = 0 in the initial condition for Y , and then Y (0) ∈ C β p(δ) . It is canonical to assume a bit of Hölder regularity for the initial condition of the KPZ equation, this is already needed for the equation on the torus [26,24]. The constraint on the growth is not so natural, in the case of smooth ξ we would expect that at least subquadratic growth is sufficient. But with our methods sublinear growth is the best we can hope for, because we need eh to be a tempered ultra-distribution.
For the smooth data Y n and initial condition h n we can solve the KPZ equation. Proof. This is a classical application of the Schauder estimates, cf. [24,Section 4].
Global existence in time follows from a partial Cole-Hopf transform, since v n = e h n −Y n solves the linear equation with continuous-in-time and smooth-in-space data.

Existence
First, we will prove an a priori estimate for the smooth solutions h n (t, Proof. Recall that the function h P,n solves Lh P,n = L(Y ,n + Y ,n ) + (X n X ,n −X n X ,n ) + X ,n X ,n + 1 2 (X ,n ) 2 Now we find an upper bound for v n . We consider the transformationũ n = exp(v n ) which solves the equation Lũ n = − [L(Y ,n +Y ,n ) + (X n X ,n −X n X ,n ) + X ,n X ,n + 1 2 (X ,n ) 2 ]ũ n +(X n +X ,n +X ,n )∂ xũ n − 1 2 (∂ xũ n ) 2 /ũ n , Lu n = − [L(Y ,n +Y ,n ) + (X n X ,n −X n X ,n ) + X ,n X ,n + 1 2 (X ,n ) 2 ]u n + (X n +X ,n +X ,n )∂ x u n , with initial condition u n (0) = exp(−[h n −Y n (0)]). Up to a sign this equation is just Hence by Lemma 2.14 u n is uniformly bounded in CC ζ e(κ) for some ζ > 0. Indeed, since β > 0 it follows that β < α+1 2 . We can conclude by the monotonicity of the logarithm.
where W is a Brownian motion and we recall that θ n is deterministic. We can then use the Feynman-Kac formula once more to derive an upper bound for the expectation on the right hand side, which leads to a lower bound for h n . Since in general θ n is only a distribution in time and also we are interested in bounds for h P,n and not for h n , we argue through the comparison principle instead.  [39] proved that the solutions are strictly positive even when started in a nonnegative, nonzero initial condition (see also [11,Theorem 5.1] for a pathwise version of this result), while in [12] there are tight estimates regarding an upper bound for the solution. These results already relied on comparison principles, but only with respect to the initial condition. While the lower bound of [11] is also pathwise, it is, along with the other quoted lower bounds bounds, only qualitative and gives no quantitative control. The price we pay for our result is that we restrict ourselves to strictly positive initial conditions, which satisfy Assumption 3.7. Now we show that the sequence h n converges to some h. In the following lemma we collect the results regarding the rough heat equation.
There exists a κ ≥ 0 such that: , where w solves the rough heat equation (3.3) on the entire space with initial condition w 0 = e h , in the sense of Definition 3.5.
Proof. The initial condition w n (0) = e h n is of the form w n 0 e Y n (0) with w n 0 converging to w 0 in C β e(l) , for some l ∈ R. Indeed this follows from Assumption 3.7 and Lemma A.1, Thus, the first result is a consequence of Proposition 5.6.
This lemma and the previous lower bound allow us to deduce the convergence of h n by exploiting the continuity of the Cole-Hopf transform. A priori it is not clear why taking the logarithm is a continuous operation, since it has a singularity in zero.
Proposition 3.14. Under Assumption 3.7 and for h n as in Proposition 3.9, there exists κ ≥ 0 such that: .

(3.7)
Moreover h = log(w), where w is the solution to the rough heat equation with initial condition e h and h P = log(w P ). In addition, we have a sub-linear bound for h P : Proof. First, we use the results from Lemma 3.10, so that we can find a C > 0 and an r ≤ 0 such that In view of this and Lemma 3.13, we can apply Lemma A.3, which guarantees that up to choosing a larger κ log(w P,n ) −→ log(w P ) = h P in L β ,α+1 .
Finally, the lower bound in L ∞ p(δ) for h P follows from Lemma 3.10. The upper bound follows from the monotonicity of the logarithm and the Cole-Hopf transform.
We have found a function h which is a candidate for being a solution to the KPZ equation on the whole real line. We have shown that h is of the form . Now we want to prove that h P is of the form for some β ∈ (0, 1). We observe that h n is already paracontrolled, since we have started with a paracontrolled solution. This allows us to control the derivative term.
Now both therms on the right-hand side of this equation converge in L β ,α e(κ) for an appropriate κ ≥ 0. Indeed, it follows from Proposition 3.14 that h P,n +Y ,n converges in L β ,α+1 e(κ) . By Lemma 2.11 this is enough to obtain convergence in L β ,α e(κ) of the spatial derivative. Now we consider the rest term h . Here we use a different argument.
as well as the equation: where we define Z as Proof. Since h n is a paracontrolled solution to the KPZ equation we know from Definition 3.3 that h ,n satisfies the equation Moreover from the previous results we know that h P,n converges to h P in L β ,α+1 e(κ) as well as that h ,n converges to h ,n in L β ,α e(κ) . At this point, since β > 0 (upon choosing a ζ small enough), we can conclude by the continuous dependence on the parameters from Proposition 5.8: , up to taking a larger κ, where h is the solution to equation (3.9). This proves the result.
The two lemmata above suffice to show that h is a paracontrolled solution to the KPZ equation. We collect all the information about h in the following theorem. Proof. That h has the correct structure follows from Proposition 3.14, Lemma 3.15 and Lemma 3.16. In addition, h P solves Equation (3.1), since h solves Equation (3.9).

Uniqueness
It is a rule-of-thumb that in order to obtain uniqueness for PDEs on the entire space some growth assumptions are needed in order to avoid solutions that do not have physical meaning. We will work under the assumption of sublinear growth in L ∞ . This is mainly due to the fact that we work within the framework of the Cole-Hopf transform. First, we show that the exponential map preserves the paracontrolled structure of a solution.
Proof. It follows from the growth assumptions on h P as well as from Lemma A.2 that w P lies in L β ,α+1 e(κ) and w P h lies in Lβ ,α e(κ) for some κ large enough andβ > (β −1/2) ∨ 0+β .
We still need to show that ).
where we have applied the paraproduct estimates from Lemmata 2.10 and 2.8. We now consider one term at a time. Let us start with the product w P (X ∂ x h ). We work with β small enough, so that we always have a non-trivial blow-up. The opposite case is simpler.
Since X ∂ x h has regularity 3α−1. Applying Lemma 2.14 we thus find: Here applying Lemma 2.13 we have that the term on the first row lies in ). The terms on the second row on the other hand lie in Mβ C 2α−1 . Hence, we can conclude that for someβ ∈ (0, 1) the statement of the Lemma is true. .
Finally, the function w = exp(h) is the solution to the RHE in the sense of Definition 3.5.
Proof. The existence of solution satisfying the required bound follows from Proposition 3.14. Let us prove uniqueness. Hence suppose that h is a solution to the KPZ with h P ∞,p(δ) < +∞. From the previous result we deduce that indeed w P = exp(h P ) is paracontrolled. We need to show that it solves the rough heat equation (3.5): since this equation has a unique paracontrolled solution our result will follow. First, we apply the chain rule to see that with all products classically well-defined. Thus w P solves: where we have marked the product that needs the paracontrolled structure of h P to be well defined with the diamond symbol. In view of the fact that ∂ x w P = w P ∂ x h P and by considering smooth approximations we see that:

Polymer measure
In this section we build the random directed polymer measure associated to white noise and study its link to the solution h of the KPZ equation. From this point onwards we will use arrows to denote time inversion with respect to the time horizon T , i.e. we

An informal calculation
Let us consider a formal solution to the SDE where W is a Brownian motion started in x 0 ∈ R. There are two issues with this SDE. The first is that ∂ x h is only a distribution. The approaches developed by Delarue and Diel [15] as well as Cannizzaro and Chouk [10] have tackled this aspect successfully. The second issue is that in our setting ∂ x h is of exponential growth, so a priori we would expect that the solution γ could explode in finite time. What gives us hope is that the exponential growth of ∂ x h is mostly due to our approach through the Cole-Hopf transform.
We exploit the random directed polymer measure associated to white noise to build a weak solution to the SDE (4.1), avoiding the use of h, and replacing it instead with elements of Y up to a rest term Y R (cf. [24,Section 7]). In the remainder of this preamble we present a formal calculation that explains the approach.
Consider the Wiener measure P x0 started in x 0 on the space C([0, T ]; R). Denote with γ the coordinate process on C([0, T ]; R). We define the measure Q x0 given by the Radon-Nikodym derivative: where h is a solution to KPZ for a (spatially) smooth noise θ ∈ LC α/2 (R; C ∞ b (R)), with extended data Y(θ) and with initial condition h. By Girsanov's theorem under this measure the coordinate process is a weak solution to the SDE (4.1). We can formally apply the Itô formula to where C > 0 is a normalizing constant. Note that unless h = 0 the polymer measure is not exactly the measure that solves the SDE (4.1). Indeed: where the rest is Finally, defining Y R as the solution to: the rest term can be rewritten as .
In this way we can do the change of measure in two steps: 1. We build the singular measure P U x0 with: started in x 0 , so that we can obtain the measures Q x0 and Q x0 as absolutely continuous perturbations by . (4.6) This approach goes back to [24,Section 7] for the equation on the torus, but due to the weighted spaces in which we have to work it becomes much more complicated in our setting and actually we cannot directly make sense of (4.5, 4.6). In the next paragraphs we shall show how to rigorously carry out the analysis and how to construct the measures P U x0 , Q x0 and Q x0 via the partial Girsanov transform that we just illustrated.

A paracontrolled approach
In order to construct the measure P U x0 we prove the existence of martingale solutions to the associated SDE: The essential tool for solving the martingale problem is to solve the backward Kolmogorv equation  This approach to SDEs with singular drift was established in the work by Delarue and Diel [15] who used rough path integrals (inspired by [26]) to solve the Kolmogorov equation. In contrast to our setting, the assumptions on the weight in [15] do not allow linear growth for Y . This is only a technical issue, but overcoming it would result in several lengthy calculations. Thus we prefer to follow Cannizzaro and Chouk [10] who formulated the approach of Delarue and Diel in the paracontrolled framework and thereby also extended it to higher dimensions. This suits our setting better and allows the reader to have a complete overview and a better understanding of the techniques at work.
The first results concern the existence of solutions to the Kolmogorov equation.
and a forcing f ∈ CC 2α−1 e(l) ([0, T ]). In this setting Equation (4.8) has a unique paracontrolled solution ϕ τ . Moreover for any M > 0, if we denote by ϕ 1 τ and ϕ 2 τ the respective solutions to the equation for two different external data Y 1 and Y 2 , initial condition ϕ 0 1 and ϕ 0 2 and forcings f 1 and f 2 such that we find that for some κ = κ(l, T ) and any ε ∈ (6a/δ+1−2α, 3α−1) : Proof. This result is a consequence of Theorem 5.5. By time reversal it suffices to solve the equation where we write ← − g (t) = g(τ −t) and the terms X t−τ , X T −τ belong to the data Y T −τ as constructed in Proposition B.2. We thus know that Y T −τ ∈ Y ζ,b kpz (see Definition B.1) for b = 2a (recall that a is the polynomial growth coefficient of our data) and some ζ > 1/2−α. Then we apply Theorem 5.5 with the coefficients chosen as follows: where the parameter f lives in the space X = CC 2α−1 ([0, T ]). An application of the Schauder estimates and the estimates for paraproducts shows that R and F satisfy the requirements of Assumption 5.1. Thus we find a solution ← − for any ε ∈ (6a/δ+2ζ, 3α−1), where both the parameter κ and the estimates on the norm of the solution can be chosen uniformly over τ as a consequence of the estimates from Theorem In the following we will show how to use the existence of solutions to the PDE to find unique martingale solutions to the martingale problem (4.7). For technical reasons in order to construct the polymer measure we will need a slightly more complicated version of the space Y kpz , in which we add as a requirement the convergence of an asymmetric product. This convergence is guaranteed in the case of space-time white noise by the result of Theorem 3.6.
kpz if the convergence holds in Y kpz and in addition the following asymmetric resonant product converges: satisfies: is a square integrable martingale under P U x0 , with respect to the canonical filtration.
We split the proof of this proposition in two lemmata, which are interesting in themselves. In the first one we derive a priori estimates for the exponential moments of a solution to the SDE.
Proof. Fix a terminal condition ϕ 0 such that ϕ 0 (x) = e l|x| δ for |x| > 1 and is smooth and bounded for |x| ≤ 1. For any τ ∈ [0, T ] and Y ∈ Y kpz with Y Y kpz ≤ M it follows from Proposition 4.2 that there exists a constant C = C(M, l, T ) such that the solution ϕ τ to Equation (4.8) with forcing f = 0 and terminal condition ϕ 0 satisfies |ϕ τ (t, x)| ≤ Ce C|x| δ . From the Itô formula and the fact that we chose a bounded noise we know that ϕ τ (t, γ t ) is a true martingale. Hence: With this result at hand we can prove tightness and convergence for the laws of the solutions associated to the SDE. Lemma 4.6. Consider a sequence Y n in Y ∞ kpz such that Y n → Y in Y poly kpz . Let W be a Brownian motion and γ n the strong solutions to the SDE (4.7) driven by the smooth and bounded drift ∂ x U n . Then there exists a measure P U x0 on C([0, T ]; R 2 ) such that, denoting with (γ, W ) the canonical process on this space: in the sense of weak convergence of measures. The process γ is the unique martingale solution to the martingale problem of Proposition 4.4 and it is ζ-Hölder continuous for any ζ < 1/2. In addition, for any process H n adapted to the filtration (F n t ) = σ(γ n s |s ≤ t) if the three processes (γ n , H n , W ) jointly converge to (γ, H, W ), then also: (γ n , H n , W, ∫ H n dW ) ⇒ (γ, H, W , ∫ HdW ) in the sense of weak convergence on C([0, T ]; R 4 ).
Proof. We articulate the proof of this lemma as follows. First, we show tightness for the law of γ n . Then we show that the weak limits of γ n are the unique martingale solution to the given martingale problem. From this we deduce the first weak convergence result.
Hence, we get from the Burkholder-Davis-Gundy inequality for any p ≥ 1 < +∞ for some κ large enough together with the result of the previous lemma to see that the second term is uniformly bounded by h p(2α+1−ε) , whereas the first term can be estimated via: Hence, we eventually find: and thus an application of Kolmogorov's continuity criterion gives for all ζ < 1/2 and all p > 1/(2ζ): This is enough to ensure the tightness of the sequence B n and also of the couple (B n , W ), as well as the Hölder continuity of the limiting process.
Step 2. Next we prove that all weak limit points of γ n are solutions to the martingale problem of Proposition 4.4. Uniqueness of such solutions can then be proven as in [15, Proof of Theorem 8]. Fix f and ϕ 0 as required. Then the solutions ϕ n to Equation (4.8) with smooth noise θ n converge to the solution ϕ in L α+1−ε e(κ) for ε ∈ (6a/δ+1−2α, 3α−1). For fixed n we have that is a martingale with respect to the filtration (F n t ) t∈[0,T ] (since W is an F n -Brownian motion) and satisfies, due to our exponential bound from Lemma 4.5: Hence the sequence is uniformly integrable and together with Lemma 4.5 and the Skorohod embedding theorem this guarantees that up to taking a subsequence almost surely and in L 1 and it follows the latter is a martingale with respect to the canonical filtration (F t ) t∈[0,T ] .
Step 3. By tightness we can show that along a subsequence (γ n k , W ) ⇒ (γ, W ). If we can prove that the joint law of (γ, W ) is uniquely defined, the joint weak convergence follows. If the drift were a smooth function we could observe that W t = γ t − t 0 ∂ x U (s, γ s ) ds, with the right hand-side being a measurable function of the process γ. In the rough setting one has to be more careful, since it is not clear how the last term is defined.
We will show that for a sequence of measurable functions F n it is possible to write (γ t , W t ) = lim n (γ t , F n (γ)). Indeed for n ∈ N one can solve the equation We can subtract the term of lowest regularity to find ϕ n = Y ,n + ψ n with ψ n solving: (∂ t + 1 2 ∆ x +(X+X )∂ x )ψ n = X ,n +(X +X)X ,n , ψ n (T ) = 0.
Since Y n → Y in Y poly kpz the resonant product X X ,n converges to X X in CC 2α−1 p(a) . Thus along the same lines of Theorem 5.5 it is possible to find a paracontrolled solution to the previous equation with the structure: for any l > 0 and ε ∈ (6a/δ+1−2α, 3α−1). Moreover, because the resonant product converges to the right limit, as n → ∞ the above solutions ψ n converge in L α+1−ε e(l+t) to the solution ψ of Similarly the solutions ϕ n,n to the equation (∂ t + 1 2 ∆ x +(X n +X ,n )∂ x )ϕ n,n = X n +X ,n , ϕ n,n (0) = 0, exhibits the same structure ϕ n,n = Y ,n + ψ n,n and by the Lipschitz dependence on the parameters in Theorem 5.5 we have that ψ n,n converges to ψ in L α+1−ε e(l+t) . Along the subsequence (n k ) k along which (γ n k , W ) converges to (γ, W ) we can now apply [33, By using ϕ n instead of ϕ n k ,n k we find that: we find the required uniqueness of the law.
Step 4. Finally, as we already noted, the convergence of the stochastic integrals along a subsequence is a consequence of [33, Theorem 2.2]

Polymer measure
Our next aim is to construct the "full" polymer measure Q x0 . In principle we would like to apply the formulas (4.5) or (4.6) for the explicit Radon-Nikodym derivative with respect to P U x0 . Unfortunately we do not have sufficient control of the growth of X R , so we need to argue differently. Even for "smooth" noises θ in LC α/2 (R; C ∞ b (R)) Equation Definition 4.7. For Y in Y ∞ kpz and consider the solution e h to the RHE for an initial condition e h = e Y (0) · w 0 , with w 0 ≥ 0, w 0 ∈ C β e(l) , for some l ∈ R, β > 0, we define: is the measure under which the coordinate process γ solves dγ = ← − X (γ)dt+dW for a Brownian motion W started in x 0 and where P x0 is the Wiener measure.
Although the above notation suggests that we use the solution h to the KPZ equation associated to a smooth noise, this is really just notation (which we chose because it fits the Gibbs measure formalism). Actually the construction of the continuum random polymer measure does not depend on the existence of the solution h: we only need to understand the solution w = e h to the RHE with some strictly positive initial condition.
The construction of the polymer measure we review in the following is already known from a work by Alberts, Khanin and Quastel [1]. We implement their strategy in our pathwise setting and we link it with the approach of Delarue and Diel [15]. The idea is to show convergence of the finite dimensional distributions by controlling the density of the transition function with respect to the Lebesgue measure. Eventually a tightness result guarantees that the limiting measure is supported in the space of continuous functions. Lemma 4.8. Fix x 0 ∈ R, Y ∈ Y kpz and h such that w 0 = e h−Y (0) ∈ C β e(l) for some l ∈ R, β > 0. Then we can define a measure Q x0 = Q x0 (Y) on R [0,T ] through its finite dimensional distributions as follows. For any n ∈ N and 0 = t 0 ≤ t 1 < . . . < t n ≤ T we define: where y → Z(s, x; t, y) is the probability density where ← − f (r) = f (T −r) and Z(s, x; t, y) solves: x; t, y) = δ(x−y).
In particular, the Z form a consistent family of probability distributions and hence the measure Q x0 is well-defined. Let us consider the case of spatially smooth noise. We show that the one-dimensional distributions described by Z coincide with those of the measure Q x0 as from Definition 4.7: the general case follows similarly. Fix a Lipschitz function g and 0 < t ≤ T , then Then the Feynman-Kac formula guarantees that u = e ( ← − so that a simple calculation shows that e h (t, * ) g( * ). Hence the claim follows. Let us pass to proving the convergence of the finite-dimensional distributions. We check the convergence of E Q m x 0 [g 1 (γ t1 ) · · · g n (γ tn )] for all globally bounded and Lipschitz functions g, with Q m x0 = Q x0 (Y m ) and 0 < t 1 < . . . < t n ≤ T . Then h m (tn, * n) .
The last term in the product Z m (s, x; t n , * n )e ← − h m (tn, * n) g n ( * n ) solves in (s, x) Equation (4.12) on [0, t n ] with terminal condition where we used the structure of a solution e h to the RHE. Since g n ← − w P,m converges to for any ε ∈ (6a/δ+1−2α, 3α−1) (see Proposition 5.6), Corollary 5.7 guarantees that Z m (s, x; t n , * n )e ← − h m (tn, * n) g n ( * n ) converges to Z(s, x; t n , * n )e ← − h (tn, * n) g n ( * n ), the latter solving Equation (4.12) on [0, t n ] with terminal condition e ← − h (tn,x) g n (x). Note that the convergence holds in a space with an explosion at time s = t n . Since we are interested in the value of the solution only at the time t n−1 < t n this does not play a role. In h (tn, * n) g n ( * n ) converges to some R n in C α+1−ε e(κ) , up to taking a possibly larger κ. Now we pass to the second-to-last term. Again Z m (s, x; t n−1 , * n−1 )g n−1 ( * n−1 )e ( ← − this solution converges once more via Corollary 5.7 to Z(s, x; t n−1 , * n−1 )g n−1 ( * n−1 )e ( ← − Iterating this procedure n times we deduce the convergence of the finite-dimensional distributions. For the exponential bound let us choose a smooth function ϕ such that ϕ(x) = exp(l|x| δ ) for |x| > 1 and ϕ is smooth and bounded for |x| ≤ 1. Then: where in the last step as before we used the bounds from Corollary 5.7 and the fact that β > 0.
Now we show that the polymer measure is supported on the space of continuous functions.

Lemma 4.10.
There exists a value a crit > 0 such that for Y n ∈ Y ∞ kpz , Y ∈ Y kpz such that Y n → Y in Y kpz for some a ≤ a crit (a being the growth parameter in Y kpz from Definition

3.2) and for a sequence of initial conditions
the sequence of measures Q n x0 is tight in C([0, T ]). Moreover any accumulation point has paths which are almost surely ζ-Hölder continuous, for any ζ < 1/2.
Proof. We want to use the Kolmogorov criterion. For this reason we fix q > 1 and s ≤ t with |t−s| ≤ 1 and will prove that Given such an estimate the tightness of the sequence as well as the Hölder continuity of the limit points follow by an application of Kolomogorov's continuity criterion. To find this estimate fix Y ∈ Y kpz and w 0 = e h−Y (0) ∈ C β e(l) and let us rewrite the expectation through the densities: where e h is the solution to the RHE with initial condition e h and external data Y. Let us proceed one integration variable at a time: we consider x 1 fixed and estimate First, we shift x 1 to zero. For this purpose we introduce the notation g x1 (x) = g(x+x 1 ) and where the latter is obtained by shifting all the extended data by x 1 . Hence we find the identity: Z(Y)(s, x+x 1 ; t, y) = Z(Y x1 )(s, x; t, y−x 1 ), then we rewrite the term under consideration as ϕ x1 (r, x) = Z(Y x1 )(r, x; t, * 2 )| * 2 | 2q e ← − h x 1 (t, * 2) , and we aim at estimating ϕ x1 (s, 0) uniformly over x 1 . Note that ϕ x1 solves Equation in the sense of Corollary 5.7. Now we exploit the parabolic scaling of the equation. Let us define λ = |t−s| and write g x1 λ (r, x) = ϕ x1 (s+λ 2 r, λx) for r ∈ [0, 1], so that where formally the term s,λ should be understood as ← − θ x1 s,λ (r, x) = θ(T −s−λ 2 r, x 1 +λx). Rigorously this means that g x1 λ solves Equation Since e h solve the RHE it is of the form for some κ sufficiently large, so that with Lemma 2.14: λ , the latter being the solution to Following the results from Corollary 5.7 the solution ψ x1 λ is of the form: where we can estimate the norm of the last term: Now choose a crit so that a crit q 1 = δ : for a ≤ a crit we can conclude that up to choosing a larger C(M ) > 0: This concludes the proof.
We collect the two previous results in the following proposition.

Proposition 4.11.
For any x 0 , l ∈ R and Y which lies in Y kpz for a ≤ a crit (see Lemma 4.10) and e h−Y (0) ∈ C β e(l) there exists a measure Q x0 on C([0, T ]; R) such that for Y n ∈ Y ∞ kpz converging Y n → Y in Y kpz and initial conditions e h n −Y n (0) → e h−Y (0) in C β e(l) , the polymer measures converge weakly: In addition, under Q x0 the sample paths are a.s. ζ-Hölder continuous for any ζ < 1/2. Now we show that the measure Q x0 we just built has a density with respect to the singular Girsanov transform P U x0 , i.e. while we are not able to construct the measure using Equation (4.5) from our formal discussion above, the equation holds a posteriori. This equation is useful because it describes the singular part of the polymer measure in terms of the solution Y R to the linear equation (4.4), and therefore it is not necessary to understand the solution to the KPZ equation or the RHE in order to study the polymer measure.
Recall that Y R was defined as the solution to Indeed we can find a paracontrolled solution Y R of the form: The equation for Y P can be solved with calculations similar to the ones leading to Proposition 5.6. Eventually we find a solution Y P ∈ L α+1 e(κ) for κ large enough. Proposition 4.12. For any Y which lies in Y poly kpz for a < a crit (see Lemma 4.10), the measure Q x0 has a density with respect to the measure P U x0 which is given by: where W is the Brownian motion started in x 0 from Lemma 4.6.
Proof. Under the above hypothesis the existence of the measure Q x0 is guaranteed by Proposition 4.11, while the existence of the measure P U x0 follows from Proposition 4.4. From the computation at the beginning of this section, which lead us to Equation . Now the left-hand side converges by Proposition 4.11, while the right-hand side converges by Lemma 4.6. So we find that: . Taking f ≡ 1 and sending M → ∞, we obtain from Fatou's lemma that Thus we can pass to the limit over M → ∞ and deduce the result by dominated convergence.

Remark 4.13.
We have discussed the construction of the measure Q x0 . This in particular allows us to build the measure Q x0 from Equation (4.6) by choosing h = 0. In addition, using the fact that the Radon-Nikodym derivative integrates to 1, we find a representation for the solution h to the KPZ equation as follows:

Variational representation
Here we show that we can solve the martingale problem (4.1) associated to the KPZ equation, and that the solution solves a stochastic control problem. The first step is to define martingale solutions in the paracontrolled setting. One main difference with respect to the definition of [15,10] is that we do not directly solve the PDE associated to the martingale problem. This is because we cannot control the growth of the drift ∂ x h at infinity sufficiently well. Instead, we solve the PDE to remove the singular part ∂ x U of the drift, and then we add the regular part ν of the drift (which later will be a control) back by hand.
Following [24,Section 7] we will denote by pm the set of progressively measurable processes on [0, T ] × C([0, T ]; R). By this we mean that ν ∈ pm if for any 0 ≤ t ≤ T the restriction of ν to times smaller than t, where F = (F t ) 0≤t≤T is the canonical filtration on C([0, T ]; R). Definition 4.14. For an element ν ∈ pm we say that a measure P on the filtered measurable space (C([0, T ]; R), (F t )) is a martingale solution to the SDE dγ t = (∂ x U + ν t )(t, γ t )dt + dW t , γ 0 = x 0 , (4.14) if the following two conditions are satisfied for the coordinate process (γ t ): 1. P(γ 0 = x 0 ) = 1.  This allows to show that the polymer measure solves the SDE (4.1).
is a martingale with respect to the measure Q x0 . In fact, consider smooth data Y n ∈ Y ∞ kpz such that Y n → Y in Y kpz . Then we can find solutions ϕ n to the PDE (4.15) with ∂ x U replaced by ∂ x U n . Since Proposition 4.2 guarantees that these solutions satisfy for a suitably chosen κ ≥ 0 and ε ∈ (6a/δ+1−2α, 3α−1), and since the uniform subexponential bound (4.11) holds true, the process where Q n x0 is the polymer measure associated to Y n , as in Proposition 4.11. The same proposition guarantees that Q n x0 ⇒ Q x0 . Hence the martingale property is preserved in the limit.
We conclude this section on the polymer measure with a variational characterization of the solution to the KPZ equation.
where the optimal control ν is Proof. We follow step by step the original proof of [24,Theorem 7.3]. First, let us define h R, ∈ L 2α+1 e(κ) for an appropriate κ ≥ 0. In addition h R is a paracontrolled solution to the equation: which by reversing time we can translate into we can use h R as a test function, according to Definition 4.14. From this point onwards we can follow exactly the proof of [24] to get to the conclusion that for any ν ∈ pm and γ ∈ M(ν, x 0 ) , where in the last line we took the supremum on both sides in the line above and then forgot the term withν. For fixed ν holds only ifν = 0. Thus the supremum is achieved in the polymer measure and equals ← − h R (0, x 0 ).

A solution theorem
We consider an abstract paracontrolled equation of the form: for some functionals R and F which we will specify later, and where Y ∈ Y kpz and ν is simply an additional parameter living in a Banach space X , which we add to treat certain applications. At an intuitive level R represents a smooth rest term, is the irregular part of a product and is the ill-posed part of a product, the latter term being the one which requires a paracontrolled structure from the solution.
Actually it will be necessary to consider slightly more general Y, allowing for an additional singularity: see Definition B.1 for the definition of Y ζ,b kpz . Now we introduce the Banach space of paracontrolled distributions that will contain the solution to Equation (5.1). Consider u 0 ∈ C β e(l) for some l ∈ R and β ∈ (2α−1, 2α+1] (recall the regularity parameter α from Table 2 and the preceding discussion) as well as a parameter ε > 0 andβ Furthermore, fix a time horizon T h ≥ 0 and Y ∈ Y ζ,b kpz ([0, T h ]) for some values ζ, b ≥ 0 which we will specify later. The parameter ε represents a small gap between the regularity of the solution we prove and the expected maximal regularity and appears essentially to deal with the global spatial well-posedness. The parameterβ quantifies the time blow-up at t = 0 in the space C 2α+1 . The parameter β quantifies the blow-up in the space C α+1 (cf. Lemma 2.14).
Then we introduce a subset which is defined by the property: Moreover, we endow D(Y) with the product topology and a product norm. With some abuse of notation we write |||u||| = (u, u , u ) D(Y) .
Since we want to compare also paracontrolled distributions which are controlled by different enhanced data we introduce the quantity Since the proof of the solution theorem relies on a contraction argument on a small time interval we also introduce the notation D 0 S (Y) for S ≤ T h for the space D(Y) where we replaced the time horizon T h with S. We also use the convention that D 0 0 (Y) = C β e(l) . This is motivated by the fact that a point in D 0 S will later be an initial condition for the solution on [S, S+τ ] for some small τ . Then if we fix 0 ≤ T < T r ≤ T h and an initial condition We endow this space with the product norm on: If we do not fix any initial condition, we write D T Tr (Y). Furthermore we remind the notation V s for the integration operator: where (P t ) is the semigroup generated by 1 2 ∆ x . Now we state the assumption on the coefficients of the equation. As a rule-of-thumb they must be Lipschitz dependent on u and locally Lipschitz dependent on Y ∈ Y ζ,b kpz . Recall that X is an arbitrary Banach space, and we write · X for its norm.
as well as ν i in X and u i 0 in C β e(l) for i = 1, 2 such that: ≤ M and we require that there exists a p ≥ 1 such that the following holds true: 1. There exists a γ > 0 such that uniformly over Y, ν and T , T r , we have that: is a Lipschitz function which satisfies: is Lipschitz continuous (for fixed Y) and satisfies uniformly over Y, Y 1 , Y 2 and T , T r : Remark 5.2. These assumptions provide a minimal working environment that is sufficient for our needs, and they could of course be generalized. In every point, the first two inequalities are necessary for the fixed-point argument, and the last one to obtain the locally Lipschitz continuous dependence of the fixed point on the parameters.
The central idea in the paracontrolled approach is that the ill-posed resonant product is well defined for paracontrolled distributions. This is the content of the next result. Tr (Y), and Y ∈ Y ζ,b kpz the following product estimate holds: Tr .
If we consider two different enhanced data Y i ∈ Y ζ,b kpz as well as initial conditions Tr we can also bound Proof. Let us prove the first estimate and assume T r = T . We define then we can compute: .
Similarly we can treat the other terms, by applying the commutation results of Lemma 2.8 to the second term and Lemma 2.10 to the third term, as well as the resonant product estimate from Lemma 2.8 for the last term. The second estimate follows similarly.
In view of the previous lemma, we can rigorously make sense of the equation under consideration. ).
Moreover, the solution depends locally Lipschitz continuously on the parameters of the equation: for two solutions u i , i = 1, 2, associated to Y i and parameters ν i we find that Proof. Fix 0 ≤ T < T r ≤ T h and u T ∈ D 0 T (Y). Let us define the following map on D T Tr (Y, u T ). For t ≥ T we write: and I (u) = u T on [0, T ]. By induction, we assume that if T > 0, then u T is a solution to the equation on [0, T ]: in particular we will use that u T = F (u T ).
For the sake of brevity we will write ||| · ||| for the norm in D T Tr (Y, u T ) and |||u T ||| for the norm of u T in D 0 T (Y). We show that (I , I , I ) maps D T Tr (Y, u T ) into itself, similar arguments then show that (I , I , I ) 2 is a contraction on the same space (the presence of the square will guarantee that also the derivative term is contractive). We proceed one term at a time.
Step 1. Let us start with I , for which Assumption 5.1 yields Regarding the first term: By the first estimate of the Proposition 2.12 we can bound it with (β , T , |||u T |||), with satisfying the bound Regarding the second term, we use the second estimate of the same proposition to get: where the term (T r −T ) γ1 is a consequence of the last estimate from Lemma 2.14 with γ 1 = (ε−4a/δ)/2 > 0. The last line follows from the third condition on F . A similar estimate holds for the ill-posed product: where in the first step we used the last estimate from Lemma 2.14 and we defined γ 2 = (ε−6a/δ−2ζ−λ)/2 and λ ∈ (0, ε−6a/δ−2ζ). We chose to subtract an additional (arbitrarily small) regularity λ in order to apply the second estimate of Lemma 2.14 in the second step and thus gain a factor ζ in the time-explosion. Hence, we eventually estimate:

([T ,Tr])
From these estimates it follows that I maps D T Moreover similar calculations, based on the Lipschitz assumptions on the coefficients, show that I is a contraction for fixed initial condition u(T ) = u T (T ), provided that Step 2. We consider the paracontrolled remainder term. Since u T is a solution to the equation on [0, T ] we find that: Proceeding as before we can estimate: where we used the same estimate as before for the rest term R and the ill-posed product. Through the bounds from Lemma 2.10 for the commutators C 3 , C 4 we can then estimate the last two terms in the sum via Tr where in the last step we used the estimate on F from Assumption 5.1. Eventually we find the following bound: Step 3. Now we can conclude that for some q large enough and the map (I , I , I ) 2 is a contraction on D T Tr (Y, u T ) and thus it has a unique fixed point. Indeed, due to the presence of the square, and our assumptions on F , the derivative I inherits the contractive property from I . Since the length T * of the interval [T , T r ] could be chosen independently of u T , we can iterate this procedure and concatenate the fixed points to get a solution on [0, T h ]. Then the exponential bound follows immediately by observing that we need to iterate approximately where in the last step we used that T ≥ T * so that we have a good bound on . The local Lipschitz dependence on the parameters follows along the same lines.
In the remainder of the section we will apply this to several concrete linear equations.

Rough heat equation
In this section we show how to solve Equation (3.5), which we recall here: This equation can be written in the form of Equation (5.1) with Our aim is clearly to apply Theorem 5.5: For this reason we have to check the requirements from Assumption 5.1. The first step is counting the regularities. Taking away the time derivative (which for technical reasons we treat differently) we find that for fixed T h > 0 and uniformly over 0 ≤ T < T r ≤ T h : so that Proposition 2.12 applied to this term and Lemma 2.13 guarantee: for any ν < 2α+1−4a/δ. Thus the third and fourth estimates from Lemma 2.14 then provide the bound: The bounds for the differences follow from the bilinearity of R and F . Hence applying Theorem 5.5 guarantees the following result.
Proposition 5.6. For l, ε, ζ, b, a, β, β ,β as in the requirements of Assuption 5.1, Equation (3.5) admits a unique paracontrolled solution with local Lipschitz dependence upon the parameters. That is, for initial conditions w 1 0 , w 2 0 and extended data Y 1 , Y 2 which satisfy the requirements of Assumption 5.1, there exist respectively two unique solutions w P 1 , w P 2 to the RHE, that satisfy: Moreover we can bound the norm of the solution in terms of the extended data as follows: ) for some q ≥ 0 large enough. In particular, if ζ = 0 these are the unique solutions to the RHE in the sense of Definition 3.5.
We can also solve a time-reversed version of the RHE, Equation (4.12). In particular, we are also interested in uniform estimates over parabolic scaling of the equation.
Consider some Y ∈ Y kpz with ξ = LY and let us write , as well as f s,λ (t, x) = f (s+λ 2 t, λx). The aim is to solve the equation: with w 0 ∈ C β e(l) for some β ∈ (2α−1, 2α+1], l ∈ R we consider solutions w of the form: with the time-reversed w P rev (t, x) = w P (τ −t, x) solving the equation w P rev (0) = w 0 , with µ = T −s−λ 2 τ . This is exactly the same equation as for the paracontrolled term of the RHE, up to translations and scaling. Following Definition 3.2 and Proposition B.2, translating the enhanced data by a factor µ and rescaling by a factor λ gives rise to a valid element Y µ,λ of Y ζ,b kpz for b = 2a and any ζ ∈ (1/2−α, α]. Hence the previous equation admits a paracontrolled solution in the sense of Proposition 5.6. We collect this information in the following result. Corollary 5.7. For any w 0 ∈ C β e(l) , for β ∈ (2α−1, 2α+1], l ∈ R and λ ∈ [0, 1], τ ∈ [0, λ −2 (T −s)], and Y ∈ Y kpz there exists a unique paracontrolled solution w P rev to the previous equation. In particular the following bound holds for some κ, q ≥ 0 and any ε ∈ (6a/δ+1−2α, 3α−1): ).
Moreover the solution depends continuously on the parameters in the following sense: for two different enhanced data Y i and initial conditions w i

Sharp equation
Now we consider the "Sharp" equation, that is Equation (3.9): We only need to check that R satisfies the properties of Assumption 5.1. Since R is constant in u this reduces to the first and third property. Due to the non-linearity in h P we work under the additional assumption that β > 0. We see that: and note that due to our assumptions on β we have that β ∈ (0, 1). In view of the regularity assumptions on the enhanced data (see Table 2) and the paraproduct estimates from Lemmata 2.8 and 2.10 we see that: ) 4 so that an application of the Schauder estimates of Proposition 2.12 guarantees that: The local Lipschitz dependence on the parameters then follows similarly by multi-linearity. Thus we can apply Theorem 5.5 and obtain the result below.
Proposition 5.8. For any l ∈ R u 0 ∈ C β e(l) , for β ∈ (0, 2α+1], and Y ∈ Y kpz , (h P , h ) ∈ X there exists a κ > l such that Equation (3.9) has a unique solution h in L β ,2α+1 e(κ) , for β as in Equation (5.4). For two initial conditions u i 0 and two extended data Y i and parameters h P i , h i where i = 1, 2, which satisfy the requirements of Assumption 5.1, there exist respectively unique solutions h 1 , h 2 to the equation and they satisfy .
Proof. Theorem 5.5 yields that the paracontrolled solution to Equation (3.9), which according to the theorem has only regularity α+1−ε, has a vanishing derivative since F = 0. Moreover Y ∈ Y ζ,b kpz for b = a and ζ = 0. Thus applying one last time the Schauder estimates to the solution h give us the bounds in L β ,2α+1 e(κ) .

A Exponential and logarithm on weighted Hölder spaces
Here we discuss the regularity of the exponential and logarithmic maps on weighted Hölder spaces.
Lemma A.1. Consider any α ∈ (0, 2) \ {1} and R,l ≥ 0. Then there exists an l = l(R) ≥ 0 such that the exponential function exp maps Moreover the exponential map is locally Lipschitz continuous, i.e. for f, g in the set above such that Proof. Due to our choice of α and the classical characterization of Besov spaces (see Corollary 2.7), we need to find a bound for the uniform norm of exp(f ) and, if α > 1, ∂ x exp(f ) as well as for the α Hölder seminorm of exp(f ) or ∂ x exp(f ), according to whether α < 1 or α > 1 respectively. Regarding the first bound, since f ∞,p(δ) ≤ R it follows directly that implying that exp(f ) ∞,e(R) ≤ 1.
Furthermore for any β ∈ (0, 1 ∧ α] we also have wheneverl > R. Moreover exp is also locally Lipschitz continuous, since for any two functions f and g as in the statement of this lemma we can write: so that applying inequalities of the kind |aã − bb| ≤ |a − b||ã| + |ã −b||b| along with an appropriate choice ofl leads to the bound: Finally, if α ∈ (1, 2), we can write ∂ x e f = e f ∂ x f , so that via the previous calculations and through an application of paraproduct estimates we deduce that ∂ x e f lies in C α−1 e(l) for l =l +l, along with the local Lipschitz continuity.
The same calculations also show that the result still holds if we introduce time dependence: Lemma A.2. Consider any α ∈ (0, 2) \ {1}, γ ∈ (0, 1) and R,l ≥ 0. Then there exists an l = l(R) ≥ 0 depending on R such that the exponential function maps: . Now we pass to a dual statement, namely the continuity of the logarithm.
Proof. As before we use the classical definition of Hölder spaces and we only treat spatial regularity, as the time regularity follows similarly. We also just discuss the local Lipschitz continuity, since then the first statement follows by choosing g = 1. For f, g ∈ A and for all ξ ≥ f (t, x) ∧ g(t, x) we find 1 ξ e(r)(x).
Thus by the mean value theorem and for l ≥l + r: where every tree Y • solves the same PDE as in Table 2 under the condition that implying that all other trees are allowed to have inhomogeneous initial conditions. To keep the notation simple we omit from writing the dependence on these inhomogeneous initial conditions. We will also sometimes omit the explicit dependence on T h and write Y ζ,b kpz . Next we will formulate translations and parabolic scaling operations on the extended data. As for the scaling, we will only zoom into small scales and therefore the scaling parameter λ will be small: λ ∈ (0, 1]. Thus, for any τ ∈ [0, T ) we write θ τ,λ (t, x) = θ(τ +λ 2 t, λx). We change the time horizon accordingly to T τ,λ = λ −2 (T −τ ). The question we want to answer is whether for given Y(θ n ) ∈ Y ∞ kpz converging to Y in Y kpz we can show also the convergence of Y(θ n τ,λ ) to some Y τ,λ : this is the content of the following result.

Proposition B.2.
For any τ, λ as above and for every Y in Y kpz and ζ ∈ ( where for a given smooth noise θ ∈ LC α/2 (R; and similarly for the elements Y (θ τ,λ ), Y (θ τ.λ ). The elements are defined respectively as the solution to Furthermore we have the estimate: Proof. We concentrate on the proof of the uniform bound. The convergence result then follows from the fact that the rescaling operator is linear. Let us write Y • τ,λ for Y • (θ τ,λ ) defined as above. Note that and similarly for Y , where Λ λ is the spatial rescaling operator with Λ λ f (x) = f (λx). We can find the required bounds for X τ,λ and for the tree terms Y • τ,λ in view of [21,Lemma A.4]. While the results from [21] do not treat weighted spaces, they hold nonetheless in the weighted setting. To see this one has to take care of the effect of rescaling, and it is only because we are zooming in (λ ≤ 1) that the scaling does not affect the weight. The most complicated object that we have to consider is the ill-posed product . First, we treat the rescaling parameter λ. Note that Y τ, where "Commutator" is defined implicitly through the formula above, by taking the first term on the right hand-side to the left. An application of [21, Lemma B1] (taking into account the remark about the weights from above) tells us that: and an application of [21, Lemma A4] tells us that: . Now we can estimate the norm of the ill-posed product uniformly by: and since ζ ≤ α, this can be bounded uniformly over λ and t by the quantity: .
It is easy to estimate the last term uniformly over t and τ . Let us consider the first term.
Here we have to take into account that Y τ (t) = Y (τ + t) − P t Y (τ ), where we recall that P t indicates convolution with the heat kernel and that it commutes with derivatives.
Since we have no a priori estimates for P t ∂ x Y (τ ) ∂ x Y τ (t) we need to apply the usual estimates for the resonant product. For that purpose note that and since 2ζ+2α−1 > 0 we can bound the norm of the ill-posed product by Now all the required properties follow promptly.

C Asymmetric approximation of a resonant product
Next we prove a result which is a slightly asymmetric version of the computations in [24,Section 9.5]. Indeed we show convergence of X ,n X to X X, that is we only regularize one of the two factors.

D Schauder estimates
In this section we review classical Schauder theory for space-time distributions. Such results are well-known in literature: we adapt them in order to deal with time-dependent weights and blow-ups at time t = 0. The method of proof we use is essentially the same of [23,Theorem 1], which is based on the construction of the Young integral. Such integral is the content of the next lemma.
Proof. Following the classical construction via the sewing lemma (cf. [16,Lemma 4.2]) we can build the integral t s f (r)dh(r) for any t ≥ s > 0. We repeat the construction in order to get a tight control on the singularity in zero and the weights involved. We will prove the result for time independent weights. The general case then follows from the identity: ∫ t 0 f (s)dh(s) = ∫ t 0 f (s ∧ t)dg(s). Define for n ≥ 0 and t n k = k/2 n I n t = +∞ k=0 f (t n k+1 )(h(t n k+1 ∧ t)−h(t n k ∧ t)), unlike the more traditional integration scheme we choose a right base-point to remove some tedious, but only technical difficulties when dealing with time blow ups. We want to estimate the following quantity We will treat only the case β > 0, since β = 0 follows similarly. We fix n and estimate one of the terms above. We will divide the estimate in two parts.
Step 1. First we look on large scales, that is |t−s| > 2 −n . To lighten the notation we write g u,v = g(u)−g(v). We also write t n ( resp. t n ) for the nearest left (resp. right) dyadic point to t: k n (t) = arg min k|k≤2 n t |t−t n k |, t n = k n /2 n , t n = t n +1/2 n .
We start by considering t ≤ 2s. In this case we will estimate the terms t β (I n+1 s,t −I n s,t ) and (t β − s β )(I n+1 s −I n s ) separately. Let us start with the first one. Since |t−s| > 2 −n we have in particular that s n < t and a close inspection of the sums reveals that: I n t,s −I n+1 t,s = f s n ,s n+1 h s n+1 ,s + kn(t)−1 k=kn(s)+1 2k+1 ,t n+1 2k + f t n ,tn+1 h tn+1,tn + f t n ,t n+1 h t,tn+1 .
This approximation has the advantage of simplifying our calculations, but the disadvantage of not being continuous in time. We want to show that V n t converges to some V t which lies in L β,η e(l+t) for η < γ−2a/δ. For this reason fix κ, ε ≥ 0 small at will such that γ+κ > 2 and ζ = (η+κ+ε)/2+a/δ < 1. We divide the proof in two steps, estimating the spatial and the temporal regularity differently. For t n , t n , t n k , k n (t) we use the same definition as in the proof of Lemma D.1.
Step 1. We show that for fixed t the sequence V n t converges in C η e(l+t) . As in the previous proof we show that: . We can write the difference of the increments as: Now we can estimate |t−s| −a/δ f C ν e(l+s) , for t ≥ s. Now we have to estimate the norm of the increment. Here the time explosions come into play: we write X u,v = u −β X u,v + (u −β −v −β )X(v), with X(t) = t β X(t) ∈ C γ/2 L ∞