A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^2$

We propose a simple construction of the solution to the continuum parabolic Anderson model on $\mathbf{R}^2$ which does not rely on any elaborate arguments and makes extensive use of the linearity of the equation. A logarithmic renormalisation is required to counterbalance the divergent product appearing in the equation. Furthermore, we use time-dependent weights in our spaces of distributions in order to construct the solution on the unbounded space $\mathbf{R}^2$.


Introduction
The goal of this note is to construct solutions to the continuous parabolic Anderson model: Here, u is a function of t ≥ 0 and x ∈ R 2 , while ξ is a white noise on R 2 . Notice that ξ is constant in time, so this is quite different from the model studied for example in [CM94,CJK13]. The difficulty of this problem is twofold. First, the product u·ξ is not classically well-defined since the sum of the Hölder regularities of u and ξ is slightly below 0. Second, our space variable x lies in the unbounded space R 2 so that one needs to incorporate weights in the Hölder spaces at stake; this causes some difficulty in obtaining the fixed point argument, since one would a priori require a larger weight for u · ξ than for u itself. The first issue is handled thanks to a renormalisation procedure which, informally, consists in subtracting an infinite linear term from the original equation. The main trick that spares us from using elaborate renormalisation theories is to introduce the "stationary" solution Y of the (additive) stochastic heat equation and to solve the PDE associated to v = ue Y instead of u. This is analogous to what was done for example in [DPD02,HPP13]. The second issue is dealt with by choosing an appropriate time-increasing weight for the solution u. Roughly speaking, if ξ is weighted by the INTRODUCTION polynomial function p a (x) = (1 + |x|) a with a small, and u s is weighted by the exponential function e s (x) = e s(1+|x|) , then t 0 P t−s * (u s · ξ)(x) ds requires a weight of order t 0 p a (x)e s (x) ds, which is smaller than e t (x). This argument already appears in [HPP13], and probably also elsewhere in the PDE literature.
The solution to the (generalised) parabolic Anderson model has already been constructed independently by Gubinelli, Imkeller and Perkowski [GIP12] and by Hairer [Hai14] in dimension 2 and, to some extent, by Hairer and Pardoux [HP14] in dimension 3. (The latter actually considers the case of dimension 1 with space-time white noise, but the case of dimension 3 with spatial noise has exactly the same scaling behaviour, so the proof given there carries through mutatis mutandis. The main difference is that some of the renormalisation constants that converge to finite limits in [HP14] may diverge logarithmically.) However, in all of these results the space variable is restricted to a torus, which is the constraint that we lift in this note. The construction that we propose here is very specific to (PAM) in dimension 2: in particular, as it stands, it unfortunately applies neither to the generalised parabolic Anderson model considered in [GIP12,Hai14], nor to the case of dimension 3.
Let us now present the main steps of our construction. First, we introduce a mollified noise ξ ǫ := ̺ ǫ * ξ, where ̺ is a compactly supported, even, smooth function on R 2 that integrates to 1, and ̺ ǫ (x) := ǫ −2 ̺( x ǫ ) for all x ∈ R 2 . In order to quantify the Hölder regularity of ξ, ξ ǫ , we introduce weighted Hölder spaces of distributions, see Section 2 below for the general definitions. Informally speaking, given a weight w and an exponent α, C α w consists of those elements of C α that grow at most as fast as w at infinity. We have the following very simple convergence result, the proof of which is given on Page 5 below.
Lemma 1.1 For any given a > 0, let p a (x) = (1 + |x|) a on R 2 as above. For every ǫ, κ > 0, ξ ǫ belongs almost surely to C −1−κ pa (R 2 ). As ǫ ↓ 0, ξ ǫ converges in probability to ξ in C −1−κ pa . From now on, a is taken arbitrarily small. Since, for any fixed ε > 0, the mollified noise ξ ε is a smooth function belonging to C α pa for any α > 0, the SPDE is well-posed, as can be seen for example by using its Feynman-Kac representation. The constant C ǫ appearing in this equation is required in order to control the limit ǫ → 0 and will be determined later on. Second, let G be a compactly supported, even, smooth function on R 2 \{0}, such that G(x) = − log |x| 2π whenever |x| ≤ 1 2 . Then, there exists a compactly supported smooth function F on R 2 that vanishes on the ball of radius 1 2 and such that, in the distributional sense, we have: (1.1) INTRODUCTION the Poisson equation driven by ξ ǫ , up to some smooth term: From now on, D x i denotes the differentiation operator with respect to the variable x i , with i ∈ {1, 2}. More generally, for every ℓ ∈ N 2 , we define D ℓ x f as the map obtained from f by differentiating ℓ 1 times in direction x 1 and ℓ 2 times in direction x 2 . We also use the notation ∇f = (D x 1 f, D x 2 f ). The following result is a consequence of Lemma 1.1 together with the smoothing effect of the convolution with G and D x i G.

Corollary 1.2 For any given
x)e Yǫ(x) for all x ∈ R 2 and t ≥ 0, and we observe that where we have introduced the renormalised process At this stage we fix the renormalisation constant C ǫ to be given by where the part denoted by O(1) converges to a constant (depending on the choice of G) as ǫ → 0. The following result, which is proven on Page 8, shows that this sequence of renormalised processes also converges in an appropriate space. We refer to Nualart [Nua06] for details on Wiener chaoses.

Proposition 1.3
For any given κ ∈ (0, 1/2), the collection of processes Z ǫ converges in probability as ǫ → 0, in the space C −κ pa (R 2 ), towards the generalised process Z defined as follows: for every test function η, Z, η is the random variable in the second homogeneous Wiener chaos associated to ξ represented by the L 2 (dz dz) function We are now able to set up a fixed point argument for the process v ǫ with controls that are uniform in ǫ. The precise statement of the theorem requires us to introduce an appropriate Banach space E r ℓ,T ; at first reading, one can replace convergence in E r ℓ,T by locally uniform convergence in (0, T ] × R 2 . For any r > 0, ℓ ∈ R and T > 0, we consider the Banach space E r ℓ,T of all continuous functions v on (0, T ] × R 2 such that |||v||| ℓ,T := sup This leads us to the main result of this article.

Theorem 1.4
Consider an initial condition u 0 ∈ C −1+4κ e ℓ and a final time T > 0. For all ℓ ′ > ℓ, the sequence of processes v ǫ converges in probability as ǫ → 0 in the space E 1+2κ ℓ ′ ,T to a limit v which is the unique solution of As a consequence, u ǫ converges in probability in E 1−κ ℓ ′ ,T towards the process u = ve −Y .

Weighted Hölder spaces
In this section, we introduce the appropriate weighted spaces that will allow us to set up a fixed point argument associated to (PAM). We work in R d for a general dimension d ∈ N, even though we will apply these results to d = 2 in the next sections.
In this article, we will consider two families of weights indexed by a, ℓ ∈ R: Observe that the constant C can be taken uniformly for all p a and e ℓ , as long as a and ℓ lie in a compact domain of R 2 . We can now consider weighted versions of the usual spaces of Hölder functions C α (R d ).
More generally, for every α > 1, we define C α w (R d ) recursively as the space of functions f which admit first order derivatives and such that We then extend this definition to negative α. To this end, we define for every r ∈ N, the space B r 1 of all smooth functions η on R d , which are compactly supported in the unit ball of R d and whose C r norm is smaller than 1. We will use the notation η λ x to denote the function y → λ −d η( y−x λ ).

Definition 2.3
For every α < 0, we set r := −⌊α⌋ and we define C α w (R d ) as the space of distributions f on R d such that In order to deal with the regularity of random processes, it is convenient to have a characterisation of C α w that only relies on a countable number of test functions. To state such a characterisation, we need some notation. For any ψ ∈ C r , we set We also define Λ n : Proposition 2.4 Let α < 0 and r > |α|. There exists a finite set Ψ of compactly supported functions in C r , as well as a compactly supported function ϕ ∈ C r such that forms an orthonormal basis of R d , and such that for any distribution ξ on R d , the following equivalence holds: ξ ∈ C α w if and only if ξ belongs to the dual of C r and Proof. This result is rather standard and is obtained by a wavelet analysis, see [Mey92,Dau88] or [Hai14,Prop. 3.20]. In these references, the spaces are not weighted, but since all the arguments needed for the proof are local, it suffices to use the fact that w(y) w(x) is bounded from above and below uniformly over all x, y such that |x − y| ≤ 1 to obtain our statement.

Remark 2.5
If ξ is a linear transformation acting on the linear span of the functions ϕ 0 x , ψ n x such that (2.1) is finite, then ξ can be extended uniquely to an element of C α w .
We are now in position to characterise the regularity of the noise.
We restrict to the first bound, since the second is simpler. For any integer p ≥ 1, we write where we used the equivalence of moments of Gaussian random variables. Recall that the L 2 norm of ψ n x is 1, that the restriction of Λ n to the unit ball of R 2 has at most of the order of 2 2n elements and that Ψ is a finite set. Taking p large enough, we deduce that the triple sum converges, so that ξ admits a modification that almost surely belongs to C α pa . We now turn to ξ ǫ − ξ α,pa : the computation is very similar, the only difference rests on the term For p large enough and α < −1, we obtain Consequently, for p large enough E ξ ǫ − ξ α,pa → 0 as ǫ ↓ 0.
Let w f and w g be two weights on R d . We have the following elementary extension of the classical theorem [BCD11, Thm 2.52].

Then there exists a continuous bilinear map
that extends the classical multiplication of smooth functions.
Proof. Let χ be a compactly supported, smooth function on R d such that k∈Z d χ(x− k) = 1 for all x ∈ R d . For simplicity, we set χ k (·) := χ(· − k). Writing · α for the α-Hölder norm without weight (i.e. with weight 1), observe that h ∈ C α w if and only if hχ k α w(k) hold uniformly over all k ∈ Z d , and h α,w is equivalent to the smallest possible bound. From [BCD11, Thm 2.52], we know that f χ k · gχ ℓ is well-defined for all k, ℓ ∈ Z d , and that the bound f χ k · gχ ℓ α f χ k α gχ ℓ β holds. Consequently, we get f χ k · gχ ℓ α w f (k)w g (ℓ) f α,w f g β,wg , uniformly over all k, ℓ ∈ Z d . Since the number of non-zero terms among { f χ k · gχ ℓ , η x , k, ℓ ∈ Z d } is uniformly bounded over all η ∈ B r 1 , all x ∈ R d and all f, g as in the statement, we deduce that f · g := k,ℓ∈Z d f χ k · gχ ℓ is well-defined and that f · g α,w f wg f α,w f g β,wg holds. Finally, the multiplication of [BCD11, Thm 2.52] extends the classical multiplication of smooth functions, therefore, from our construction, it is plain that this property still holds in our case.
Let now P t (x) := (2πt) − d 2 e −|x| 2 /4t be the heat kernel in dimension d. We write P t * f for the spatial convolution of P t with a function/distribution f on R d . We have the following regularisation property which is a slight variant of well-known facts.
Lemma 2.7 For every β ≥ α and every f ∈ C α e ℓ , we have uniformly over all ℓ in a compact set of R and all t in a compact set of [0, ∞).
Proof. We use a decomposition of the heat kernel P t (x) = P + (t, x) + P − (t, x) where P − is smooth and P + is supported in the unit ball centred at 0. Using the decay properties of the heat kernel, the statement regarding P − is easy to check. Concerning the singular part, one writes P + = n≥0 P n where each P n is a smooth function supported in the parabolic annulus {(t, x) : 2 −n−1 ≤ |t| 1 2 + |x| ≤ 2 −n+1 } and such that P n (t, x) = 2 dn P 0 (2 2n t, 2 n x). Then, we get uniformly over all η ∈ B r 1 , all x, y ∈ R d , all t > 0, all n ≥ 0 and all k ∈ N 2 . Notice that P n (t, ·) vanishes as soon as n ≥ 1 − 1 2 log 2 t. Consequently, , so that the statement follows by interpolation.

Bounds on Y and Z
Let us collect a few facts on the behaviour of smooth functions with a singularity at the origin; we refer to [Hai14,Sec. 10.3] for proofs. For any smooth function K : R d \{0} → R and any real number ζ, we define where the first supremum runs over k ∈ N d and |k| = i k i . We say that K is of order ζ if |||K||| ζ;m < ∞ for all m ∈ N. Recall ̺ ǫ from the introduction, and define K ǫ = K * ̺ ǫ . If K is of order ζ ∈ (−d, 0) then for all m ∈ N, there exists C > 0 such that |||K ǫ ||| ζ;m ≤ C|||K||| ζ;m , uniformly over ǫ ∈ (0, 1]. Furthermore, for all ζ ∈ [ζ − 1, ζ), there exists a constant C > 0 such that If K 1 and K 2 are of order ζ 1 and ζ 2 respectively, then K 1 K 2 is of order ζ = ζ 1 + ζ 2 and we have the bound where C is a positive constant. Assume that ζ 1 ∧ ζ 2 > −d. We set ζ = ζ 1 + ζ 2 + d. If ζ < 0, then K 1 * K 2 is of order ζ and we have the bound (3.1) On the other hand, if ζ ∈ R + \N and K 1 , K 2 are compactly supported, then the function is of order ζ and a bound similar to (3.1) holds, but with the constant C depending on the size of the supports in general. We will apply these bounds to the function G defined in the introduction. Since G is smooth on R 2 \{0}, compactly supported and satisfies G(x) = − log |x| 2π in a neighbourhood of the origin, it is a function with a singularity of order ζ, for all ζ < 0, according to our definition. From now on, we set ̺ * 2 = ̺ * ̺ and we assume without loss of generality that ̺, ̺ * 2 are supported in the unit ball of R 2 . κ ∈ (0, 1). We have the bounds
Proof. By translation invariance, it suffices to consider x = 0. The random variables Z(η λ ), Z ǫ (η λ ) and Z ǫ (η λ ) − Z(η λ ) all belong to the second homogeneous Wiener chaos associated with the noise ξ. This is because the constant C ǫ has been chosen to cancel the 0-th Wiener chaos component of |∇Y ǫ | 2 . We start with the second bound of the statement: so that the bounds at the beginning of the section yield the desired result. The first bound of the statement follows by replacing G ǫ by G in the expression above. We turn to the proof of the third bound. To that end, we write so that, once again, the bounds on the behaviour of singular functions at the origin yield the asserted bound.
Proof of Proposition 1.3. Let L denote an arbitrary element among Z, Z ǫ and Z − Z ǫ . Using the equivalence of moments of elements in inhomogeneous Wiener chaoses of finite order, we obtain When L is equal to Z or Z ǫ , Lemma 3.1 ensures that E[L(ψ n x ) 2 ] 2 −2n+κn uniformly over all x, n, and ǫ. Moreover, #(Λ n ∩ B(k, 1)) 2 2n , so that This quantity is finite for α = −κ and p large enough. Therefore, Z and Z ǫ belong to uniformly over all x, n and ǫ. Then, the same arguments as before yield so that, choosing for instance α = −3κ and p large enough, one gets the bound E Z − Z ǫ −3κ,pa ǫ κ 2 uniformly over all ǫ ∈ (0, 1], thus concluding the proof.
Proof of Corollary 1.2. Since G is compactly supported and coincides with the Green function of the Laplacian in a neighbourhood of the origin, the classical Schauder estimates [Sim97] imply that for any α ∈ R, the bounds hold uniformly over all f ∈ C α . Recall the functions χ k , k ∈ Z d from the proof of Theorem 2.6. Since G is compactly supported, we deduce from the bounds above that uniformly over all k ∈ Z d and all f ∈ C α w . For fixed x, only a bounded number of {χ k (x), k ∈ Z d } are non-zero, uniformly over all x ∈ R d . Since f = k∈Z d f χ k , we deduce that uniformly over all f ∈ C α w . This being given, the statement is a direct consequence of Lemma 1.1.

Picard iteration
Fix κ ∈ (0, 1 2 ), and let the parameter a appearing in the weight p a be any value in (0, κ 2 ). Let g, h (1) , h (2) ∈ C −κ pa and f ∈ C −1+4κ e ℓ be given. We define the map v → M T,f (v) as follows: In this equation, there is an implicit summation over i ∈ {1, 2}. This convention will be in force for the rest of the article.
Proof. First, Lemma 2.7 ensures that P t * f 1+2κ,e ℓ+t t −1+κ f −1+4κ,e ℓ uniformly over all t in any given compact interval of R + . Second, using Theorem 2.6 and the simple inequality uniformly over all s, t in a compact set of R + and all ℓ in a compact set of R. Then, by Lemma 2.7 and using a < κ/2, we obtain , where ℓ * 0 = ℓ 0 + T * /2. Since ℓ * 0 ≤ ℓ 0 + T , the map M T * ,f * is again a contraction on E ℓ * 0 ,T * , so that it admits a unique fixed point v * * ∈ E ℓ * 0 ,T * . We define v s := v * s for all s ∈ (0, T * /2] and v s := v * * s−T * /2 for all s ∈ (T * /2, 3T * /2]. A simple calculation shows that v is a fixed point of M 3T * 2 ,f and that v ∈ E ℓ 0 ,3T * /2 . Suppose thatv is another fixed point. By the uniqueness of the fixed point on (0, T * ], we deduce that v * andv coincide on this interval. Moreover, a simple calculation shows that (v s+ T * 2 , s ∈ (0, T * ]) is necessarily a fixed point of M T * ,f * so that it coincides with v * * . Iterating this argument ensures existence and uniqueness of the fixed point on any interval [0, T ].
We are now in position to prove the main result of this article.
Proof of Theorem 1.4. Let u 0 be an element in C −1+4κ e ℓ for a given ℓ ∈ R. Let f ǫ := u 0 e Yǫ . By Corollary 1.2 and Theorem 2.6, f ǫ converges to f = u 0 e Y in C −1+4κ e ℓ ′′ for any ℓ ′′ > ℓ. Let v ǫ be the unique fixed point of M T,fǫ with g ǫ = Z ǫ − F * ξ ǫ and h (i) ǫ := −2D x i Y ǫ . By Corollary 1.2 and Proposition 1.3, we know that g ǫ , h (i) ǫ converge in probability to in C −κ pa . Notice that the convergence of F * ξ ǫ towards F * ξ is a consequence of Lemma 1.1, since F is a compactly supported, smooth function. Therefore, Proposition 4.1 ensures that v ǫ converges in probability in E 1+2κ ℓ ′′ ,T to the unique fixed point v of the map M T,f associated to g, h (1) , h (2) . Moreover, Theorem 2.6 ensures that, for any ℓ ′ > ℓ ′′ , u ǫ = v ǫ e −Yǫ converges to u = ve −Y in the space E 1−κ ℓ ′ ,T .