A simple construction of the continuum parabolic Anderson model on R 2
aa r X i v : . [ m a t h . P R ] M a y A simple construction of the continuumparabolic Anderson model on R May 23, 2017
Martin Hairer and Cyril Labb´e University of Warwick, Email:
[email protected] University of Warwick, Email:
Abstract
We propose a simple construction of the solution to the continuum parabolic Ander-son model on R which does not rely on any elaborate arguments and makes exten-sive use of the linearity of the equation. A logarithmic renormalisation is requiredto counterbalance the divergent product appearing in the equation. Furthermore, weuse time-dependent weights in our spaces of distributions in order to construct thesolution on the unbounded space R . The goal of this note is to construct solutions to the continuous parabolic Andersonmodel: ∂ t u = ∆ u + u · ξ , u ( , x ) = u ( x ) . (PAM)Here, u is a function of t ≥ and x ∈ R , while ξ is a white noise on R . 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 notclassically well-defined since the sum of the H ¨older regularities of u and ξ is slightlybelow . Second, our space variable x lies in the unbounded space R so that oneneeds to incorporate weights in the H ¨older spaces at stake; this causes some difficultyin obtaining the fixed point argument, since one would a priori require a larger weightfor 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 maintrick 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 PDEassociated to v = ue Y instead of u . This is analogous to what was done for exam-ple in [DPD02, HPP13]. The second issue is dealt with by choosing an appropriatetime-increasing weight for the solution u . Roughly speaking, if ξ is weighted by the I NTRODUCTION polynomial function p a ( x ) = ( | x | ) a with a small, and u s is weighted by the ex-ponential function e s ( x ) = e s ( | x | ) , then R t P t − s ∗ ( u s · ξ )( x ) ds requires a weight oforder R t 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 con-structed independently by Gubinelli, Imkeller and Perkowski [GIP12] and by Hairer[Hai14] in dimension and, to some extent, by Hairer and Pardoux [HP14] in di-mension . (The latter actually considers the case of dimension with space-timewhite noise, but the case of dimension with spatial noise has exactly the same scal-ing behaviour, so the proof given there carries through mutatis mutandis . The maindifference is that some of the renormalisation constants that converge to finite limitsin [HP14] may diverge logarithmically.) However, in all of these results the spacevariable is restricted to a torus, which is the constraint that we lift in this note. Theconstruction that we propose here is very specific to (PAM) in dimension : in particu-lar, as it stands, it unfortunately applies neither to the generalised parabolic Andersonmodel considered in [GIP12, Hai14], nor to the case of dimension . Let us also men-tion the work of Hu [Hu02] who considers a different equation: the usual product u · ξ in (PAM) is replaced by the Wick product.Let us now present the main steps of our construction. First, we introduce a mol-lified noise ξ ε := ̺ ε ∗ ξ , where ̺ is a compactly supported, even, smooth function on R that integrates to , and ̺ ε ( x ) := ε − ̺ ( xε ) for all x ∈ R . In order to quantify theH ¨older regularity of ξ, ξ ε , we introduce weighted H ¨older spaces of distributions, seeSection 2 below for the general definitions. Informally speaking, given a weight w and an exponent α , C αw 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 whichis given on Page 5 below. Lemma 1.1
For any given a > , let p a ( x ) = ( | x | ) a on R as above. For every ε, κ > , ξ ε belongs almost surely to C − − κ p a ( R ) . As ε ↓ , ξ ε converges in probabilityto ξ in C − − κ p a . From now on, a is taken arbitrarily small. Since, for any fixed ε > , the mollifiednoise ξ ε is actually a smooth function belonging to C α p a for any α > , the SPDE ∂ t u ε = ∆ u ε + u ε ( ξ ε − C ε ) , u ε ( , x ) = u ( x ) , (PAM ε )is well-posed, as can be seen for example by using its Feynman-Kac representation.The constant C ε appearing in this equation is required in order to control the limit ε → and will be determined later on.Second, let G be a compactly supported, even, smooth function on R \{ } , suchthat G ( x ) = log | x | π whenever | x | ≤ . Then, there exists a compactly supportedsmooth function F on R that vanishes on the ball of radius and such that, in thedistributional sense, we have: ∆ G ( x ) = δ ( x ) + F ( x ) . (1.1) NTRODUCTION With these notations at hand, we introduce the process Y ε ( x ) := G ∗ ξ ε ( x ). By con-struction, Y ε is a smooth stationary process on R that coincides with the solution ofthe Poisson equation driven by ξ ε , up to some smooth term: ∆ Y ε ( x ) = ξ ε ( x ) + F ∗ ξ ε ( x ) . From now on, D x i denotes the differentiation operator with respect to the variable x i ,with i ∈ { , } . More generally, for every ℓ ∈ N , we define D ℓx f as the map obtainedfrom f by differentiating ℓ times in direction x and ℓ times in direction x . Wealso use the notation ∇ f = ( D x f, D x f ). The following result is a consequence ofLemma 1.1 together with the smoothing effect of the convolution with G and D x i G . Corollary 1.2
For any given κ ∈ ( , / ) , the sequence of processes Y ε (resp. D x i Y ε )converges in probability as ε → in the space C − κ p a ( R ) (resp. C − κ p a ( R ) ) towards theprocess Y (resp. D x i Y ) defined by Y := G ∗ ξ , D x i Y := D x i G ∗ ξ . We introduce v ε ( t, x ) := u ε ( t, x ) e Y ε ( x ) for all x ∈ R and t ≥ , and we observe that ∂ t v ε = ∆ v ε + v ε ( Z ε − F ∗ ξ ε ) − ∇ v ε · ∇ Y ε , v ε ( , x ) = u ( x ) e Y ε ( x ) ,where we have introduced the renormalised process Z ε ( x ) := |∇ Y ε ( x ) | − C ε . At this stage we fix the renormalisation constant C ε to be given by C ε := E (cid:2) |∇ Y ε | (cid:3) = − π log ε + O ( ) , (1.2)where the part denoted by O ( ) converges to a constant (depending on the choice of G and ̺ ) as ε → , we refer to the end of Section 3 for the calculation. The followingresult, which is proven on Page 9, shows that this sequence of renormalised processesalso converges in an appropriate space. We refer to Nualart [Nua06] for details onWiener chaoses. Proposition 1.3
For any given κ ∈ ( , / ) , the collection of processes Z ε convergesin probability as ε → , in the space C − κ p a ( R ) , towards the generalised process Z de-fined as follows: for every test function η , h Z, η i is the random variable in the secondhomogeneous Wiener chaos associated to ξ represented by the L ( dz d ˜ z ) function ( z, ˜ z ) Z X i =1 , D x i G ( z − x ) D x i G ( ˜ z − x ) η ( x ) dx . We are now able to set up a fixed point argument for the process v ε with controls thatare uniform in ε . The precise statement of the main result of this article requires somenotation: in this introduction, we provide a weaker but more readable version of thestatement and we refer to Section 4 for the details. W EIGHTED
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Theorem 1.4
Let u be a H¨older distribution with regularity better than − , and thatgrows at most exponentially fast at infinity. The sequence of processes v ε convergesuniformly on all compact sets of ( , ∞ ) × R , in probability as ε → , to a limit v which is the unique solution of ∂ t v = ∆ v + v ( Z − F ∗ ξ ) − ∇ v · ∇ Y , v ( , x ) = u ( x ) e Y ( x ) . As a consequence, u ε converges in probability towards the process u = ve − Y . In this section, we introduce the appropriate weighted spaces that will allow us to setup 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. Definition 2.1
A function w : R d → ( , ∞ ) is a weight if there exists a positiveconstant C > such that C − ≤ sup | x − y |≤ w ( x ) w ( y ) ≤ C .
In this article, we will consider two families of weights indexed by a, ℓ ∈ R :p a ( x ) := ( | x | ) a , e ℓ ( x ) := exp ( ℓ ( | x | ) ) . 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 . We can now consider weighted versions of the usualspaces of H ¨older functions C α ( R d ). Definition 2.2
For α ∈ ( , ), C αw ( R d ) is the space of functions f : R d → R such that k f k α,w := sup x ∈ R d | f ( x ) | w ( x ) + sup | x − y |≤ | f ( x ) − f ( y ) | w ( x ) | x − y | α < ∞ . More generally, for every α > , we define C αw ( R d ) recursively as the space of func-tions f which admit first order derivatives and such that k f k α,w := sup x ∈ R d | f ( x ) | w ( x ) + d X i =1 k D x i f k α − ,w < ∞ . We then extend this definition to negative α . To this end, we define for every r ∈ N ,the space B r of all smooth functions η on R d , which are compactly supported in theunit ball of R d and whose C r norm is smaller than . We will use the notation η λx todenote the function y λ − d η ( y − xλ ) . EIGHTED
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For every α < , we set r := −⌊ α ⌋ and we define C αw ( R d ) as the spaceof distributions f on R d such that k f k α,w := sup x ∈ R d sup η ∈B r sup λ ∈ ( , ] | f ( η λx ) | w ( x ) λ α < ∞ . In order to deal with the regularity of random processes, it is convenient to have acharacterisation of C αw that only relies on a countable number of test functions. Tostate such a characterisation, we need some notation. For any ψ ∈ C r , we set ψ nx ( y ) := 2 nd ψ (( y − x ) n , . . . , ( y d − x d ) n ) , x, y ∈ R d , n ≥ . We also define Λ n := { ( − n k i ) i =1 ...d : k i ∈ Z } . Proposition 2.4
Let α < and r > | α | . There exists a finite set Ψ of compactlysupported functions in C r , as well as a compactly supported function ϕ ∈ C r such that { ϕ x , x ∈ Λ } ∪ { ψ nx , n ≥ , x ∈ Λ n , ψ ∈ Ψ } forms an orthonormal basis of R d , andsuch that for any distribution ξ on R d , the following equivalence holds: ξ ∈ C αw if andonly if ξ belongs to the dual of C r and sup n ≥ sup ψ ∈ Ψ sup x ∈ Λ n |h ξ, ψ nx i| w ( x ) − nd − nα + sup x ∈ Λ | (cid:10) ξ, ϕ x (cid:11) | w ( x ) < ∞ . (2.1) 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, butsince 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 | ≤ to obtain our statement. (cid:3) Remark 2.5 If ξ is a linear transformation acting on the linear span of the functions ϕ x , ψ nx such that (2.1) is finite, then ξ can be extended uniquely to an element of C αw .We are now in position to characterise the regularity of the noise. Proof of Lemma 1.1.
We work in dimension d = 2 . Set α = − − κ with κ > . ByProposition 2.4, it suffices to show that almost surelysup n ≥ sup ψ ∈ Ψ sup x ∈ Λ n |h ξ, ψ nx i| − n ( α ) p a ( x ) . , sup x ∈ Λ | (cid:10) ξ, ϕ x (cid:11) | p a ( x ) . . We restrict to the first bound, since the second is simpler. For any integer p ≥ , wewrite E " sup n ≥ sup ψ ∈ Ψ sup x ∈ Λ n (cid:16) |h ξ, ψ nx i| − n ( α +1 ) p a ( x ) (cid:17) p . X n ≥ X ψ ∈ Ψ X x ∈ Λ n np ( α +1 ) p a ( x ) p ( E h ξ, ψ nx i ) p W EIGHTED
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OLDER SPACES . X n ≥ X ψ ∈ Ψ X x ∈ Z np ( α +1 ) p a ( x ) p n . At the first line, we used the equivalence of moments of Gaussian random variables.At the second line, we used the following facts: the restriction of Λ n to the unit ball of R has at most of the order of n elements, the L norm of ψ nx is and p a is a weight.Recall that α < − , Ψ is a finite set and p a ( x ) = ( | x | ) a . Taking p large enough,we deduce that the triple sum converges, so that ξ admits a modification that almostsurely belongs to C α p a . We now turn to k ξ ε − ξ k α, p a : the computation is very similar,the only difference rests on the term E h ξ − ξ ε , ψ nx i = k ψ n − ̺ ε ∗ ψ n k L . ∧ ( ε n ) . (2.2)Let n be the smallest integer such that − n ≤ ε . For p large enough, we obtain E " sup n ≥ sup ψ ∈ Ψ sup x ∈ Λ n (cid:16) |h ξ − ξ ε , ψ nx i| − n ( α +1 ) p a ( x ) (cid:17) p . X x ∈ Z X n ≥ n +2 np ( α +1 ) p a ( x ) p ( ∧ ε p np ) . X n Let f ∈ C αw f and g ∈ C βw g where α < and β > with α + β > .Then there exists a continuous bilinear map ( f, g ) f · g from C αw f × C βw g into C αw f w g that extends the classical multiplication of smooth functions. Remark 2.7 The space C α defined in Section 2 coincides with the usual Besov space B α ∞ , ∞ . Indeed, they enjoy the same characterisation in a wavelet analysis, see [Hai14,Prop 3.20] and [Mey92, Section 6.10]. Proof. Let χ be a compactly supported, smooth function on R d such that P k ∈ Z d χ ( x − k ) = 1 for all x ∈ R d . For simplicity, we set χ k ( · ) := χ ( · − k ). Writing k · k α forthe α -H ¨older norm without weight (i.e. with weight ), observe that h ∈ C αw if andonly if k hχ k k α . w ( k ) hold uniformly over all k ∈ Z d , and k h k α,w is equivalent tothe smallest possible bound. From [BCD11, Thm 2.52], we know that f χ k · gχ ℓ iswell-defined for all k, ℓ ∈ Z d , and that the bound k f χ k · gχ ℓ k α . k f χ k k α k gχ ℓ k β holds. Consequently, we get k f χ k · gχ ℓ k α . w f ( k ) w g ( ℓ ) k f k α,w f k g k β,w g , OUNDS ON Y AND Z uniformly over all k, ℓ ∈ Z d . Since the number of non-zero terms among {h f χ k · gχ ℓ , η x i , k, ℓ ∈ Z d } is uniformly bounded over all η ∈ B r , all x ∈ R d and all f, g as in the statement, we deduce that f · g := P k,ℓ ∈ Z d f χ k · gχ ℓ is well-defined andthat k f · g k α,w f w g . k f k α,w f k g k β,w g holds. Finally, the multiplication of [BCD11,Thm 2.52] extends the classical multiplication of smooth functions, therefore, fromour construction, it is plain that this property still holds in our case. (cid:3) Let now P t ( x ) := ( πt ) − d e −| x | / t 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 havethe following regularisation property which is a slight variant of well-known facts. Lemma 2.8 For every β ≥ α and every f ∈ C α e ℓ , we have k P t f k β, e ℓ . t − β − α k f k α, e ℓ ,uniformly over all ℓ in a compact set of R and all t in a compact set of [ , ∞ ) .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 , we refer the readerto Lemma 5.5 in [Hai14] for instance. Using the decay properties of the heat kernel,the statement regarding P − is easy to check. Concerning the singular part, one writes P + = P n ≥ P n where each P n is a smooth function supported in the parabolic annu-lus { ( t, x ) : 2 − n − ≤ | t | + | x | ≤ − n +1 } and such that P n ( t, x ) = 2 dn P ( n t, n x ).Then, we get |h f, η λx ( · − y ) i| . λ α e ℓ ( x + y ) , |h f, D kx P n ( t, · − y ) i| . − n ( α −| k | ) e ℓ ( y ) ,uniformly over all η ∈ B r , all x, y ∈ R d , all t > , all n ≥ and all k ∈ N . Noticethat P n ( t, · ) vanishes as soon as n ≥ − log t . Consequently, |h P + ( t ) ∗ f, η λx i| . e ℓ ( x )( λ α ∧ t α ) , |h f, D kx P + ( t, · − x ) i| . e ℓ ( x ) t α −| k | ,so that the statement follows by interpolation. (cid:3) Y and Z Let us collect a few facts on the behaviour of smooth functions with a singularityat the origin; we refer to [Hai14, Sec. 10.3] for proofs. For any smooth function K : R d \{ } → R and any real number ζ , we define ||| K ||| ζ ; m = sup | k |≤ m sup x ∈ R d k x k | k |− ζ | D kx K ( x ) | ,where the first supremum runs over k ∈ N d and | k | = P i k i . We say that K is oforder ζ if ||| K ||| ζ ; m < ∞ for all m ∈ N . Recall ̺ ε from the introduction, and define K ε = K ∗ ̺ ε . If K is of order ζ ∈ ( − d, ) then for all m ∈ N , there exists C > B OUNDS ON Y AND Z such that ||| K ε ||| ζ ; m ≤ C ||| K ||| ζ ; m , uniformly over ε ∈ ( , ]. Furthermore, for all ¯ ζ ∈ [ ζ − , ζ ), there exists a constant C > such that ||| K − K ε ||| ¯ ζ ; m ≤ Cε ζ − ¯ ζ ||| K ||| ζ ; m +1 . If K and K are of order ζ and ζ respectively, then K K is of order ζ = ζ + ζ and we have the bound ||| K K ||| ζ ; m ≤ C ||| K ||| ζ ; m ||| K ||| ζ ; m ,where C is a positive constant.Assume that ζ ∧ ζ > − d . We set ζ = ζ + ζ + d . If ζ < , then K ∗ K is oforder ζ and we have the bound ||| K ∗ K ||| ζ ; m ≤ C ||| K ||| ζ ; m ||| K ||| ζ ; m . (3.1)On the other hand, if ζ ∈ R + \ N and K , K are compactly supported, then the func-tion K ( x ) = ( K ∗ K )( x ) − X | k | <ζ x k k ! D kx ( K ∗ K )( ) ,is of order ζ and a bound similar to (3.1) holds, but with the constant C depending onthe 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 \{ } , compactly supported and satisfies G ( x ) = log | x | π in a neigh-bourhood of the origin, it is a function with a singularity of order ζ , for all ζ < ,according to our definition. From now on, we set ̺ ∗ = ̺ ∗ ̺ and we assume withoutloss of generality that ̺ , ̺ ∗ are supported in the unit ball of R . Lemma 3.1 Fix κ ∈ ( , ) . We have the bounds E h | Z ( η λx ) | i . λ − κ , E h | Z ε ( η λx ) | i . λ − κ , E h | Z ε ( η λx ) − Z ( η λx ) | i . λ − κ ε κ ,uniformly over all ε, λ ∈ ( , ) , all x ∈ R and all η ∈ B r .Proof. By translation invariance, it suffices to consider x = 0 . The random variables Z ( η λ ), Z ε ( η λ ) and Z ε ( η λ ) − Z ( η λ ) all belong to the second homogeneous Wienerchaos associated with the noise ξ . This is because the constant C ε has been chosen tocancel the -th Wiener chaos component of |∇ Y ε | . We start with the second boundof the statement: E h | Z ε ( η λ ) | i = X i =1 Z z, ˜ z (cid:16) Z η λ ( x ) D x i G ε ( z − x ) D x i G ε ( ˜ z − x ) dx (cid:17) dz d ˜ z = X i =1 Z Z η λ ( x ) η λ ( x ′ ) (cid:16) ( D x i G ε ) ∗ ( D x i G ε )( x − x ′ ) (cid:17) dx dx ′ , OUNDS ON Y AND Z so that the bounds at the beginning of the section yield the desired result. The firstbound of the statement follows by replacing G ε by G in the expression above. Weturn to the proof of the third bound. To that end, we write E h | Z ε ( η λ ) − Z ( η λ ) | i = X i =1 Z Z η λ ( x ) η λ ( x ′ ) H ε,i ( x − x ′ ) dx dx ′ ,where H ε,i ( y ) = (cid:16) ( D x i ( G ε − G ) ) ∗ D x i G ε (cid:17) · (cid:16) ( D x i ( G ε + G ) ) ∗ D x i G ε (cid:17) ( y ) − (cid:16) ( D x i ( G ε − G ) ) ∗ D x i G (cid:17) · (cid:16) ( D x i ( G ε + G ) ) ∗ D x i G (cid:17) ( y ) ,so that, once again, the bounds on the behaviour of singular functions at the originyield the asserted bound. (cid:3) Proof of Proposition 1.3. Let L denote an arbitrary element among Z , Z ε and Z − Z ε .Using the equivalence of moments of elements in inhomogeneous Wiener chaoses offinite order, we obtain E " sup n ≥ sup x ∈ Λ n (cid:16) L ( ψ nx )p a ( x ) − nα − n (cid:17) p . X k ∈ Z p a ( k ) p X n ≥ X x ∈ Λ n ∩ B ( k, ) E [ L ( ψ nx ) ] p − nα p − np . When L is equal to Z or Z ε , Lemma 3.1 ensures that E [ L ( ψ nx ) ] . − n + κn uniformlyover all x , n , and ε . Moreover, ( Λ n ∩ B ( k, )) . n , so that E " sup n ≥ sup x ∈ Λ n (cid:16) L ( ψ nx )p a ( x ) − nα − n (cid:17) p . X k ∈ Z p a ( k ) p X n ≥ np ( α + κ ) +2 n . This quantity is finite for α = − κ and p large enough. Therefore, Z and Z ε belong to C − κ p a .Regarding Z − Z ε , Lemma 3.1 ensures that E [( Z − Z ε )( ψ nx ) ] . ε κ − n +5 κn uni-formly over all x , n and ε . Then, the same arguments as before yield E " sup n ≥ sup x ∈ Λ n (cid:16) ( Z − Z ε )( ψ nx )p a ( x ) − nα − n (cid:17) p . X k ∈ Z p a ( k ) p X n ≥ ε κp np ( α +5 κ ) +2 n ,so that, choosing for instance α = − κ and p large enough, one gets the bound E (cid:2) k Z − Z ε k − κ, p a (cid:3) . ε κ uniformly over all ε ∈ ( , ], thus concluding the proof. (cid:3) Proof of Corollary 1.2. Since G is compactly supported and coincides with the Greenfunction of the Laplacian in a neighbourhood of the origin, the classical Schauderestimates [Sim97] imply that for any α ∈ R , the bounds k G ∗ f k α +2 . k f k α , k D x i G ∗ f k α +1 . k f k α , P ICARD ITERATION hold uniformly over all f ∈ C α . Recall the functions χ k , k ∈ Z d from the proof ofTheorem 2.6. Since G is compactly supported, we deduce from the bounds above that k G ∗ ( f χ k ) k α +2 . w ( k ) k f k α,w , k D x i G ∗ ( f χ k ) k α +1 . w ( k ) k f k α,w ,uniformly over all k ∈ Z d and all f ∈ C αw . 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 = P k ∈ Z d f χ k , wededuce that k G ∗ f k α +2 ,w . k f k α,w , k D x i G ∗ f k α +1 ,w . k f k α,w ,uniformly over all f ∈ C αw . This being given, the statement is a direct consequence ofLemma 1.1. (cid:3) We conclude this section with the computation of the renormalisation constant C ε . Recall that ̺ , ̺ ∗ and G are compactly supported. We let G ε be the compactlysupported, smooth function G ∗ ̺ ε . We have C ε = E (cid:2) |∇ Y ε | (cid:3) = X i =1 , Z x ∈ R D x i G ε ( x ) D x i G ε ( x ) dx = − Z x ∈ R G ε ( x ) ∆ G ε ( x ) dx ,where we used a simple integration by parts to get the last identity. By (1.1), we have ∆ G ε = ̺ ε + F ε , where F ε = F ∗ ̺ ε . The latter is a compactly supported, smoothfunction that vanishes on the centred ball of radius / − ε . Hence, uniformly over all ε ∈ ( , ], the function G ε F ε is smooth and compactly supported so that its integralis uniformly bounded. On the other hand, since ̺ is even, ̺ ∗ integrates to and G ( x ) = π log | x | for all x ∈ B ( , / ), we get − Z G ε ( x ) ̺ ε ( x ) dx = − Z G ( x ) ̺ ∗ ε ( x ) dx = 12 π log ε − − π Z log | x | ̺ ∗ ( x ) dx . The first term on the right gives the diverging term of the renormalisation constant,while the second term is finite. This concludes the computation. For any r > , ℓ ∈ R and T > , we consider the Banach space E rℓ,T of all continuousfunctions v on ( , T ] × R such that ||| v ||| ℓ,T,r := sup t ∈ ( ,T ] k v ( t, · ) k r, e ℓ + t t − κ < ∞ . This being given, we have the following precise statement of our main result. Theorem 4.1 Let ℓ ∈ R and T > . Consider an initial condition u ∈ C − κ e ℓ . Forall ℓ ′ > ℓ , the sequence of processes v ε converges in probability as ε → in the space E κℓ ′ ,T to a limit v which is the unique solution of ∂ t v = ∆ v + v ( Z − F ∗ ξ ) − ∇ v · ∇ Y , v ( , x ) = u ( x ) e Y ( x ) . As a consequence, u ε converges in probability in E − κℓ ′ ,T towards the process u = ve − Y . ICARD ITERATION The rest of this section is devoted to the proof of this result. Fix κ ∈ ( , ), and letthe parameter a appearing in the weight p a be any value in ( , κ ). Let g, h ( ) , h ( ) ∈C − κ p a and f ∈ C − κe ℓ be given. We define the map v 7→ M T,f v as follows: M T,f v ( t ) = Z t P t − s ∗ (cid:16) v s · g + D x i v s · h ( i ) (cid:17) ds + P t ∗ f . In this equation, there is an implicit summation over i ∈ { , } . This convention willbe in force for the rest of the article. Proposition 4.2 Take ℓ ∈ R . For any given g, h ( ) , h ( ) ∈ C − κ p a and any f ∈ C − κ e ℓ ,the map M T,f admits a unique fixed point v ∈ E κℓ ,T . Furthermore, the solution map ( g, h ( ) , h ( ) , f ) v is continuous.Proof. The parameter r in the space E rℓ,T is taken to be equal to κ . Since thisvalue is fixed until the end of the proof, we do not write the subscript r in the associatednorm.First, Lemma 2.8 ensures that k P t ∗ f k κ, e ℓ + t . t − κ k f k − κ, e ℓ uniformly overall t in any given compact interval of R + . Second, using Theorem 2.6 and the simpleinequality sup x ∈ R p a ( x )e ℓ + s ( x )e ℓ + t ( x ) ≤ e − a (cid:16) at − s (cid:17) a ,we obtain k v s · g + D x i v s · h ( i ) k − κ, e ℓ + t . ( t − s ) − a k v s k κ, e ℓ + s ( || g || − κ, p a + || h ( i ) || − κ, p a ) . ( t − s ) − a s − κ ||| v ||| ℓ,T ( || g || − κ, p a + || h ( i ) || − κ, p a ) ,uniformly over all s, t in a compact set of R + and all ℓ in a compact set of R . Then,by Lemma 2.8 and using a < κ/ , we obtain (cid:13)(cid:13)(cid:13)(cid:13) Z t P t − s ∗ (cid:16) v s · g + D x i v s · h ( i ) (cid:17) ds (cid:13)(cid:13)(cid:13)(cid:13) κ, e ℓ + t (4.1) . Z t ( t − s ) − − κ s − κ ds ||| v ||| ℓ,T ( || g || − κ, p a + || h ( i ) || − κ, p a ) . t − κ T − κ ||| v ||| ℓ,T ( || g || − κ, p a + || h ( i ) || − κ, p a ) ,uniformly over all t ∈ ( , T ]. This ensures that M T,f ( v ) ∈ E κℓ,T . Furthermore wehave |||M T,f v − M T,f ¯ v ||| ℓ,T . T − κ ||| v − ¯ v ||| ℓ,T ( k g k − κ, p a + k h ( i ) k − κ, p a ) , (4.2)uniformly over all ℓ in a compact set of R , all T in a compact set of R + , all f ∈C − κe ℓ and all v, ¯ v ∈ E ℓ,T . (Here and below we write E ℓ,T instead of E κℓ,T forconciseness.) Consequently, there exists T ∗ > such that M T ∗ ,f is a contraction on P ICARD ITERATION E ℓ,T ∗ , uniformly over all ℓ ∈ [ ℓ , ℓ + T ] and all f ∈ C − κe ℓ . Fix an initial condition f ∈ C − κe ℓ . To obtain a fixed point for the map M T,f , we proceed by iteration.The map M T ∗ ,f admits a unique fixed point v ∗ ∈ E ℓ ,T ∗ . If T ∗ ≥ T , we are done.Otherwise, set f ∗ := v ∗ T ∗ / ∈ C κe ℓ ∗ , where ℓ ∗ = ℓ + T ∗ / . Since ℓ ∗ ≤ ℓ + T , themap M T ∗ ,f ∗ is again a contraction on E ℓ ∗ ,T ∗ , so that it admits a unique fixed point v ∗∗ ∈ E ℓ ∗ ,T ∗ . We define v s := v ∗ s for all s ∈ ( , T ∗ / ] and v s := v ∗∗ s − T ∗ / for all s ∈ ( T ∗ / , T ∗ / ]. A simple calculation shows that v is a fixed point of M T ∗ ,f andthat v ∈ E ℓ , T ∗ / . Suppose that ¯ v is another fixed point. By the uniqueness of thefixed point on ( , T ∗ ], we deduce that v ∗ and ¯ v coincide on this interval. Moreover,a simple calculation shows that ( ¯ v s + T ∗ , s ∈ ( , T ∗ ]) is necessarily a fixed point of M T ∗ ,f ∗ so that it coincides with v ∗∗ . Iterating this argument ensures existence anduniqueness of the fixed point on any interval [ , T ].We turn to the continuity of the solution map with respect to f , g and h ( i ) . Let ¯ M be the map associated with ¯ g and ¯ h ( i ) . For any initial conditions f and ¯ f in C − κ e ℓ ,both M T,f and ¯ M T, ¯ f admit a unique fixed point v and ¯ v . Furthermore, we have v t − ¯ v t = (cid:16) M T,f v − M T,f ¯ v (cid:17) t + Z t P t − s ∗ (cid:16) ¯ v s ( g − ¯ g ) + D x i ¯ v s ( h ( i ) − ¯ h ( i ) ) (cid:17) ds + P t ∗ ( f − ¯ f ) . Using (4.1) and (4.2), we deduce that ||| v − ¯ v ||| ℓ,T . T − κ ||| v − ¯ v ||| ℓ,T ( k g k − κ, p a + k ¯ g k − κ, p a + k h ( i ) k − κ, p a + k ¯ h ( i ) k − κ, p a )+ T − κ ||| v ||| ℓ,T ( k ¯ g − g k − κ, p a + k ¯ h ( i ) − h ( i ) k − κ, p a )+ k f − ¯ f k − κ,ℓ ,uniformly over all ℓ in a compact set of R and all T in a compact set of R + . Fix R > .There exists T > such that ||| v − ¯ v ||| ℓ,T . k f − ¯ f k − κ,ℓ + T − κ ( k ¯ g − g k − κ, p a + k ¯ h ( i ) − h ( i ) k − κ, p a ) ,uniformly over all ℓ in a compact set of R and all g, ¯ g, h, ¯ h such that ||| v ||| ℓ,T , k g k − κ, p a , k ¯ g k − κ, p a , k h ( i ) k − κ, p a and k ¯ h ( i ) k − κ, p a are smaller than R . This yields the continuity ofthe solution map on ( , T ]. By iterating the argument as above, we obtain continuityon any bounded interval. (cid:3) We are now in position to prove the main result of this article. Proof of Theorem 4.1. Let u be an element in C − κ e ℓ for a given ℓ ∈ R . Let f ε := u e Y ε . By Corollary 1.2 and Theorem 2.6, f ε converges to f = u e Y in C − κ e ℓ ′′ for any ℓ ′′ > ℓ . Let v ε be the unique fixed point of M T,f ε with g ε = Z ε − F ∗ ξ ε and h ( i ) ε := − D x i Y ε . By Corollary 1.2 and Proposition 1.3, we know that g ε , h ( i ) ε converge in probability to g = Z − F ∗ ξ , h ( i ) = − D x i Y , ICARD ITERATION in C − κ p a . Notice that the convergence of F ∗ ξ ε towards F ∗ ξ is a consequence ofLemma 1.1, since F is a compactly supported, smooth function. Therefore, Proposi-tion 4.2 ensures that v ε converges in probability in E κℓ ′′ ,T to the unique fixed point v of the map M T,f associated to g, h ( ) , h ( ) . 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