A spatial version of the Itô-Stratonovich correction
aa r X i v : . [ m a t h . P R ] J u l The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2012
A SPATIAL VERSION OF THE IT ˆO–STRATONOVICHCORRECTION
By Martin Hairer and Jan Maas University of Warwick and University of Bonn
We consider a class of stochastic PDEs of Burgers type in spatialdimension 1, driven by space–time white noise. Even though it is wellknown that these equations are well posed, it turns out that if oneperforms a spatial discretization of the nonlinearity in the “wrong”way, then the sequence of approximate equations does converge toa limit, but this limit exhibits an additional correction term.This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with thesolution itself. This can be understood as a consequence of the factthat for any fixed time, the law of the solution is locally equivalentto Wiener measure, where space plays the role of time. In this sense,the correction term is similar to the usual Itˆo–Stratonovich correctionterm that arises when one considers different temporal discretizationsof stochastic ODEs.
1. Introduction.
In this work, we give a rigorous analysis of the behaviorof stochastic Burgers equations in one spatial dimension under various ap-proximation schemes. It was recently argued in [12] that if the approximationscheme fails to satisfy a certain symmetry condition, then one expects theapproximations to converge to a modified equation, with the appearance ofan additional correction term in the limit. This correction term is somewhatsimilar to the Itˆo–Stratonovich correction that appears in the study of SDEswhen one compares centred and one-sided approximations. The present ar-ticle provides a rigorous justification of the results observed in [12], at least
Received November 2010; revised March 2011. Supported by the EPSRC Grants EP/E002269/1 and EP/D071593/1, a Wolfson Re-search Merit Award of the Royal Society and a Philip Leverhulme prize of the LeverhulmeTrust. Supported by Rubicon Grant 680-50-0901 of the Netherlands Organisation for Scien-tific Research (NWO).
AMS 2000 subject classifications.
Primary 60H15; secondary 35K55, 60H30, 60H35.
Key words and phrases.
Itˆo–Stratonovich correction, stochastic Burgers equation, spa-tial discretizations, Wiener chaos.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2012, Vol. 40, No. 4, 1675–1714. This reprint differs from the original inpagination and typographic detail. 1
M. HAIRER AND J. MAAS in the case where the nonlinearity of the equation is of gradient type, andtherefore the limiting equation is well posed in a classical sense.More precisely, we will consider in this work stochastic PDEs of the form ∂ t u ( x, t ) = ν ∂ x u ( x, t ) + F ( u ( x, t )) + ( ∇ G )( u ( x, t )) ∂ x u ( x, t ) + ξ ( x, t ) , (1.1)where u = u ( x, t ) is an R n -valued function, with x ∈ [0 , π ], t ≥
0. In thisequation, ν >
F, G : R n → R n areassumed to be C , and the stochastic forcing term ξ consists of independentspace–time white noises in each component of R n . For the sake of simplicity,we endow this equation with periodic boundary conditions, but we do notexpect this to alter our results significantly.Endowed with an initial condition u ∈ C ([0 , π ]; R n ), (1.1) is locally wellposed [9], provided that we rewrite the nonlinearity as ∂ x G ( u ) and considersolutions either in the weak or the mild form [7]. (Note that our noise term is not the gradient of space–time white noise, as in [3]. Therefore, our solutionsare actually α -H¨older continuous functions for all α < .) The aim of thisarticle is to show that this well-posedness is much less stable than one mayimagine at first. Indeed, if we set D + ε u ( x ) = u ( x + ε ) − u ( x ) ε , and consider the family u ε of solutions to the approximate equation ∂ t u ε = ν ∂ x u ε + F ( u ε ) + ∇ G ( u ε ) D + ε u ε + ξ, then our main result, Theorem 1.6 below, implies that u ε ⇒ ¯ u , where ¯ u isthe solution to (1.1), but with F replaced by¯ F ( u ) = F ( u ) − ν ∆ G ( u ) , (1.2)where ∆ is the usual Laplacian on R n . Remark 1.1.
The correction term in (1.2) is reminiscent of the Wong–Zakai correction [17], which arises if the driving Brownian motion in a stochas-tic ODE or PDE is approximated by stochastic processes of bounded vari-ation. This correction term is due to the temporal roughness of the drivingBrownian motion and does not appear if the noise is additive.Our correction term is a consequence of the spatial roughness of the so-lutions and appears even though we consider SPDEs with additive noise. Infact, an explicit calculation allows to check that the local quadratic variation(in space) of the solution u to (1.1) is precisely given by 1 / (2 ν ). Therefore,one can interpret the correction term appearing in (1.2) as precisely beingequal to − times the quadratic covariation between u and ∇ G ( u ). Recallthat this is exactly the correction term that appears when one switches be-tween Itˆo and Stratonovich integral in the usual setting of stochastic calcu-lus. See also [12] for a heuristic argument for computing the correction term. SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Remark 1.2.
This correction term is a purely stochastic effect and iscompletely unrelated to the fact that our discretization scheme is not anupwind scheme (see [5, 15]). In the absence of noise, we would still have theregularizing property from the nonvanishing viscosity, so that pretty muchany “reasonable” numerical scheme would converge to the correct solution.If D + ε is replaced by D − ε , defined by D − ε u ( x ) = ( u ( x ) − u ( x − ε )) /ε , thena similar result is true, but the sign in front of the correction term in (1.2)changes. We will actually consider a much more general class of approxima-tions to (1.1), where we also allow both the linear operator ∂ x and the noiseterm ξ to be replaced by approximate versions that are still translation-invariant, but modified at the lengthscale ε .1.1. Statement of the main result.
For ε >
0, we consider approximatingstochastic PDEs of the type ∂ t u ε = ν ∆ ε u ε + F ( u ε ) + ∇ G ( u ε ) D ε u ε + ξ ε . Since our system is invariant under spatial translations, it seems natural torestrict ourselves to a class of approximations that enjoys the same prop-erty. Throughout this article, we will therefore use approximate differentialoperators ∆ ε and D ε , as well as an approximate space–time white noise ξ ε given by their Fourier transforms: d ∆ ε u ( k ) = − k f ( ε | k | ) b u ( k ) , d D ε u ( k ) = ikg ( εk ) b u ( k ) , b ξ ε ( k ) = h ( ε | k | ) b ξ ( k ) . Several natural discretizations arising in numerical analysis are of this form(see the examples below Theorem 1.6). We will make the following standingassumptions on these objects.
Assumption 1.3.
The function f : [0 , ∞ ) → [0 , + ∞ ] is twice differen-tiable at 0 with f (0) = 1 and f ′ (0) = 0. Furthermore, there exists q ∈ (0 , f ( k ) ≥ q for all k > f ( k ) = + ∞ for some values of k , we use the convention exp( − t ∞ ) = 0for every t >
0. In this case, the semigroup generated by ∆ ε is not stronglycontinuous, but this is of no consequence for our analysis. Assumption 1.4.
There exists a signed Borel measure µ such that Z R e ikx µ ( dx ) = ikg ( k ) , and such that µ ( R ) = 0 , | µ | ( R ) < ∞ , Z R xµ ( dx ) = 1 , Z R | x | | µ | ( dx ) < ∞ . (1.3)In particular, we have ( D ε u )( x ) := ε R R u ( x + εy ) µ ( dy ). M. HAIRER AND J. MAAS
Assumption 1.5.
The function h is bounded and such that h /f is ofbounded variation. Furthermore, h is twice differentiable at the origin with h (0) = 1 and h ′ (0) = 0.Let ¯ u be the solution to the equation ∂ t ¯ u = ν ∂ x ¯ u + ¯ F (¯ u ) + ∇ G (¯ u ) ∂ x ¯ u + ξ. (1.4)In this equation, ¯ F := ( F − Λ∆ G )and Λ ∈ R is a correction constant given byΛ def = 12 πν Z R + Z R (1 − cos( yt )) h ( t ) t f ( t ) µ ( dy ) dt. (1.5)Note that a straightforward calculation shows that Λ is indeed well defined,as a consequence of the fact that h /f is bounded by assumption and that | µ | has a finite second moment.Before we state our main result, note that the equation (1.4) is locally wellposed in L ∞ , see [2, 4, 6, 9, 10]. As a consequence, it has a well-defined blow-up time τ ∗ (possibly infinite) such that, almost surely, lim t → τ ∗ k ¯ u ( t ) k L ∞ =+ ∞ on the event { τ ∗ < ∞} . With this notation, we are now ready to statethe main result of this paper. Theorem 1.6.
Let κ > and let u ε and ¯ u have initial conditions asin Theorem 2.2. There exists a sequence of stopping times τ ε satisfying lim ε → τ ε = τ ∗ in probability, and such that lim ε → P (cid:16) sup t ≤ τ ε k u ε ( t ) − ¯ u ( t ) k L ∞ > ε / − κ (cid:17) = 0 . Remark 1.7.
In order to avoid further technical complications, we con-sider sequences of initial conditions that have the property that the initialcondition for u ε “behaves like” the solution u ε ( t ) for positive times. In fact,the initial condition for u ε is a smooth perturbation of the stationary solu-tion to the linearized equation for u ε . We refer to Section 2 for more details.Before we proceed, we list some of the most common examples of dis-cretizations that do fit our framework. For a, b ≥ a + b >
0, it isnatural to discretize the derivative operator by choosing µ := δ a − δ − b a + b . This is also the discretization that was considered in [12]. As far as thediscretizations of the noise and the Laplacian are concerned, there are atleast three natural choices.
SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION No discretization.
This is the case f = h = 1 where only the nonlinearityis discretized. With this choice, one can check that the correction factor isgiven by Λ = ν a − ba + b . Finite difference discretization.
In this case, we divide the interval [0 , π ]into N equally sized intervals. For convenience, we assume that N is oddand we set(∆ ε u )( x ) = 1 ε ( u ( x + ε ) + u ( x − ε ) − u ( x )) , ε = 2 πN . We furthermore identify a function u with the trigonometric polynomial ofdegree ( N − / u at the gridpoints. This corresponds to thechoice f ( k ) = ( k sin ( k/ , k ∈ [0 , π ),+ ∞ , k ∈ [ π, ∞ ), h = [0 ,π ) . The natural choice for the discretization of the derivative operator in thiscase is to choose a and b to be integers, so that discretization takes place onthe gridpoints. With this choice, it can be shown that the correction factoris identical to that obtained in the previous case. Note however that this is not the case if the discretization of the derivative operator is not adaptedto the gridsize. Galerkin discretization.
In this case, we approximate ∆ and ξ by onlykeeping those Fourier modes that appear in the approximation by trigono-metric polynomials. This corresponds to the choice f ( k ) = (cid:26) , k ∈ [0 , π ),+ ∞ , k ∈ [ π, ∞ ), h = [0 ,π ) . The correction factor Λ is then given byΛ = cos( πa ) + πa Si( πa ) − cos( πb ) − πb Si( πb )2 π ν ( a + b ) , where Si t = R t xx dx .The rest of this paper is structured as follows. In Section 2, we introducenotation, we give a refined formulation of the main result and present anoutline of the proof of the main result (Theorem 2.2). In Section 3, we proveseveral useful bounds on the approximating semigroups and the approxima-tions of the gradient. Section 4 is devoted to several estimates for stochasticconvolutions, the most crucial one being Proposition 4.6, which is responsi-ble for the correction term appearing in the limiting equation. Most of thework is performed in Section 5, where convergence of various approximatingequations is proved. M. HAIRER AND J. MAAS
2. Proof of the main result.
In order to shorten notation, we introducethe semigroups S and S ε , defined as rescaled versions of the heat semigroupand its approximation: S ( t ) def = e − t (1 − ν∂ x ) , S ε ( t ) def = e − t (1 − ν ∆ ε ) , where we define S ε by Fourier analysis, that is, d S ε u ( k ) = e − t (1+ νk f ( ε | k | )) b u ( k ) , making use of the convention e −∞ = 0.Since we will always work with the mild formulation, it will be convenientto have a notation for the convolution (in time) of a function with one ofthe semigroups. We will henceforth write( S ∗ w )( t ) def = Z t S ( t − s ) w ( s ) ds. Let ( W ( t )) t ∈ R be a two-sided cylindrical Wiener process on H def = L ([0 , π ] , R n ) (see [7, 10] for precise definitions) and let Q ε be the bounded operatoron H defined as a Fourier multiplier by d Q ε u ( k ) = h ( ε | k | ) b u ( k ) . (We assume that it acts independently on each component.) Finally, wedefine the H -valued processes ψ and e ψ by ψ ( t ) = Z t −∞ S ( t − s ) dW ( s ) , e ψ ( t ) = Z t −∞ S ε ( t − s ) Q ε dW ( s ) , so that, in the notation of the previous section, they are the stationary solutions to the linear equations ∂ t ψ = ( ν∂ x − ψ + ξ, ∂ t e ψ = ( ν ∆ ε − e ψ + ξ ε . With this notation at hand, we can rewrite the equations for ¯ u and u ε inthe mild form as¯ u ( t ) = S ( t ) v + ψ ( t ) + S ∗ ( ¯ F (¯ u ) + ∇ G (¯ u ) ∂ x ¯ u )( t ) , (2.1) u ε ( t ) = S ε ( t ) v + e ψ ( t ) + S ε ∗ ( F ( u ε ) + ∇ G ( u ε ) D ε u ε )( t ) . (2.2) Remark 2.1.
Note that we have used here a common initial condition v for the difference ¯ u − ψ and u ε − e ψ . As a consequence, the two equations do not start with the same initial condition! However, as ε →
0, the initialcondition of u ε converges to that of ¯ u . The reason for not starting with thesame initial condition is mostly of technical nature.It will be convenient to define for any 0 < γ < χ , ψ γ def = ( I − Π ε − γ ) ψ, ψ γ def = Π ε − γ ψ, ψ χγ def = (Π ε − χ − Π ε − γ ) ψ. SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION The expressions e ψ γ , e ψ γ and e ψ χγ are defined analogously. Here Π N denotesthe projection onto the low-frequency components of the Fourier expansion,defined by Π N e n def = | n |≤ N e n , where e n ( x ) = (2 π ) − / e inx .We set ¯ v def = ¯ u − ψ, e v def = u ε − e ψ. In the proof, it will be convenient to work with the functions ¯ v γ and e v γ defined by ¯ v γ def = ¯ v + ψ γ = ¯ u − ψ γ , e v γ def = e v + e ψ γ = u ε − e ψ γ . It follows from (2.1) and (2.2) that these functions satisfy the followingequations:¯ v γ ( t ) = S ( t ) v + ψ γ ( t ) + S ∗ ( ¯ F (¯ v γ + ψ γ ) + ∂ x ( G (¯ v γ + ψ γ )))( t ) , (2.3) e v γ ( t ) = S ε ( t ) v + e ψ γ ( t )(2.4) + S ε ∗ ( F ( e v γ + e ψ γ ) + ∇ G ( e v γ + e ψ γ ) D ε ( e v γ + e ψ γ ))( t ) . For large parts of this article, it will be convenient to work in the fractionalSobolev space H α for some α > , so that H α ⊂ L ∞ . Recall that H α denotesthe space of (equivalence classes of) functions u = P j ∈ Z u k e k on [0 , π ] with u k ∈ C , for which k u k α := X k ∈ Z | u k | (1 + k ) α < ∞ . Furthermore, we will need to use a high-frequency cut-off, which will smoothenout the solutions at a scale ε χ for some χ >
1. It turns out that a reasonablechoice for these parameters is given by α = , γ = , χ = , (2.5)and we will fix these values from now on. With this notation at hand, thefollowing theorem, which is essentially a more precise reformulation of The-orem 1.6, is a more precise statement of our main result. Here and in therest of the paper we write k u k β to denote the norm of an element u in thefractional Sobolev space H β for β ∈ R . Theorem 2.2.
Let κ > be an arbitrary (small) exponent and let v ∈ H β for all β < . There exists a sequence of stopping times τ ε satisfying τ ε → τ ∗ in probability as ε → , such that lim ε → P (cid:16) sup t ≤ τ ε k u ε ( t ) − ¯ u ( t ) k L ∞ > ε / − κ (cid:17) = 0 . (2.6) M. HAIRER AND J. MAAS
In fact, we have the bounds lim ε → P (cid:16) sup t ≤ τ ε k e v γ ( t ) − ¯ v γ ( t ) k α > ε / − κ (cid:17) = 0 , (2.7) lim ε → P (cid:16) sup t ≤ τ ε k e ψ γ ( t ) − ψ γ ( t ) k L ∞ > ε / − κ (cid:17) = 0 . (2.8) Remark 2.3.
We emphasize again that the initial conditions ¯ u (0) and u ε (0) are slightly different. In fact, one has u ε (0) = ¯ u (0) + e ψ (0) − ψ (0). Remark 2.4.
The rate is not optimal. By adjusting the parameters α , γ and χ in an optimal way, and by sharpening some of the arguments inour proof, one could achieve a slightly better rate. However, we do notbelieve that any rate obtained in this way would reflect the true speed ofconvergence, so we keep with the values (2.5) that yield simple fractions. Remark 2.5.
From a technical point of view, the general methodologyfollowed in this section and the subsequent sections is inspired from [11],where a somewhat similar phenomenon was investigated. Besides the struc-tural differences in the equations considered here and in [11], the main tech-nical difficulties that need to be overcome for the present work are the fol-lowing:(1) In [11], it is possible to simply subtract the stochastic convolution ψ (or e ψ ) and work with the equation for the remainder. Here, we instead sub-tract only the highest Fourier modes of ψ . The reason for this choice is thatit entails that ¯ v γ → ¯ u as ε →
0. This allows us to linearize the nonlinearityaround ¯ v γ in order to exhibit the desired correction term. As a consequence,our a priori regularity estimates are much worse than those in [11] and ourconvergence rates are worse. The main reason why we need this complicationis that our approximate derivative D ε does not satisfy the chain rule.(2) All of our fixpoint arguments need to be performed in the fractionalSobolev space H α , for some α > . This is in contrast to [11] where someof the arguments could be performed first in L ∞ , and then lifted to H α by a standard bootstrapping argument. These bootstrapping arguments failhere, since the nonlinearity of our approximating equation contains an ap-proximate derivative, which gives rise to correction terms which are not easyto control.(3) In one crucial step where a Gaussian concentration inequality is em-ployed in [11], it was necessary that the stochastic convolutions belong to H α for some α > . This is the case in [11] as a consequence of the extra reg-ularizing effect caused by a small fourth-order term present in the linearpart. This additional regularizing effect is not always present in the currentwork. We therefore perform another truncation in Fourier space, at veryhigh frequencies. This is the purpose of the exponent χ . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Note also that Proposition 4.6 is the analogue of Proposition 4.1 in [11]. Onedifference is that we have a much cleaner separation of the probabilistic andthe analytical aspects of this result.By a standard Picard fixed point argument (see, e.g., [10]) it can be shownthat (2.1) admits a unique mild solution ¯ u defined on a random time interval[0 , τ ∗ ]. Moreover, the spatial regularity of ψ and ¯ u equals that of a Brownianpath, in the sense that ψ ( t ) and ¯ u ( t ) are continuous and belong to H β forany β < and any t >
0, but not to H / . We shall take advantage of thefact that the process ¯ v is much more regular. In fact, ¯ v ( t ) ∈ H β almostsurely for any β < and any t >
0, but one does not expect it to belongto H / in general. This follows immediately from the mild formulation (2.1)combined with a standard bootstrapping argument. It follows from theseconsiderations that, for every fixed time horizon T , the stopping time τ K ∗ := T ∧ inf { t : k ¯ v ( t ) k α ∨ k ¯ u ( t ) k L ∞ ≥ K } converges in probability to τ ∗ ∧ T as K → ∞ .It will be shown in Section 4 that a number of functionals of ψ and e ψ scale in the following way: k e ψ χγ ( t ) k L ∞ . ε γ/ − κ , k e ψ γ ( t ) k L ∞ . ε γ/ − κ , k ψ γ ( t ) k L ∞ . ε γ/ − κ , k e ψ χ ( t ) k L ∞ . ε χ/ − κ , k ψ γ ( t ) k α . ε − γ ( α − / − κ , k e ψ χγ ( t ) k α . ε − χ ( α − / − κ , k e ψ γ ( t ) − ψ γ ( t ) k α . ε − γ ( α +3 / − κ , Θ ε ( e ψ γ ( t )) . ε − − κ , Θ ε ( e ψ χ ( t )) . ε χ − − κ , k Λ − Ξ ε ( e ψ χγ ( t )) k − α . ε / − κ , where the quantities Θ ε and Ξ ε are defined byΘ ε ( u ) def = Z R y k b D εy u k L | µ | ( dy ) , Ξ ε ( u ) def = Z R εy b D εy u ⊗ b D εy uµ ( dy ) . Note that all of these relations are of the form Ψ εi ( t ) . ε α i − κ for some ex-pression Ψ εi depending on ε and some exponent α i . In the proof, it will beconvenient to impose this behavior by means of a hard constraint. For thispurpose, we introduce the stopping time τ K , which is defined for K > τ K def = τ K ∗ ∧ inf { t : ∃ i : Ψ εi ( t ) ≥ ε α i − κ } . (2.9)From now on, we will write C K to denote a constant which may depend on K (and T ) and is allowed to change from line to line. Similarly, κ will be a posi-tive universal constant which is sufficiently small and whose value is allowedto change from line to line. However, the final value of κ is independent of ε , K and T .The remainder of this section is devoted to the proof of Theorem 2.2. M. HAIRER AND J. MAAS
Proof of Theorem 2.2.
Most of the work in the proof consists ofbounding the difference between e v γ and ¯ v γ in H α . This bound will be ob-tained in several steps, using the intermediate processes v ( i ) ε , i = 1 , . . . , v (1) ε ( t ) = S ( t ) v + ψ γ ( t ) + S ∗ ( ¯ F ( v (1) ε ) + ∂ x G ( v (1) ε ))( t ) , (2.10a) v (2) ε ( t ) = S ( t ) v + ψ γ ( t ) + S ∗ ( ¯ F ( v (2) ε ) + D ε G ( v (2) ε ))( t ) , (2.10b) v (3) ε ( t ) = S ε ( t ) v + e ψ γ ( t ) + S ε ∗ ( ¯ F ( v (3) ε ) + D ε G ( v (3) ε ))( t ) , (2.10c) v (4) ε ( t ) = S ε ( t ) v + e ψ γ ( t )(2.10d) + S ε ∗ ( F ( v (4) ε + e ψ χγ ) + ∇ G ( v (4) ε + e ψ χγ ) D ε ( v (4) ε + e ψ χγ ))( t ) . At this stage, we stress that the main difficulty of the proof consists ofshowing that v (3) ε and v (4) ε are close (see Proposition 5.5 below). Showingthe smallness of the remaining differences v ( j ) ε − v ( j +1) ε is relatively straight-forward and follows by applying standard SPDE techniques. The main in-gredient in this part of the proof is an estimate which compares the squareof the approximate derivative of e ψ χγ to the correction term, in a suitableSobolev space of negative order. The estimate is purely probabilistic and ul-timately relies on the fact that the quantity that we wish to control belongsto the second order Wiener chaos. It can be found in Proposition 4.6, whichwe consider to be the core of the paper.Recall the definition of the stopping time τ K given in (2.9). With thisdefinition at hand, we set τ K = τ K as well as v (0) ε def = ¯ v γ and v (5) ε def = e v γ , andwe define recursively a sequence of stopping times τ Kj with j = 1 , . . . , τ Kj = τ Kj − ∧ inf { t : k v ( j ) ε ( t ) − v ( j − ε ( t ) k α ≥ K } . (2.11)With this notation at hand, Propositions 5.1–5.7 state that, for all fixedvalues K, κ > j = 1 , . . . ,
5, one haslim ε → P (cid:16) sup t ≤ τ Kj k v ( j ) ε ( t ) − v ( j − ε ( t ) k α > ε / − κ (cid:17) = 0 . (2.12)Combining all of these bounds, we conclude immediately that, for everyfixed time horizon T > K and κ , we havelim ε → P (cid:16) sup t ≤ τ K k e v γ ( t ) − ¯ v γ ( t ) k α > ε / − κ (cid:17) = 0 . This is formally very close to (2.7), except that we still have the values
T, K > τ K .Since τ ∗ ∧ T → τ ∗ as T → ∞ and since we already argued that τ K ∗ → τ ∗ ∧ T as K → ∞ , the bound (2.7) follows if we are able to show that, for every SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION fixed choice of K and T , lim ε → P ( τ K = τ K ∗ ) = 0 . (2.13)Since the statement of our theorem is stronger, the smaller the value of κ , wecan assume without loss of generality that κ < . In this case, lim ε → ε / − κ =0, so that (2.12) and (2.11) together imply thatlim ε → P ( τ Kj = τ Kj − ) = 0for j = 1 , . . . ,
5, from which we conclude that lim ε → P ( τ K = τ K ) = 0.In order to finish the proof of (2.7), it now suffices to show thatlim ε → P ( τ K = τ K ∗ ) = 0. Fix an arbitrary T > κ >
0. It then followsfrom Propositions 4.3, 4.4 and 4.5 that for each of the terms Ψ εj appearingin (2.9), there exists a constant C j > E sup t ∈ [0 ,T ] Ψ εj ( t ) ≤ C j ε α j − κ/ , uniformly for all ε ≤
1. It then follows from Chebychev’s inequality that P ( τ K = τ K ∗ ) ≤ X j P (cid:16) sup t ∈ [0 ,T ] Ψ εj ( t ) ≥ ε α j − κ (cid:17) ≤ X j C j ε κ/ , from which the claim follows.Since (2.6) follows from (2.7) and (2.8), the proof of the theorem is com-plete if we show that (2.8) holds. Since it follows from Proposition 4.3 andChebychev’s inequality thatlim ε → P (cid:16) sup t ≤ T k e ψ γ ( t ) − ψ γ ( t ) k L ∞ > ε / − κ (cid:17) = 0for every T >
0, this claim follows at once. (cid:3)
3. Analytic tools.
Products and compositions of functions in Sobolev spaces.
In thissubsection, we collect some well-known bounds for products and composi-tions of functions in Sobolev spaces. As is usual in the analysis literature, weuse the notation Φ . Ψ as a shorthand for “there exists a constant C suchthat Φ ≤ C Ψ.” These estimates will be useful in order control the variousterms that arise in the Taylor expansion of the nonlinearity that will beperformed in Section 5 below.
Lemma 3.1.
Let r, s, t ≥ be such that r ∧ s > t and r + s > + t . (1) For f ∈ H r and g ∈ H s , we have f g ∈ H t and k f g k t . k f k r k g k s . (3.1) M. HAIRER AND J. MAAS (2)
For f ∈ H r and g ∈ H − t , we have f g ∈ H − s and k f g k − s . k f k r k g k − t . (3.2) Proof.
This result is very well known. A proof of (3.1) can be found,for example, in [10], Theorem 6.18, and (3.2) follows by duality. (cid:3)
Lemma 3.2.
Let s ∈ ( , . There exists C > such that for any u ∈ H s and any G ∈ C ( R n ; R n ) satisfying k G u k C := sup {| G ( x ) | + |∇ G ( x ) | : | x | ≤ k u k L ∞ } < ∞ , we have k G ◦ u k s ≤ C k G u k C (1 + k u k s ) . Proof.
Let τ h be the shift operator defined by τ h u ( x ) := u ( x − h ). It iswell known (see, e.g., [8] or, for functions defined on R n , [1], Theorem 7.47)that the expression k u k L + (cid:18)Z h t − s sup | h | We will frequently use the fact that for α ≥ β and T > 0, there exists a constant C > k S ( t ) u k α ≤ Ct − ( α − β ) / k u k β (3.4)for any ε ∈ (0 , t ∈ [0 , T ] and u ∈ H β . This is a straightforward consequenceof standard analytic semigroup theory [10, 14]. Since the generator of S isselfadjoint in all of the H s , it is also straightforward to prove (3.4) by hand.As a consequence, we have: Lemma 3.3. Let α, β ∈ R be such that ≤ α − β < and let T > .There exists C > such that for all t ∈ [0 , T ] and u ∈ C ([0 , t ]; H β ) we have (cid:13)(cid:13)(cid:13)(cid:13)Z t S ( t − s ) u ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) α ≤ Ct − ( α − β ) / sup s ∈ [0 ,t ] k u ( s ) k β . (3.5) Proof. It suffices to integrate the bound (3.4). (cid:3) SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION The following bounds measure how well S ε approximates S in these in-terpolation spaces. The general philosophy is that every power of ε has tobe paid with one spatial derivative worth of regularity. This type of power-counting is a direct consequence of the fact that the function f that measureshow much ∆ ε differs from ∂ x , is evaluated at ε | k | in the definition of ∆ ε .The precise bounds are the following: Lemma 3.4. Let κ ∈ [0 , . For T > there exists C > such that forany t ∈ [0 , T ] , ε ∈ (0 , , and u ∈ H β , we have k S ε ( t ) u − S ( t ) u k α ≤ Cε κ t − ( α − β + κ ) / k u k β ( β ≤ α + 2 κ ) , (3.6) k S ε ( t ) u k α ≤ Ct − ( α − β ) / k u k β ( β ≤ α ) . (3.7) Proof. We set ¯ f def = f − ν = 1 for notational simplicity,since the case ν = 1 is virtually identical. The assumptions on f imply that | ¯ f ( εn ) | ≤ cε n whenever n < δ/ε and δ is some sufficiently small constant.Using the mean value theorem and the fact that we can assume δ < n < δ/ε and κ ∈ [0 , | exp( − tn ¯ f ( εn )) − | ≤ (2 ∧ ctε n ) e ctε n ≤ Ct κ/ ε κ n κ e ctδ n ≤ Cε κ t κ/ n κ e cδ t (1+ n ) . Inserting this bound into the identity( S ε ( t ) u − S ( t )) e n = ( e − tn ¯ f ( εn ) − e − t (1+ n ) e n , it then follows from (3.4) that k Π δ/ε ( S ε ( t ) − S ( t )) u k α ≤ Cε κ t κ/ k S ((1 − δ c ) t ) u k α +2 κ (3.8) ≤ Cε κ t − ( α − β + κ ) / k u k β , provided that we choose δ sufficiently small so that δ c ≤ , say.On the other hand, note that( I − Π δ/ε )( S ε ( t ) u − S ( t )) e n = {| n | >δ/ε } ( e − tn ¯ f ( εn ) − e − t (1+ n ) e n . Recall that ¯ f ( εn ) ≥ q − n , and that q ∈ (0 , C such that | exp( − tn ¯ f ( εn )) − | e − t (1+ n ) ≤ Ce − q (1+ n ) t . Moreover, for any κ > {| n | >δ/ε } ≤ | εn/δ | κ . It thus follows, us-ing (3.4) again, that k ( I − Π δ/ε )( S ε ( t ) − S ( t )) u k α ≤ Cε κ k S ( qt ) u k α + κ ≤ Cε κ t − ( α − β + κ ) / k u k β . The bound (3.6) now follows by combining this inequality with (3.8). In-equality (3.7) follows by combining the special case κ = 0 with (3.4). (cid:3) M. HAIRER AND J. MAAS Estimates for the gradient term. In this section, we similarly showhow well the operator D ε approximates ∂ x . Again, the guiding principle isthat every power of ε “costs” the loss of one derivative. However, we arealso going to use the fact that D ε is a bounded operator. In this case, wecan gain up to one spatial derivative with respect to the operator ∂ x , but wehave to “pay” with the same number of inverse powers of ε . The rigorousstatement for the latter fact is the following lemma. Lemma 3.5. Let β ∈ R and α ∈ [0 , . There exists C > such that forall ε ∈ (0 , and u ∈ H β the estimate k D ε u k β − α ≤ Cε α − k u k β holds. Proof. Using the assumption that M := | µ | ( R ) < ∞ , together withJensen’s inequality and Fubini’s theorem, we obtain k D ε u k L ≤ ε Z (cid:18)Z R | u ( x + εy ) || µ | ( dy ) (cid:19) dx ≤ Mε Z Z R | u ( x + εy ) | | µ | ( dy ) dx = M ε k u k L . On the other hand, assuming for the moment that u is smooth, we use theassumption that µ ( R ) = 0, and apply Jensen’s inequality and Minkowski’sintegral inequality to obtain k D ε u k L = 1 ε Z (cid:18)Z R u ( x + εy ) µ ( dy ) (cid:19) dx = 1 ε Z (cid:18)Z R Z εy u ′ ( x + z ) dz µ ( dy ) (cid:19) dx ≤ Mε Z Z R (cid:18)Z εy | u ′ ( x + z ) | dz (cid:19) | µ | ( dy ) dx ≤ Mε Z R (cid:18)Z εy (cid:18)Z | u ′ ( x + z ) | dx (cid:19) / dz (cid:19) | µ | ( dy )= M k u ′ k L Z R y | µ | ( dy ) ≤ C k u k . Using complex interpolation, it follows that k D ε u k L ≤ Cε α − k u k α for every α ∈ [0 , D ε commuteswith every Fourier multiplier. (cid:3) The announced approximation result on the other hand is the followinglemma. SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Lemma 3.6. Let β ∈ R and α ∈ [0 , . There exists C > such that forall ε ∈ (0 , and u ∈ H β the estimate k D ε u − ∂ x u k β − − α ≤ Cε α k u k β holds. Proof. In view of (1.3) we have, assuming for the moment that u issmooth, ( D ε − ∂ x ) u ( x ) = 1 ε Z R Z εy Z w u ′′ ( x + z ) dz dw µ ( dy ) . Integrating against a test function ϕ and applying Fubini’s theorem, wearrive at (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ( x )( D ε − ∂ x ) u ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε Z R Z εy Z w k ϕ k − β k u k β dz dw | µ | ( dy ) ≤ Cε k ϕ k − β k u k β Z R | y | | µ | ( dy ) , which implies that k ( D ε − ∂ x ) u k β − ≤ Cε k u k β . On the other hand, Lemma 3.5 implies that k ( D ε − ∂ x ) u k β − ≤ C k u k β , and the result then follows as before by interpolating between these esti-mates. (cid:3) As an immediate corollary of these bounds, we obtain the following usefulfact. Corollary 3.7. Let β ∈ [0 , . There exists C > such that for ε ∈ (0 , , u ∈ H β , and G ∈ C ( R n ) we have k D ε G ( u ) − ∂ x G ( u ) k − ≤ Cε β k G u k C (1 + k u k β ) , where k G u k C is defined as in Lemma 3.2. Proof. Using Lemmas 3.6 and 3.2, we obtain k D ε G ( u ) − ∂ x G ( u ) k − ≤ Cε β k G ( u ) k β ≤ Cε β k G u k C (1 + k u k β ) , which is the stated claim. (cid:3) 4. Probabilistic tools. In this section, we prove some sharp estimatesfor certain expressions involving stochastic convolutions. Our main tool isthe following version of Kolmogorov’s continuity criterion, which follows M. HAIRER AND J. MAAS immediately from the one given, for example, in [16]. The reason why westate condition (4.1) in this form, is that it is automatically satisfied (byhypercontractivity) for random fields taking values in a Wiener chaos offixed (finite) order. Lemma 4.1. Let ( ϕ ( t )) t ∈ [0 , n be a Banach space-valued random fieldhaving the property that for any q ∈ (2 , ∞ ) there exists a constant K q > such that ( E k ϕ ( t ) k q ) /q ≤ K q ( E k ϕ ( t ) k ) / , (4.1) ( E k ϕ ( s ) − ϕ ( t ) k q ) /q ≤ K q ( E k ϕ ( s ) − ϕ ( t ) k ) / for all s, t ∈ [0 , n . Furthermore, suppose that the estimate E k ϕ ( s ) − ϕ ( t ) k ≤ K | s − t | δ holds for some K , δ > and all s, t ∈ [0 , n . Then, for every p > thereexists C > such that E sup t ∈ [0 , n k ϕ ( t ) k p ≤ C ( K + E k ϕ (0) k ) p/ . Throughout this subsection, we shall use θ k and e θ k for the Fourier coeffi-cients of ψ and e ψ , so that ψ ( t ) = X k ∈ Z θ k ( t ) e k , e ψ ( t ) = X k ∈ Z e θ k ( t ) e k . With this notation at hand, we first state the following approximationbound, which shows that we can again trade powers of k for powers of ε ,provided that we look at the difference squared: Lemma 4.2. For t ≥ , k ∈ Z and ε ∈ (0 , , we have E | e θ k ( t ) − θ k ( t ) | ≤ C ( k − ∧ ε k ) . (4.2) Proof. We write again ¯ f = f − ν = 1 for simplicity. TheItˆo isometry then implies that E | e θ k ( t ) − θ k ( t ) | = C Z ∞ e − t (1+ k ) (1 − h ( ε | k | ) e − tk ¯ f ( ε | k | ) ) dt ≤ C Z ∞ e − t (1+ k ) (1 − e − tk ¯ f ( ε | k | ) ) dt (4.3) + C Z ∞ e − t (1+ k ) e − tk ¯ f ( ε | k | ) (1 − h ( ε | k | )) dt def = I + I . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Let δ > I with | εk | ≤ δ . Since f is twice differentiable near the origin, we canfind δ small enough so that | ¯ f ( | εk | ) | ≤ cε k for some c > 0. Therefore, for t ≥ | − e − tk ¯ f ( ε | k | ) | ≤ ctε k e ctε k ≤ ctε k e cδ tk , (4.4)so that | I | ≤ Cε k Z ∞ t e − t (1+ k )+2 cδ k t dt. If we ensure that δ is small enough so that 2 cδ ≤ 1, we obtain | I | ≤ Cε k Z ∞ t e − k t dt ≤ Cε k ≤ C ( k − ∧ ε k ) , where the last inequality follows from the fact that | εk | ≤ δ by assumption.To treat the case | εk | > δ , we use the fact that by assumption there exists q ∈ (0 , 1] such that f ≥ q , so that | I | ≤ Z ∞ e − tk (1 − e − tk ( q − ) dt ≤ C Z ∞ e − tqk dt (4.5) ≤ Ck − ≤ C ( k − ∧ ε k ) . The bound on I works in pretty much the same way, using the fact thatthe assumptions on h imply that | − h ( ε | k | ) | ≤ C (1 ∧ ε k ) . Using again the fact that f ≥ q , we then obtain I ≤ C Z ∞ e − tqk (1 ∧ ε k ) dt ≤ C ( k − ∧ ε k )as required. (cid:3) We continue with a sequence of propositions, in which the estimatesobtained in the previous lemma are used to establish various bounds forstochastic convolutions. Proposition 4.3. Let < γ < χ . For κ > and ε ∈ (0 , we have E sup t ∈ [0 ,T ] k ψ γ ( t ) k L ∞ ≤ Cε γ/ − κ , E sup t ∈ [0 ,T ] k e ψ γ ( t ) k L ∞ ≤ Cε γ/ − κ , E sup t ∈ [0 ,T ] k e ψ χγ ( t ) k L ∞ ≤ Cε γ/ − κ , E sup t ∈ [0 ,T ] k e ψ γ ( t ) − ψ γ ( t ) k L ∞ ≤ Cε / − κ . M. HAIRER AND J. MAAS Proof. We start with the proof of the second estimate. Observe that e θ k is a complex one-dimensional stationary Ornstein–Uhlenbeck process withvariance h / (2(1 + νk f )) and characteristic time 1 + νk f . This impliesthat E | e θ k ( t ) | = h ( ε | k | )2(1 + νk f ( ε | k | )) ≤ C (1 ∧ k − )(4.6)and E | e θ k ( t ) − e θ k ( s ) | ≤ Ch ( ε | k | ) | t − s | ≤ C | t − s | . (4.7)These bounds imply that, on the one hand, E | e θ k ( t ) e k ( x ) − e θ k ( s ) e k ( y ) | ≤ C E | e θ k ( t ) | + C E | e θ k ( s ) | ≤ C (1 ∧ k − ) , while on the other hand, one has E | e θ k ( t ) e k ( x ) − e θ k ( s ) e k ( y ) | ≤ C E | e θ k ( t ) − e θ k ( s ) | + Ck | x − y | E | e θ k ( s ) | ≤ C | t − s | + C | x − y | . Combining these inequalities we find that, for every κ ∈ [0 , E | e θ k ( t ) e k ( x ) − e θ k ( s ) e k ( y ) | ≤ C (1 ∧ k − ) − κ/ ( | t − s | + | x − y | ) κ/ . Since the e θ k ’s are independent except for the reality condition e θ − k = e θ k , weinfer that E | e ψ γ ( t, x ) − e ψ γ ( s, y ) | ≤ C X | k | >ε − γ E | e θ k ( t ) e k ( x ) − e θ k ( s ) e k ( y ) | ≤ C ( | t − s | + | x − y | ) κ/ X | k | >ε − γ (1 ∧ k − ) − κ/ ≤ Cε (1 − κ ) γ ( | t − s | + | x − y | ) κ/ . Arguing similarly, we obtain E | e ψ γ (0 , | ≤ C X | k | >ε − γ E | e θ k (0) | ≤ C X | k | >ε − γ (1 ∧ k − ) ≤ Cε γ . The result now follows by combining these two bounds with Lemma 4.1.The proof of the first and third estimates being very similar, we do notreproduce them here. In order to prove the last estimate, we use Lemma 4.2to obtain E | e θ k ( t ) − θ k ( t ) | ≤ C ( k − ) / κ/ ( ε k ) / − κ/ ≤ Cε − κ k − − κ . This bound then replaces (4.6), and the rest of the proof is again analogousto the proof of the second estimate. (cid:3) SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Proposition 4.4. Let ζ > . For κ > and ε ∈ (0 , , we have E sup t ∈ [0 ,T ] k ψ ζ ( t ) k α ≤ Cε − ζ ( α − / − κ (cid:18) α > (cid:19) , (4.8) E sup t ∈ [0 ,T ] k e ψ ζ ( t ) − ψ ζ ( t ) k α ≤ Cε − ζ ( α +3 / − κ (cid:18) α > − (cid:19) . (4.9) Proof. In view of the estimates E | θ k ( t ) | ≤ Ck − , E | θ k ( t ) − θ k ( s ) | ≤ C | t − s | , (4.10)we obtain E k ψ ζ ( t ) − ψ ζ ( s ) k α ≤ C | t − s | κ X | k |≤ ε − ζ (1 + k ) α − κ ≤ C | t − s | κ ε − ζ ( α − / κ ) and E k ψ ζ (0) k α ≤ Cε − ζ ( α − / . Inequality (4.8) thus follows from Lemma 4.1.In order to prove (4.9), we argue similarly, but the estimates are slightlymore involved. Write δ k := e θ k − θ k so that e ψ ζ − ψ ζ = P | k |≤ ε − ζ δ k e k . Us-ing (4.7) and (4.10), we have for s, t ≥ E | δ k ( t ) − δ k ( s ) | ≤ C | t − s | . Combining this bound with Lemma 4.2, we infer that for κ ∈ [0 , ), E | δ k ( t ) − δ k ( s ) | ≤ C ( k − ) κ ( ε k ) − κ | t − s | κ = Cε − κ k − κ | t − s | κ . For κ ∈ (0 , α + ), we thus obtain E k ( e ψ ζ − ψ ζ )( t ) − ( e ψ ζ − ψ ζ )( s ) k α ≤ C | t − s | κ ε − κ X | k |≤ ε − ζ (1 + k ) α +1 − κ ≤ C | t − s | κ ε − ζ (2 α +3) − κ and similarly E sup t ∈ [0 ,T ] k e ψ ζ ( t ) − ψ ζ ( t ) k α ≤ Cε − ζ (2 α +3) − κ . The desired estimate (4.9) now follows from Lemma 4.1. (cid:3) Proposition 4.5. Let ζ > . For every κ > there exists C > suchthat E sup t ∈ [0 ,T ] Θ( e ψ ζ ( t )) ≤ Cε − ζ − + − κ for all ε ∈ (0 , , where we wrote ( ζ − + def = 0 ∨ ( ζ − . M. HAIRER AND J. MAAS Proof. As in the proof of Propositions 4.3 and 4.4, we shall applyKolmogorov’s continuity criterion from Lemma 4.1, this time for L -valuedrandom fields. It follows from (4.6) that E k b D εy ( e ψ ζ ( t ) − e ψ ζ ( s )) k L = X | k | >ε − ζ (cid:12)(cid:12)(cid:12)(cid:12) e ikεy − εy (cid:12)(cid:12)(cid:12)(cid:12) E | e θ k ( t ) − e θ k ( s ) | ≤ C X k>ε − ζ − cos( kεy ) | εky | . Note that, up to a factor ε | y | , this sum can be interpreted as a Riemannsum for the function H ( t ) def = t − (1 − cos( t )). In fact, since H ( t ) ≤ ∧ t − ), ε | y | X k>ε − ζ − cos( kεy ) | kεy | = X k>ε − ζ ε | y | H ( kεy ) ≤ Z ∞ ε − ζ (1 ∧ t − ) dt (4.11) ≤ Cε ( ζ − + . It thus follows that E k b D εy ( e ψ ζ ( t ) − e ψ ζ ( s )) k L ≤ C | εy | − ε ( ζ − + . (4.12)On the other hand, (4.6) and (4.7) imply that E | e θ k ( t ) − e θ k ( s ) | ≤ C (1 ∧ k − ) / | t − s | / , and therefore E k b D εy ( e ψ ζ ( t ) − e ψ ζ ( s )) k L = X | k | >ε − ζ (cid:12)(cid:12)(cid:12)(cid:12) e ikεy − εy (cid:12)(cid:12)(cid:12)(cid:12) E | e θ k ( t ) − e θ k ( s ) | ≤ C | εy | − | t − s | / X | k | >ε − ζ (1 ∧ k − ) / (4.13) ≤ C | εy | − | t − s | / . Combining (4.12) and (4.13), we find that E k b D εy ( e ψ ζ ( t ) − e ψ ζ ( s )) k L ≤ | εy | − − κ | t − s | κ/ ε (1 − κ )( ζ − + . Similarly, we obtain E k b D εy e ψ ζ (0) k L = X | k | >ε − ζ (cid:12)(cid:12)(cid:12)(cid:12) e ikεy − εy (cid:12)(cid:12)(cid:12)(cid:12) E | e θ k (0) | ≤ C X | k | >ε − ζ − cos( kεy ) | εky | ≤ C | εy | − ε ( ζ − + . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION In view of Lemma 4.1, the latter two estimates imply that E sup t ∈ [0 ,T ] k b D εy e ψ ζ ( t ) k L ≤ C | εy | − − κ ε (1 − κ )( ζ − + . Using this bound, the desired result for Θ( e ψ ζ ( t )) can be obtained easily,since E sup t ∈ [0 ,T ] Θ( e ψ ζ ( t )) = E sup t ∈ [0 ,T ] Z R | y | k b D εy e ψ ζ ( t ) k L | µ | ( dy ) ≤ Z R | y | E sup t ∈ [0 ,T ] k b D εy e ψ ζ ( t ) k L | µ | ( dy ) ≤ Cε − − κ +(1 − κ )( ζ − + Z R | y | − κ | µ | ( dy ) ≤ Cε − − κ +(1 − κ )( ζ − + . The result now follows by rescaling κ . (cid:3) The next and final result of this section involves the term which gives riseto the correction term in the limiting equation. Before stating the result, weintroduce the notationΞ yε ( u ) def = εy b D εy u ⊗ b D εy u, Λ y def = 12 πν Z R + h ( t ) t f ( t ) (1 − cos( yt )) dt and Λ yε def = X ε − γ Let α > , γ ≤ and χ ≥ . For ε ∈ (0 , , we thenhave E sup t ∈ [0 ,T ] k Λ − Ξ ε ( e ψ χγ ( t )) k − α ≤ Cε / . Proof. The proof is an application of Lemma 4.1 with ξ = Λ − Ξ ε ( e ψ χγ ).For brevity, we shall write A := Ξ ε ( e ψ χγ ) and A y := Ξ yε ( e ψ χγ ). We divide theproof into several steps. M. HAIRER AND J. MAAS Step 1. First, we claim that ξ ( t ) = Λ − A ( t ) satisfies the condition (4.1)concerning the equivalence of all q -moments.To see this, note that e ψ χγ admits the representation e ψ ( t ) = P k α k ( t ) e k where each α k ( t ) is a Gaussian random vector in R n . As a consequence, forevery y ∈ R , each component of Λ yε − A y is a polynomial of Gaussian randomvariables of degree at most two. It thus belongs to the direct sum of Wienerchaoses of order ≤ ε − A , since each Wiener chaosis a closed subspace of the space of square integrable random variables. Theclaim thus follows from the well-known equivalence of moments for Hilbertspace-valued Wiener chaos (see, e.g., [13]). Step 2. In this step, we estimate how well Λ yε approximates Λ y . Since | − cos x | ≤ C (1 ∧ x ), we have the bound | Λ yε,k | ≤ C ( εy ∧ ( εk ) − ) forsome constant C . As an immediate consequence, we have the bound (cid:12)(cid:12)(cid:12)(cid:12) Λ yε − X k ≥ Λ yε,k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ε − γ y + ε χ − ) . (4.14)Define now the function Φ y ( t ) = (1 − cos( yt )) h ( t )2 πνt f ( t ) , so that, since h /f is bounded by assumption, we obtain the bound | Λ yε,k − ε Φ y ( εk ) | ≤ C εy k . Combining this bound with (4.14), we have (cid:12)(cid:12)(cid:12)(cid:12) Λ yε − X k ≥ ε Φ y ( εk ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ε − γ y + ε χ − ) . At this stage, we recall that for any function Φ of bounded variation, onehas the approximation (cid:12)(cid:12)(cid:12)(cid:12)X k ≥ ε Φ( εk ) − Z ∞ Φ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k Φ k BV , where k Φ k BV denotes the variation of Φ over R + . Furthermore, for anypair Φ, Ψ, we have the bound k ΦΨ k BV ≤ k Φ k L ∞ k Ψ k BV + k Ψ k L ∞ k Φ k BV . (4.15)If we set Ψ y ( t ) = (1 − cos( yt )) /t , we have k Ψ y k BV = Z ∞ | Ψ ′ y ( t ) | dt = Z ∞ | yt sin yt + 2 cos yt − | t dt ≤ C | y | Z ∞ (cid:18) ∧ y t (cid:19) dt ≤ Cy . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Since Ψ(0) = y / 2, a similar bound holds for its L ∞ norm, and we concludefrom (4.15) that k Φ y k BV ≤ Cy . It follows immediately that we have the bound | Λ yε − Λ y | ≤ C ( ε − γ y + εy + ε χ − ) . (4.16) Step 3. We now use these bounds in order to obtain control over k Λ − A k − α for a fixed time t ≥ e ψ ( x, t ) = X k ∈ Z h ( ε | k | ) √ p νk f ( ε | k | ) η k ( t ) e k ( x ) , where the η k are a sequence of i.i.d. C n -valued Ornstein–Uhlenbeck processeswith E ( η k ( t ) ⊗ η ℓ ( s )) = E t − sk δ k, − ℓ I, E tk = exp( − (1 + νk f ( ε | k | )) | t | ) , and satisfying the reality condition η − k = ¯ η k . Here, I denotes the identitymatrix. We will also use the notational shortcut A tk,ℓ def = η k ( t ) ⊗ η ℓ ( t ) . Set now q kε = e ikεy − √ h ( ε | k | ) p νk f ( ε | k | ) , as a shorthand. With all of this notation in place, it follows from the defini-tion of Λ yε that A y ( t ) − Λ yε I = X ε − γ < | k | , | ℓ |≤ ε − χ q kε q ℓε ( A tk,ℓ − δ k, − ℓ I ) e k + ℓ . As a consequence, we have the identity E k Λ yε I − A y ( t ) k − α = X k ∈ Z (1 + | k | ) − α X ℓ,m q ℓε q k − ℓε ¯ q mε ¯ q k − mε × E tr(( A tℓ,k − ℓ − δ k, I )( ¯ A tm,k − m − δ k, I )) , where the second sum ranges over all ℓ, m ∈ Z for which ℓ, k − ℓ, m, k − m belong to ( ε − γ , ε − χ ]. A straightforward case analysis allows to check that E tr(( A tℓ,k − ℓ − δ k, I )( ¯ A tm,k − m − δ k, I )) = nδ ℓ,m + n δ ℓ,k − m , (4.17)so that E k Λ yε I − A y ( t ) k − α ≤ C X k ∈ Z (1 + | k | ) − α X ℓ ∈ Z | q ℓε | | q k − ℓε | . M. HAIRER AND J. MAAS Note now that there exists a constant C such that the bound | q kε | ≤ C √ ε (cid:18) | y | ∧ ε | k | (cid:19) ≤ Cε (1 − β ) / | k | − β/ | y | − β/ is valid for all ε < k ∈ Z , y ∈ R , and β ∈ [0 , C > E k Λ yε I − A y ( t ) k − α ≤ C X ℓ,m ≥ | q ℓε | | q mε | | ℓ + m | α ≤ C X ℓ,m ≥ | q ℓε | | q mε | | ℓ | α | m | α ≤ Cε − β X ℓ,m ≥ | y | − β | ℓ | α + β | m | α + β ≤ Cε | y | , where we made the choice β = to obtain the last bound, using the factthat α > by assumption. Combining this bound with (4.16), the constraints γ ≤ and χ ≥ , and using the fact that µ has finite fourth moment, wehave E k Λ I − A ( t ) k − α ≤ C Z E k Λ y − A y ( t ) k − α | µ | ( dy ) ≤ C Z E k Λ yε − A y ( t ) k − α | µ | ( dy ) + Cε ≤ Cε. Step 4. Finally, we shall estimate E k A ( t ) − A ( s ) k − α . Similarly to (4.17),this involves the identity E tr( A tℓ,k − ℓ ¯ A sm,k − m ) = nδ k, + ( nδ l,m + n δ l,k − m ) E t − sℓ E t − sk − m . As a consequence, we infer that D kℓm ( t, s ) def = E tr(( A tℓ,k − ℓ − A sℓ,k − ℓ )( ¯ A tm,k − m − ¯ A sm,k − m ))= 2( nδ ℓ,m + n δ ℓ,k − m )(1 − E t − sℓ E t − sk − m ) . It thus follows that for any δ ∈ [0 , D kℓm ( t, s ) ≤ C ( δ ℓ,m + δ ℓ,k − m )(1 ∧ (2 + νℓ f ( ε | ℓ | ) + ν ( k − m ) f ( ε | k − m | )) | t − s | ) ≤ C ( δ ℓ,m + δ ℓ,k − m ) | t − s | δ (1 + ℓ δ f ( ε | ℓ | ) δ + ( k − m ) δ f ( ε | k − m | ) δ ) . Using this bound, we obtain E k A y ( t ) − A y ( s ) k − α = X k ∈ Z (1 + | k | ) − α X ℓ,m q ℓε q k − ℓε ¯ q mε ¯ q k − mε D kℓm ( t, s ) SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION ≤ X k ∈ Z (1 + | k | ) − α X ℓ | q ℓε | | q k − ℓε | ( D kℓℓ ( t, s ) + D k,ℓ,k − ℓ ( t, s )) ≤ C | t − s | δ X k ∈ Z (1 + | k | ) − α X ℓ | q ℓε | | q k − ℓε | × (1 + ℓ δ f ( ε | ℓ | ) δ + | k − ℓ | δ f ( ε | k − ℓ | ) δ ) . Note that this expression is almost the same as in Step 3. Using the cal-culations done there and taking into account that h and h/f are boundedfunctions, we infer that E k A y ( t ) − A y ( s ) k − α = C | t − s | δ X ℓ,m ≥ | q ℓε | | q mε | | ℓ | α − δ | m | α − δ ≤ C | t − s | δ ε | y | , and therefore, using Jensen’s inequality (which can be applied since | µ | hasfinite mass), and Fubini’s theorem, E k A ( t ) − A ( s ) k − α = E (cid:13)(cid:13)(cid:13)(cid:13)Z R ( A y ( t ) − A y ( s )) µ ( dy ) (cid:13)(cid:13)(cid:13)(cid:13) − α ≤ C Z R E k A y ( t ) − A y ( s ) k − α | µ | ( dy ) ≤ Cε | t − s | δ Z R | y || µ | ( dy ) ≤ Cε | t − s | δ , which is the desired bound.The result follows by combining these steps with Lemma 4.1. (cid:3) 5. Convergence of the approximations. This last section is devoted tothe convergence result itself. Recall that we are considering a number of in-termediate processes v ( j ) ε with j = 1 , . . . , j th subsection yield-ing a bound on k v ( j ) ε − v ( j − ε k α . To prove these bounds, we shall introducein each step a stopping time that forces the difference between the processesconsidered in that step to remain bounded. We then show that this differ-ence actually vanishes as ε → From ¯ v γ to v (1) ε . Define τ K := τ K ∧ inf { t ≤ T : k v (1) ε ( t ) − ¯ v γ ( t ) k α ≥ K } . We shall show that for t ≤ τ K , the H α -norm of v (1) ε ( t ) − ¯ v γ ( t ) is controlledby the L ∞ -norm of ψ γ , which is of order ε γ/ − κ for any κ > 0, as shownin Section 4. The proof uses the mild formulations of the equations for v (1) ε and ¯ v γ ( t ) as well as the regularizing properties of the semigroup S . Note M. HAIRER AND J. MAAS that the next proposition would still be true if we had replaced the H α -norm in the definition of τ K by the L ∞ -norm. However, in the proof ofProposition 5.2 below it will be important to have a bound on v (1) ε in H α . Proposition 5.1. For κ > , we have lim ε → P (cid:16) sup t ≤ τ K k v (1) ε ( t ) − ¯ v γ ( t ) k α > ε γ/ − κ (cid:17) = 0 . Proof. Let 0 ≤ s ≤ t ≤ τ ∗ . It follows from (2.3) and (2.10a) that ̺ ε := v (1) ε − ¯ v γ satisfies the equation ̺ ε ( t ) = S ( t − s ) ̺ ε ( s ) + Z ts S ( t − r )( σ + ∂ x σ ε )( r ) dr, where σ ε := ¯ F (¯ v γ + ̺ ε ) − ¯ F (¯ v γ + ψ γ ) ,σ ε := G (¯ v γ + ̺ ε ) − G (¯ v γ + ψ γ ) . Lemma 3.3 yields the estimate k ̺ ε ( t ) k α ≤ k ̺ ε ( s ) k α + C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k ( σ ε + ∂ x σ ε )( r ) k − ≤ k ̺ ε ( s ) k α + C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k σ ε ( r ) k L ∞ + k σ ε ( r ) k L ∞ . Since ¯ v γ , ̺ ε , and ψ γ are bounded in L ∞ -norm for r ≤ τ K , and F, G are C ,it follows that k σ ε ( r ) k L ∞ + k σ ε ( r ) k L ∞ ≤ C K k ̺ ε ( r ) k L ∞ + C K k ψ γ ( r ) k L ∞ , from which we infer that k ̺ ε ( t ) k α ≤ k ̺ ε ( s ) k α + C ′ K ( t − s ) (1 − α ) / sup r ∈ ( s,t ) ( k ̺ ε ( r ) k L ∞ + k ψ γ ( r ) k L ∞ ) . Choose δ K > C ′ K δ (1 − α ) / K ≤ , and set for k ≥ r k := sup {k ̺ ε ( t ) k α : t ∈ [ kδ K ∧ τ K , ( k + 1) δ K ∧ τ K ] } . Taking into account that H α ⊆ L ∞ , we obtain the inequality r k +1 ≤ r k + 12 r k +1 + 12 sup t ∈ [0 ,T ] k ψ γ ( t ) k L ∞ , which reduces to r k +1 ≤ r k + sup t ∈ [0 ,T ] k ψ γ ( t ) k L ∞ . Combined with the estimate r ≤ r ≤ δ K ∧ τ K k ψ γ ( r ) k L ∞ , SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION which can be derived similarly, it then follows thatsup t ∈ [0 ,τ K ] k ̺ ε ( t ) k α ≤ sup ≤ k ≤ T/δ K r k ≤ C K sup t ∈ [0 ,T ] k ψ γ ( t ) k L ∞ , which together with Proposition 4.3 implies the desired result. (cid:3) From v (1) ε to v (2) ε . For the purpose of this section, we define thestopping time τ K := τ K ∧ inf { t ≤ T : k v (2) ε ( t ) − v (1) ε ( t ) k α ≥ K } as well as the exponent e α def = (1 − γ ) α + γ . Proposition 5.2. For κ > , we have lim ε → P (cid:16) sup t ≤ τ K k v (2) ε ( t ) − v (1) ε ( t ) k α > ε e α − κ (cid:17) = 0 . Proof. Let 0 ≤ s ≤ t ≤ τ ∗ and note that ̺ ε := v (2) ε − v (1) ε satisfies ̺ ε ( t ) = S ( t − s ) ̺ ε ( s ) + Z ts S ( t − r ) σ ε ( r ) dr, where σ ε := ¯ F ( v (2) ε ) − ¯ F ( v (1) ε ) + D ε ( G ( v (1) ε + ̺ ε )) − ∂ x G ( v (1) ε ) . From the definition of τ K , we know that v (1) ε and ̺ ε are bounded in L ∞ by a constant depending on K . Moreover, we have the bound k v (1) ε k α ≤ C K ε − γ ( α − / − κ . Using these facts together with Corollary 3.7 we obtain,for r ≤ τ K , k σ ε k − ≤ k ¯ F ( v (2) ε ) − ¯ F ( v (1) ε ) k L ∞ + k ( D ε − ∂ x ) G ( v (1) ε ) k − + k D ε ( G ( v (1) ε + ̺ ε ) − G ( v (1) ε )) k − ≤ C K k ̺ ε k L ∞ + C K ε α (1 + k v (1) ε k α ) + k G ( v (1) ε + ̺ ε ) − G ( v (1) ε ) k L ∞ ≤ C K ( ε e α − κ + k ̺ ε k L ∞ ) , hence k ̺ ε ( t ) k α ≤ k ̺ ε ( s ) k α + C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k σ ε ( r ) k − ≤ k ̺ ε ( s ) k α + C K ( t − s ) (1 − α ) / sup r ∈ ( s,t ) ( ε e α − κ + k ̺ ε ( r ) k L ∞ ) . M. HAIRER AND J. MAAS Arguing as in the proof of Proposition 5.1, it follows thatsup t ∈ [0 ,τ K ] k ̺ ε ( t ) k α ≤ C K ε e α − κ , which immediately yields the desired result. (cid:3) From v (2) ε to v (3) ε . Define τ K := τ K ∧ inf { t ≤ T : k v (3) ε ( t ) − v (2) ε ( t ) k α ≥ K } . In this case, the singularity ( t − s ) − α/ which arises in the proof below,prevents us from arguing as in Proposition 5.1. We nevertheless have thefollowing proposition. Proposition 5.3. For κ > , we have lim ε → P (cid:16) sup t ≤ τ K k v (3) ε ( t ) − v (2) ε ( t ) k α > ε ζ − κ (cid:17) = 0 , where the exponent ζ is given by ζ def = e α ∧ ( − α ) ∧ (2 − γ ( α + )) = . Remark 5.4. The exponent ζ arises by collecting the bounds (5.2),(5.3), and (5.5). Proof of Proposition 5.3. Let 0 ≤ s ≤ t ≤ τ ∗ . It follows from (2.10b)and (2.10c) that ̺ ε := v (3) ε − v (2) ε satisfies ̺ ε ( t ) = S ε ( t − s ) ̺ ε ( s ) + ( S ε ( t − s ) − S ( t − s )) v (2) ε ( s )(5.1) + R ( s, t ) + R ( s, t ) , where R ( s, t ) def = ( e ψ γ ( t ) − ψ γ ( t )) − ( S ε ( t − s ) e ψ γ ( s ) − S ( t − s ) ψ γ ( s ))and R ( s, t ) := Z ts ( S ε ( t − r ) − S ( t − r ))( ¯ F ( v (3) ε ( r )) + D ε G ( v (3) ε ( r ))) dr + Z ts S ( t − r )( ¯ F ( v (3) ε ( r )) − ¯ F ( v (2) ε ( r ))+ D ε G ( v (3) ε ( r )) − D ε G ( v (2) ε ( r ))) dr. We shall first prove a bound on R ( s, t ). Using both inequalities fromLemma 3.4, we obtain k ( S ε ( t − s ) e ψ γ ( s ) − S ( t − s ) ψ γ ( s )) k α ≤ k S ε ( t − s )( e ψ γ ( s ) − ψ γ ( s )) k α + k ( S ε ( t − s ) − S ( t − s )) ψ γ ( s ) k α ≤ C k ( e ψ γ ( s ) − ψ γ ( s )) k α + Cε k ψ γ ( s ) k α +2 , SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION and therefore kR ( s, t ) k α ≤ k ( e ψ γ ( t ) − ψ γ ( t )) k α + C k ( e ψ γ ( s ) − ψ γ ( s )) k α + Cε k ψ γ ( s ) k α +2 . It thus follows from Proposition 4.4 that E sup s,t ∈ [0 ,T ] kR ( s, t ) k α ≤ Cε − γ ( α +3 / − κ . (5.2)We shall now prove a bound on R ( s, t ). For this purpose, we note thatthe definitions of the various stopping times imply that v (2) ε ( t ) is boundedin H α -norm by C K ε − γ ( α − / − κ . Using this fact, together with Lemmas 3.4,3.5 and 3.2, we obtain (cid:13)(cid:13)(cid:13)(cid:13)Z ts ( S ε ( t − r ) − S ( t − r ))( F ( v (3) ε ( r )) + D ε G ( v (3) ε ( r ))) dr (cid:13)(cid:13)(cid:13)(cid:13) α ≤ ε α Z ts ( t − r ) − (1+ α ) / k F ( v (3) ε ( r )) + D ε G ( v (3) ε ( r )) k α − dr ≤ Cε α ( t − s ) (1 − α ) / sup r ∈ [ s,t ] ( k F ( v (3) ε ( r )) k α + k G ( v (3) ε ( r )) k α ) ≤ C K ε α ( t − s ) (1 − α ) / (cid:16) r ∈ [ s,t ] k v (3) ε ( r ) k α (cid:17) ≤ C K ε α ( t − s ) (1 − α ) / (cid:16) r ∈ [ s,t ] ( k v (2) ε ( r ) k α + k ̺ ε ( r ) k α ) (cid:17) ≤ C K ε e α − κ ( t − s ) (1 − α ) / . Furthermore, taking into account the L ∞ -bounds on v (2) ε and ̺ ε enforcedby the stopping times, Lemma 3.5 implies that (cid:13)(cid:13)(cid:13)(cid:13)Z ts S ( t − r )( F ( v (3) ε ( r )) − F ( v (2) ε ( r )) + D ε G ( v (3) ε ( r )) − D ε G ( v (2) ε ( r ))) dr (cid:13)(cid:13)(cid:13)(cid:13) α ≤ C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k F ( v (3) ε ( r )) − F ( v (2) ε ( r ))+ D ε G ( v (3) ε ( r )) − D ε G ( v (2) ε ( r )) k − ≤ C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) ( k F ( v (3) ε ( r )) − F ( v (2) ε ( r )) k L ∞ + k G ( v (3) ε ( r )) − G ( v (2) ε ( r )) k L ∞ ) ≤ C K ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k ̺ ε ( r ) k L ∞ . M. HAIRER AND J. MAAS It thus follows that kR ( s, t ) k α ≤ C ′ K ( t − s ) (1 − α ) / (cid:16) ε e α − κ + sup r ∈ ( s,t ) k ̺ ε ( r ) k L ∞ (cid:17) , (5.3)where we gave the constant a name, since it will be reused below.Choose δ K ∈ (0 , 1) sufficiently small so that C ′ K ( δ (1 − α ) / K + δ α/ K ) ≤ . For k ≥ ℓ k := kδ K ∧ τ K , and for k ≥ r k := sup {k ̺ ε ( t ) k α : t ∈ [ ℓ k − , ℓ k +1 ] } . Our next aim is to find a bound for r . Observe that, when s = 0, (5.1)simplifies to ̺ ε ( t ) = ( S ε ( t ) − S ( t )) v + ( e ψ γ ( t ) − ψ γ ( t )) + R (0 , t )(5.4)with R defined as previously. Using Lemma 3.4 and the definition of τ K ,we obtain k ( S ε ( t ) − S ( t )) v k α ≤ Cε / − α − κ k v k / − κ ≤ C K ε / − α − κ . (5.5)Since t ≤ δ K and C ′ K δ (1 − α ) / K ≤ , it follows from (5.3) and (5.4) that r ≤ C K ε / − α − κ + sup t ∈ [0 ,T ] k e ψ γ ( t ) − ψ γ ( t ) k α + 12 ( ε e α − κ + r ) , hence, by definition of τ K , r ≤ C K ε (3 / − α ) ∧ (2 − γ ( α +3 / ∧ e α ε − κ = C K ε ζ − κ , (5.6)where ζ is defined as in the statement of the result.Next, we shall prove a recursive bound for r k . Note that the nonnegativityof the function f in the definition of S ε implies that k S ε ( t − s ) ̺ ε ( s ) k α ≤ k ̺ ε ( s ) k α . Furthermore, by Lemma 3.4 and the fact that k v (2) ε k α ≤ C K ε − γ ( α − / − κ before time τ K , we have k ( S ε ( t − s ) − S ( t − s )) v (2) ε ( s ) k α ≤ Cε α ( t − s ) − α/ k v (2) ε ( s ) k α ≤ C K ( t − s ) − α/ ε e α − κ . Combining these bounds with (5.1) and (5.3), we find that k ̺ ε ( t ) k α ≤ k ̺ ε ( s ) k α + C K ( t − s ) − α/ ε e α − κ + kR ( s, t ) k α + C ′ K ( t − s ) (1 − α ) / (cid:16) ε e α − κ + sup r ∈ ( s,t ) k ̺ ε ( r ) k α (cid:17) . Taking k ≥ s = ℓ k − , and t ∈ [ ℓ k , ℓ k +2 ], it then follows, since | t − s | ∈ [ δ K , δ K ] and C ′ K δ (1 − α ) / ≤ , that k ̺ ε ( t ) k α ≤ k ̺ ε ( ℓ k − ) k α + C K ε e α − κ + kR ( ℓ k − , t ) k α + ε e α − κ + r k +1 . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Taking the supremum over t ∈ [ ℓ k , ℓ k +2 ], we obtain r k +1 ≤ r k + C K ε e α − κ + sup s,t ∈ [0 ,T ] kR ( s, t ) k α + 12 r k +1 , hence r k +1 ≤ r k + C K ε e α − κ + 2 sup s,t ∈ [0 ,T ] kR ( s, t ) k α . (5.7)It readily follows from (5.6) and (5.7) thatsup t ∈ [0 ,τ K ] k ̺ ε ( t ) k α = sup ≤ k ≤⌈ T/δ K ⌉ r k ≤ C K (cid:16) ε ζ − κ + ε e α − κ + sup s,t ∈ [0 ,T ] kR ( s, t ) k α (cid:17) , hence the result follows in view of the bound on R ( s, t ). (cid:3) From v (3) ε to v (4) ε . Define τ K := τ K ∧ inf { t ≤ T : k v (4) ε ( t ) − v (3) ε ( t ) k α ≥ K } . Proposition 5.5. For κ > , we have lim ε → P (cid:16) sup t ≤ τ K k v (4) ε ( t ) − v (3) ε ( t ) k α > ε ξ − κ (cid:17) = 0 , where ξ def = γ ∧ (cid:18)e α − (cid:19) ∧ (cid:18) − χ (cid:18) α − (cid:19)(cid:19) = 18 . Remark 5.6. Similarly to above, the exponent ξ arises from the bounds(5.9)–(5.14). Proof of Proposition 5.5. Let 0 ≤ s ≤ t ≤ τ ∗ . It follows from (2.10d)and (2.10c) that ̺ ε := v (4) ε − v (3) ε satisfies ̺ ε ( t ) = S ε ( t − s ) ̺ ε ( s ) + Z ts S ε ( t − r ) σ ε ( r ) dr, where σ ε := F ( v (4) ε + e ψ χγ ) − F ( v (3) ε )+ ∇ G ( v (4) ε + e ψ χγ ) D ε ( v (4) ε + e ψ χγ ) − D ε G ( v (3) ε ) + Λ∆ G ( v (3) ε ) . The definition of D ε , together with (1.3), implies that for any function u thefollowing identity holds: D ε G ( u )( x )= ∇ G ( u ( x )) D ε u ( x ) M. HAIRER AND J. MAAS + Z R εy D G ( u ( x ))[ b D εy u ( x ) , b D εy u ( x )] µ ( dy )(5.8) + Z R ε y Z Z t Z s D G ((1 − r ) u ( x ) + ru ( x + εy )) × [ b D εy u ( x ) , b D εy u ( x ) , b D εy u ( x )] dr ds dt µ ( dy ) , where the operator b D ε is defined by taking µ := δ − δ in the definitionof D ε , that is, b D ε u ( x ) = ε − ( u ( x + ε ) − u ( x )). As a consequence, we maywrite σ ε = F ( v (4) ε + e ψ χγ ) − F ( v (3) ε ) + D ε ( G ( v (4) ε + e ψ χγ ) − G ( v (3) ε ))+ (Λ∆ G ( v (3) ε ) − A ( u (4) ε , u (4) ε )) − B = F ( v (4) ε + e ψ χγ ) − F ( v (3) ε ) + D ε ( G ( v (4) ε + e ψ χγ ) − G ( v (3) ε )) − A ( v (4) ε , v (4) ε ) − A ( v (4) ε , e ψ χγ ) + (Λ∆ G ( v (3) ε ) − A ( e ψ χγ , e ψ χγ )) − B, where we have used A ( v, w )( x ) def = Z R εy D G ( u (4) ε ( x ))[ b D εy v ( x ) , b D εy w ( x )] µ ( dy ) ,B ( x ) def = Z R ε y Z Z t Z s D G ((1 − r ) u (4) ε ( x ) + ru (4) ε ( x + εy )) × [ b D εy u (4) ε ( x ) , b D εy u (4) ε ( x ) , b D εy u (4) ε ( x )] dr ds dt µ ( dy )and u (4) ε := v (4) ε + e ψ χγ .Our next aim is to prove the estimates (5.9)–(5.14) below in order tobound k σ ε k − . First term. Since v (4) ε , e ψ χγ and ̺ ε are bounded in L ∞ by definition of τ K ,it follows that k F ( v (4) ε + e ψ χγ ) − F ( v (3) ε ) k − ≤ C k F ( v (4) ε + e ψ χγ ) − F ( v (4) ε − ̺ ε ) k L ∞ ≤ C K ( k e ψ χγ k L ∞ + k ̺ ε k L ∞ )(5.9) ≤ C K ( ε γ/ − κ + k ̺ ε k α ) . Second term. We use Lemma 3.5 and the fact that v (4) ε , e ψ χγ and ̺ ε arebounded in L ∞ by definition of τ K , to estimate k D ε ( G ( v (4) ε + e ψ χγ ) − G ( v (3) ε )) k − ≤ C k G ( v (4) ε + e ψ χγ ) − G ( v (4) ε − ̺ ε ) k L ∞ ≤ C K ( k e ψ χγ k L ∞ + k ̺ ε k L ∞ )(5.10) ≤ C K ( ε γ/ − κ + k ̺ ε k α ) . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Third and fourth term. First, we note that for arbitrary functions v, w ,one has k A ( v, w ) k − ≤ C k A ( v, w ) k L ≤ Cε k D G ( u (4) ε ) k L ∞ p Θ ε ( v )Θ ε ( w ) . Since k u (4) ε ( t ) k L ∞ ≤ C K for t ≤ τ K , we have k D G ( u (4) ε ) k L ∞ ≤ C K . Furthermore, we observe that k v (4) ε k α ≤ C K ε − γ ( α − / − κ before time τ K .Using this bound together with Lemma 3.5 and (1.3), we estimateΘ ε ( v (4) ε ) = Z R y k b D εy v (4) ε k L | µ | ( dy ) ≤ C K Z R y | εy | α − k v (4) ε k α | µ | ( dy ) ≤ C K Z R y | εy | α − ε − γ ( α − / − κ | µ | ( dy ) ≤ C K ε e α − − κ . Moreover, by definition of the stopping time τ K we haveΘ ε ( e ψ χγ ) ≤ Θ ε ( e ψ γ ) ≤ C K ε − − κ . Putting everything together, we obtain k A ( v (4) ε , v (4) ε ) k − ≤ C K ε e α − − κ (5.11)and k A ( v (4) ε , e ψ χγ ) k − ≤ C K ε e α − / − κ . (5.12) Fifth term. Finally, we estimate Λ∆ G ( v (3) ε ) − A ( e ψ χγ , e ψ χγ ). By definitionof τ Kε , we have k e ψ χγ k α ≤ C K ε − χ ( α − / − κ before time τ K . Since k v (4) ε k α ≤ C K ε − γ ( α − / − κ as observed before, we thus have k u (4) ε k α ≤ C K ε − χ ( α − / − κ . Furthermore, since α > , there exists a constant C > k Λ∆ G ( u (4) ε ) − A ( e ψ χγ , e ψ χγ ) k − α = k tr( D G ( u (4) ε )(Λ I − Ξ ε ( e ψ χγ ))) k − α ≤ C k D G ( u (4) ε ) k α k Λ I − Ξ ε ( e ψ χγ ) k − α . Since the stopping time τ K enforces that k Λ I − Ξ ε ( e ψ χγ ) k − α ≤ C K ε / , weinfer that k Λ∆ G ( u (4) ε ) − A ( e ψ χγ , e ψ χγ ) k − α ≤ C K ε / − χ ( α − / − κ . M. HAIRER AND J. MAAS Since u (4) ε − v (3) ε = ̺ ε + e ψ χγ , we have by definition of τ K , k ∆ G ( u ε ) − ∆ G ( v (3) ε ) k − α ≤ k ∆ G ( u ε ) − ∆ G ( v (3) ε ) k L ∞ ≤ C K k ̺ ε k L ∞ + C K k e ψ χγ k L ∞ ≤ C K k ̺ ε k α + C K ε γ/ − κ . Putting these bounds together, we obtain k Λ∆ G ( v (3) ε ) − A ( e ψ χγ , e ψ χγ ) k − α (5.13) ≤ C K ( ε / − χ ( α − / − κ + ε γ/ − κ + k ̺ ε k α ) . Sixth term. To estimate B , we use the fact that k u (4) ε ( t ) k L ∞ ≤ C K for t ≤ τ K , so that one has the bound k B k L ≤ C K Z Z R ε y | b D εy u (4) ε ( x ) | | µ | ( dy ) dx. We will split this expression into two parts, using the fact that u (4) ε = e ψ χγ + v (4) ε . First, using the fact that Θ( e ψ χγ ) ≤ C K ε − κ before time τ K by definitionof the stopping time τ K , we find that Z Z R ε y | b D εy e ψ χγ ( x ) | | µ | ( dy ) dx ≤ k e ψ χγ k L ∞ Z Z R εy | b D εy e ψ χγ ( x ) | | µ | ( dy ) dx = 2 ε k e ψ χγ k L ∞ Θ( e ψ χγ ) ≤ C K ε − κ k e ψ χγ k L ∞ ≤ C K ε γ/ − κ . Second, using the fact that H / ⊆ L , Lemma 3.5 and the fact that k v (4) ε k α ≤ C K ε − γ ( α − / − κ , we obtain Z Z R ε y | b D εy v (4) ε ( x ) | | µ | ( dy ) dx ≤ Cε Z R | y | k b D εy v (4) ε k / | µ | ( dy ) ≤ Cε α − / Z R | y | α − / | µ | ( dy ) k v (4) ε k α ≤ C K ε e α − / − κ . It thus follows that k B k L ≤ C K ε γ/ − κ + C K ε e α − / − κ . (5.14) SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Combining inequalities (5.9)–(5.13), we find thatsup r ∈ ( s,t ) k σ ε ( r ) k − ≤ C K (cid:16) ε γ/ − κ + ε e α − / − κ + ε / − χ ( α − / − κ + sup r ∈ ( s,t ) k ̺ ε ( r ) k α (cid:17) , and the result now follows as in Proposition 5.2. (cid:3) From v (4) ε to e v γ . Define τ K := τ K ∧ inf { t ≤ T : k e v γ ( t ) − v (4) ε ( t ) k α ≥ K } . Proposition 5.7. For κ > , we have lim ε → P (cid:16) sup t ≤ τ K k e v γ ( t ) − v (4) ε ( t ) k α > ε / χ − / − κ (cid:17) = 0 . Proof. Let 0 ≤ s ≤ t ≤ τ ∗ . It follows from (2.4) and (2.10d) that ̺ ε := e v γ − v (4) ε satisfies ̺ ε ( t ) = Z ts S ε ( t − r ) σ ε ( r ) dr, where σ ε := ∇ G ( e v γ + e ψ γ ) D ε ( e v γ + e ψ γ ) − ∇ G ( e v γ + e ψ χγ − ̺ ε ) D ε ( e v γ + e ψ χγ − ̺ ε )+ F ( e v γ + e ψ γ ) − F ( e v γ + e ψ χγ − ̺ ε ) . In order to estimate σ ε , we use (5.8) to write σ ε = D ε G ( e v γ + e ψ γ ) − D ε G ( e v γ + e ψ χγ − ̺ ε ) − Z R εy D G ( u ε ) − D G ( u (4) ε ))[ b D εy u ε , b D εy u ε ] µ ( dy ) − Z R εy D G ( u (4) ε )[ b D εy ( u ε + u (4) ε ) , b D εy ( u ε − u (4) ε )] µ ( dy ) − ε ( R ε ( u ε , u ε ) − R ε ( u (4) ε , u (4) ε ))+ F ( e v γ + e ψ γ ) − F ( e v γ + e ψ χγ − ̺ ε )=: σ ε, + · · · + σ ε, , where u ε := e v γ + e ψ γ , u (4) ε := v (4) ε + e ψ χγ and R ε ( u , u )( x ) := Z R ε y Z Z t Z s D G ((1 − r ) u ( x ) + ru ( x + εy )) × [ b D εy u , b D εy u , b D εy u ] dr ds dt dµ ( y ) . We shall now estimate σ ε, , . . . , σ ε, individually. M. HAIRER AND J. MAAS First term. First, we observe that e v γ , e ψ γ , e ψ χγ and ̺ ε are bounded in L ∞ before time τ K . Using Lemma 3.5, the embedding H α ⊆ L ∞ , and the defi-nition of the stopping time to bound k e ψ χ k L ∞ , we obtain k σ ε, k − = k D ε ( G ( e v γ + e ψ γ ) − G ( e v γ + e ψ χγ − ̺ ε )) k − ≤ C k G ( e v γ + e ψ γ ) − G ( e v γ + e ψ χγ − ̺ ε ) k L ∞ (5.15) ≤ C K ( k e ψ χ k L ∞ + k ̺ ε k L ∞ ) ≤ C K ( ε χ/ − κ + k ̺ ε k α ) . Second term. Using Lemma 3.5 and the fact that ε γ ( α − / κ k e v γ k α isbounded before time τ K , we estimateΘ ε ( e v γ ) = Z R y k b D εy e v γ k L | µ | ( dy ) ≤ C K Z R y | εy | α − k e v γ k α | µ | ( dy ) ≤ C K Z R y | εy | α − ε − γ ( α − / − κ | µ | ( dy ) ≤ C K ε e α − − κ , and by the definition of the stopping time τ K ,Θ ε ( e ψ γ ) ≤ C K ε − − κ . As a consequence,Θ ε ( u ε ) ≤ ε ( e v γ ) + Θ ε ( e ψ γ )) ≤ C K ( ε e α − − κ + ε − − κ ) ≤ ε − − κ . (5.16)Note that k u ε k L ∞ and k u (4) ε k L ∞ are bounded before time τ K . Using that L ⊆ H − , we obtain k σ ε, k − ≤ k σ ε, k L ≤ ε k D G ( u ε ) − D G ( u (4) ε ) k L ∞ Θ ε ( u ε ) ≤ C K ε − κ k u ε − u (4) ε k L ∞ (5.17) ≤ C K ε − κ ( k ̺ ε k L ∞ + k e ψ χ k L ∞ ) ≤ C K ε − κ ( k ̺ ε k α + ε χ/ − κ ) . Third term. By Lemma 3.5, we haveΘ ε ( ̺ ε ) = Z R y k b D εy ̺ ε k L | µ | ( dy )(5.18) ≤ C Z R y | εy | α − k ̺ ε k α | µ | ( dy ) ≤ Cε α − k ̺ ε k α . SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION Observe that u ε + u (4) ε = 2 e v γ − ̺ ε + e ψ γ + e ψ χγ and u ε − u (4) ε = e ψ χ + ̺ ε . Takinginto account that ε κ Θ ε ( e ψ γ ) , ε κ Θ ε ( e ψ γ ) , ε − χ + κ Θ ε ( e ψ χ ) , k ̺ ε k α are all bounded before time τ K , we obtainΘ ε ( u ε + u (4) ε ) ≤ C (Θ ε ( e v γ ) + Θ ε ( ̺ ε ) + Θ ε ( e ψ γ ) + Θ ε ( e ψ χγ ))(5.19) ≤ C K ( ε e α − − κ + ε α − + ε − − κ ) ≤ C K ε − − κ and Θ ε ( u ε − u (4) ε ) ≤ C (Θ ε ( e ψ χ ) + Θ ε ( ̺ ε ))(5.20) ≤ C K ε χ − − κ + Cε α − k ̺ ε k α . Using that k u (4) ε k L ∞ ≤ C K before time τ K , we obtain k σ ε, k − ≤ k σ ε, k L ≤ ε k D G ( u (4) ε ) k L ∞ q Θ ε ( u ε + u (4) ε )Θ ε ( u ε − u (4) ε ) ≤ C K ( ε χ/ − / − κ + ε α − / − κ k ̺ ε k α ) . Fourth term. We shall show that k σ ε, k − ≤ C K ( ε − κ k ̺ ε k α + ε χ/ − / − κ ) . (5.21)First, we use the L ∞ -bound on u (4) ε enforced by the stopping time, toobtain the pointwise bound | R ε ( u ε , u ε ) − R ε ( u ε , u (4) ε ) |≤ C K Z R ε y ( | b D εy u ε | + | b D εy u ε || b D εy u (4) ε | + | b D εy u (4) ε | ) × | b D εy ( u ε − u (4) ε ) || µ | ( dy ) ≤ C K Z R εy ( | b D εy u ε | + | b D εy u (4) ε | ) | b D εy ( u ε − u (4) ε ) || µ | ( dy ) . In view of (5.20) it thus follows that k R ε ( u ε , u ε ) − R ε ( u ε , u (4) ε ) k L ≤ C K ε Z R y k ( | b D εy u ε | + | b D εy u (4) ε | ) | b D εy ( u ε − u (4) ε ) |k L | µ | ( dy ) ≤ C K ε Z R y ( k b D εy u ε k L + k b D εy u (4) ε k L ) k b D εy ( u ε − u (4) ε ) k L | µ | ( dy ) ≤ C K ε q (Θ ε ( u ε ) + Θ ε ( u (4) ε ))Θ ε ( u ε − u (4) ε ) . M. HAIRER AND J. MAAS Using (5.16), (5.18), and the definition of the stopping time to bound Θ ε ( e ψ χ ),we find thatΘ ε ( u ε ) + Θ ε ( u (4) ε ) ≤ C (Θ ε ( u ε ) + Θ ε ( ̺ ε ) + Θ ε ( e ψ χ )) ≤ C K ( ε − − κ + ε α − + ε χ − − κ ) ≤ ε − − κ . Using (5.20), we thus obtain k R ε ( u ε , u ε ) − R ε ( u ε , u (4) ε ) k L ≤ C K ( ε χ/ − / − κ + ε α − / − κ k ̺ ε k α ) . (5.22)Furthermore, taking into account that k u ε − u (4) ε k L ∞ ≤ C K ( k e ψ χ k L ∞ + k ̺ ε k L ∞ ) ≤ C K ( ε χ/ − κ + k ̺ ε k α ) , we have by (5.16), k R ε ( u ε , u (4) ε ) − R ε ( u (4) ε , u (4) ε ) k L ≤ C K ε k u ε − u (4) ε k L ∞ Z R y k| b D εy u (4) ε | k L | µ | ( dy ) ≤ C K ε k u ε − u (4) ε k L ∞ Z R y k b D εy u (4) ε k L | µ | ( dy )(5.23) = C K ε k u ε − u (4) ε k L ∞ Θ ε ( u (4) ε ) ≤ C K ( ε χ/ − κ + ε − κ k ̺ ε k α ) . The claim follows by adding (5.22) and (5.23) and using the embedding L ⊆ H − . Fifth term. As in the first step, we have k σ ε, k − = k F ( e v γ + e ψ γ ) − F ( e v γ + e ψ χγ − ̺ ε ) k − ≤ C K ( k e ψ χ k L ∞ + k ̺ ε k L ∞ )(5.24) ≤ C K ( ε χ/ − κ + k ̺ ε k α ) . Combining the five estimates, we obtain k ̺ ε ( t ) k α ≤ C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) k σ ε ( r ) k − ≤ C ( t − s ) (1 − α ) / sup r ∈ ( s,t ) ( ε − κ k ̺ ε ( r ) k α + ε χ/ − / − κ ) . The result now follows as in the proof of Proposition 5.1. (cid:3) Acknowledgments. We are grateful to Hendrik Weber, Jochen Voß andAndrew Stuart for numerous discussions on this and related problems. Wethank the anonymous referee for useful comments. Part of this work hasbeen carried out while J. Maas was visiting the Courant Institute and the SPATIAL VERSION OF THE IT ˆO–STRATONOVICH CORRECTION University of Warwick. He thanks both institutions for their kind hospitalityand support. REFERENCES [1] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces , 2nd ed. Pure andApplied Mathematics (Amsterdam) . Elsevier, Amsterdam. MR2424078[2] Bertini, L. , Cancrini, N. and Jona-Lasinio, G. (1994). The stochastic Burgersequation. Comm. Math. Phys. Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations fromparticle systems. Comm. Math. Phys. Brze´zniak, Z. , Capi´nski, M. and Flandoli, F. (1991). Stochastic partial differ-ential equations and turbulence. Math. Models Methods Appl. Sci. Courant, R. , Isaacson, E. and Rees, M. (1952). 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