ANOTHER LOOK INTO THE WONG ZAKAI THEOREM FOR STOCHASTIC HEAT EQUATION

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1 ANOTHER LOOK INTO THE WONG ZAKAI THEOREM FOR STOCHASTIC HEAT EQUATION YU GU, LI-CHENG TSAI Abstract. Consider the heat equation driven by a smooth, Gaussian random potential: tu ε = uε + uεξε cε, t >, x R, 2 where ξ ε converges to a spacetime white noise, and c ε is a diverging constant chosen properly. For any n, we prove that u ε converges in L n to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux HP5, for the special case of the stochastic heat equation.. Introduction and main result The study of stochastic PDEs has witnessed significant progress in recent years. Several theories have been developed to make sense of singular equations with multiplication of distributions, see Hai3, Hai4, GIP5, Kup6, OW6 and the references therein. One example is the Wong Zakai theorem for stochastic PDEs HL5, HP5, CS7, which is an infinite dimensional analogue of WZ65a, WZ65b, SV72. In this article, we revisit this problem for a special case: the Stochastic Heat Equation SHE in one space dimension: t U = 2 xxu + Uξ, t >, x R,. where ξ is a spacetime white noise, built on an underlying probability space Ω, F, P. The SHE. has played an important role in the study of directed polymers and random growth phenomena. On the one hand, the solution of. gives the partition function of a directed polymer in a noisy environment. On the other hand, via the inverse Hopf Cole transform, the equation. yields the celebrated Kardar Parisi Zhang equation KPZ86, which describes the height function of a certain type of random growth phenomenon. Even though the physical phenomena described by SHE goes to higher dimensions, we focus on one dimension here. In fact, d = 2 and d > 2 corresponds to the so-called critical and supercritical cases, and sit beyond any existing theory. There, however, has been works on the cases in d 2 where the noise is tuned down to zero suitably with a scaling parameter. See Fen2, Fen6, CSZ7, GRZ7, MU7. Throughout this article, we fix a bounded continuous initial condition u x C b R, and a mollifier φ Cc R 2 ; R + with a unit total mass φ dtdx =. Using this mollifier, we construct the mollified noise as ξ ε t, x = φ ε t s, x yξs, ydyds, φ ε t, x = ε φ t 3 ε, x 2 ε..2 R 2 Given the smooth function ξ ε, consider the equation t u ε = 2 xxu ε + u ε ξ ε c ε, t >, x R..3 For the analogous equation where ξ ε is white-in-time, regularized in space, and interpreted in the Itô s sense, Bertini and Cancrini BC95 showed that, for c ε =, the solution u ε converges to the solution of the SHE. When the noise is regularized in both space and time, a non-zero, divergent constant c ε arises. Our main result states that, for a suitable and explicit choice of c ε, which depends explicitly on φ, the solution u ε of.3 converges pointwisely in L n Ω to the solution of SHE., for any n. It is a classical result that. admits a unique weak and mild solution starting from U, x = u x. Also, for fixed ε > and for almost every realization of ξ ε, it is standard by Feynman Kac formula to show that the PDE.3 admits a unique classical solution. Theorem.. Let c ε = c ε + 2 σ2 with c, σ given by 2.7 and 2.. Let u ε and U denote the respective solutions of.3 and., both with initial condition u x. Then, for any t, x R + R and n, the

2 2 YU GU, LI-CHENG TSAI random variable u ε t, x converges in L n Ω to Ut, x, i.e., E u ε t, x Ut, x n, as ε..4 As mentioned earlier, analogs of Theorem. have already been established in different settings. Hairer and Pardoux HP5 established the Wong Zakai theorem for a general class of semi-linear equations on the torus, with the SHE being a special case. This result was later extended to non-gaussian noise by Chandra and Shen CS7, and the problem on the whole line R was studied by Hairer and Labbé HL5. In a related direction, Bailleul, Bernicot, and Frey BBF7 have studied similar stochastic PDEs via paracontrolled calculus. All the aforementioned works build on the recently developed theory of regularity structure and paracontrolled calculus Hai4, GIP5. In this article, we present a more probabilistic proof of Theorem.. The proof is short, entirely contained within the scope of classical stochastic analysis. This offers a different perspective of the Wong Zakai theorem for the SHE. For example, the renormalizing constant c ε is identified in terms of the first and second moments of certain additive functionals of Brownian motions. See the discussion in Section 2. Our proof also shows that the solution of t u ε = 2 xxu ε + εu ε ξ ε converges to t ū = 2 xxū + ūc, which can be viewed as a homogenization type result the large, highly-oscillatory random potential εξ ε is being replaced by the constant c in the limit. For a general class of mixing random potentials, the same result was proved by Pardoux and Piatnitski PP2. Outline and conventions. In Section 2, we use the Feynman Kac formula to analyze moments of u ε t, x. These formula are expressed in terms of functionals of some auxiliary Brownian motions. We then establish various properties of these functionals. In Section 3, we prove Theorem. using the results established in Section 2. This is done by first showing that u ε t, x is a Cauchy sequence in L 2 Ω, and then identifying the limit via a Wiener chaos expansion. Throughout the paper, we denote the Fourier transform of f by fξ = fxe iξx dx. R We use Ca,... to denote a generic, deterministic, finite constant that may change from line to line, but depends only on the designated variables a,.... This is not to be confused with the renormalization constant c ε. Also, we use r ε = r ε t t to denote a generic random process, that uniformly converges to zero, i.e., sup r ε t h ε, for some deterministic h ε..5 t R + Acknowledgments. YG was partially supported by the NSF through DMS-633. LCT was partially supported by a Junior Fellow award from the Simons Foundation, and by the NSF through DMS Feynman Kac Formula and Brownian Functionals A main tool in this article is the Feynman Kac formula, which expresses the solution of the PDE.3 as u ε t, x = E B u t x + Bt exp ξ ε t s, x + Bsds c ε t. 2. Here Bt is a standard Brownian motion starting from the origin, independent of the driving noise ξ. In fact, we will be considering several independent Brownian motions. We expand the probability space Ω, F, P to a larger one Ω Σ, F F B, P P B to include several independent Brownian motions B, B, B 2,..., W, W, W 2,..., independent of ξ. We will use E B to denote the expectation on Σ. Also, we will often work with the marginal probability space Ω, F, P or Σ, F B, P B. Several functionals of the Brownian motions enter our analysis via 2.. More precisely, the n-th moment of u ε t, x is expressed in terms of functionals of Brownian motions. We begin with the first moment. To this end, define the covariance function Rt, x := φt s, x yφ s, ydsdy, R ε t, x := ε R t 3 ε, x 2 ε = Eξ εt, xξ ε,. R 3

3 THE WONG ZAKAI THEOREM FOR SHE 3 Recall that φ is compactly supported. Without loss of generality, throughout this article we assume that φ is supported in 2, 2 in t, i.e., φt, =, t 2, and hence Rt, =, t. With ξ ε t, x being a Gaussian process, averaging over ξ in 2. gives Eu ε t, x = E B u t 2 x + Bt exp 2 E ξ ε t s, x + Bsds c ε t = E B u t s 2.2 x + Bt exp R ε s u, Bs Budsdu c ε t. Before progressing to the formula for higher moments, let us use 2.2 to explain how the renormalizing constant c ε comes into play. To this end, take u x = for simplicity. In this case the solution U of the limiting SHE. satisfies E Ut, x =. For the convergence in.4 to hold, we must choose c ε so that t s E B exp R ε s u, Bs Budsdu c ε t, as ε. To this end, consider the centered double-integral process X ε t := t s R ε s u, Bs Bududs t s It is more convenient to express X ε in microscopic coordinates. ε Bε 2 law t t = Btt to write X ε t law = X mi t := ε ε = ε ε 2 t s ε 2 t Rs u, Bs Bududs ε E B R ε s u, Bs Bududs. 2.3 ε 2 t s That is, we use the scaling property E B Rs u, Bs Bududs s Ru, Bs Bs u EB Ru, Bs Bs u du ds. 2.4 Since Ru, x = whenever u, the u-integral in 2.3 goes over u, for all s. Dropping those values of s < in the integral gives Xε mi t = ε ε 2 t where, recall that, r ε t denotes a generic process satisfying.5, X s := X sds + r ε t, 2.5 Ru, Bs Bs udu c, 2.6 c := E B Ru, Budu = E B Ru, Bs Bs udu, 2.7 and we take B to be a two-sided Brownian motion in so that the resulting expression is defined for all s including s,. It is straightforward to verify that {X s} s is stationary in s, bounded, E B X s =, 2.8 with X s s s, X s s s being independent whenever s s. 2.9 Thus it is natural to expect X ε t to converges to σ W t, with W being a standard Brownian motion, and σ 2 := E B X sx ds. 2. In light of these discussions, we find that c ε := c ε + 2 σ2 as in Theorem. is the reasonable choice in order for u ε to converge to the solution U to the SHE. With X ε t and σ defined in the preceding, we rewrite the formula 2.2 in a more compact form as Eu ε t, x = E B u x + Bt exp X ε t 2 σ2 t + r ε t. 2. Similar calculations give formulas of higher moments: n n Eu ε t, x n = E B u x + B j t exp X j,ε t 2 σ2 t + r ε t + i<j n Y i,j,ε t. 2.2

4 4 YU GU, LI-CHENG TSAI Here B,..., B n are independent Brownian motions; the process X j,ε t is obtained by replacing B with B j in 2.3; and Y i,j,ε t is given by Y i,j,ε t := t t R ε s u, B i s B j udsdu. 2.3 law law Indeed, X j,ε t t = Xε t t. Likewise, writing Y ε t := Y,2,ε t, we have Y i,j,ε t t = Yε t t, for all i < j. We will also need to consider Eu ε t, xu ε2 t, x, i.e., the second moment calculated at different values of ε. A similar calculation gives the formula 2 2 Eu ε t, xu ε2 t, x = E B u x + B j t exp Y ε,ε 2 t + X j,εj t 2 σ2 t + r εj t, 2.4 where t t Y ε,ε 2 t := R ε,ε 2 s u, B s B 2 udsdu, 2.5 R ε,ε 2 t, x := φ ε t s, x yφ ε2 s, ydsdy = Eξ ε t, xξ ε2,. R Exponential moments. We first establish bounds on exponential moments of X ε t and Y ε,ε 2 t. Proposition 2.. For any λ, t >, we have sup ε, E B e λxεt + sup ε,ε 2, E B e λyε,ε 2 t <. Proof. For X ε t, we appeal to the microscopic coordinates, using to write X ε t law = X mi t = ε ε ε 2 t X sds + r ε t = ε ε 2 t In view of the finite range property 2.9 of X, we decompose Xε mi t = ε Xk + ε Xk, k I odd k I even X sds + r ε t. where X k := k+ X sds, and I k even := { k ε 2 t, even}, and I odd := { k ε 2 t, odd}. This gives E B e λxεt = E B e λrεt exp λε Xk exp λε Xk k I even Cλ, t E B exp 2λε k I even Xk k I odd E B exp 2λε k I odd Xk = Cλ, t k t/ε 2 E B e 2λε X k. By 2.8, we know that X k is stationary with zero mean, so E B e 2λε X k = E B e 2λε X t/ε 2 Cλ, t. k t/ε 2 From this we conclude the desired exponential moment bound on X ε t: sup E B e λxεt Cλ, t <. ε, We now turn to Y ε,ε 2 t. Denoting by R ε,ε 2 the Fourier transform of R ε,ε 2 in the x-variable, we express Y ε,ε 2 t via Fourier transform as Y ε,ε 2 t = 2π Rε,ε 2 s u, ξe iξbs B2u dξ dsdu.,t 2 R

5 THE WONG ZAKAI THEOREM FOR SHE 5 From this we calculate the n-th moment of Y ε,ε 2 t as n E B Y ε,ε 2 t n = 2π Rε,ε 2 s j u j, ξ j,t 2n R n n E B e iξjbsj n E B e iξjb2uj dsdudξ. Let us first focus on the integral over u, t n. We write n n R ε,ε 2 s j u j, ξ j E B e iξjb2uj du =,t n,t n R n 2.6 n n R ε,ε 2 s j u j, x j e iξjxj E B e iξjb2uj dudx. The exponents are purely imaginary. We hence bound those exponentials by in absolute value, and use,t R R ε,ε 2 s u, xdudx to get n n R ε,ε 2 u j s j, ξ j E B e iξjb2uj du.,t n Inserting this into 2.6 gives n E B Y ε,ε 2 t n 2π,t n E B e iξjbsj dsdξ. 2.7 n R n The last integral in 2.7 is in fact the n-th moment of Brownian localtime at the origin. More precisely, let Lt, x; B denote the localtime process of a Brownian motion B, it is a standard result that n 2π,t n E B e iξjbs dsdξ = E B Lt, ; B n. 2.8 n R n Informally speaking, this formula is obtained by interpreting Lt, ; B as t δb sds, where δ denotes the Dirac function, and taking Fourier transform, similarly to the preceding. The prescribed informal procedure is rigorously implemented by taking a sequence approximating the Dirac function. We omit the details here as the argument is standard. Now, combine , and sum over n. We arrive at E B e λyε,ε 2 t E B e λlt,;b. As the Brownian localtime has finite exponential moments, i.e., Ee λlt,;b < for any λ, t >, we obtain the desired exponential moment bound on Y ε,ε 2 t: This completes the proof. sup E B e λyε,ε 2 t Cλ, t <. ε,ε 2, 2.2. Weak convergence. In this section, we derive the distributional limit of X j,ε and Y ε,ε 2. First, since the covariance function R ε,ε 2 t, x converges to Dirac function δtδx, we expect the process Y ε,ε 2 t defined in 2.5 to converge to the mutual intersection localtime of B and B 2. More precisely, recalling that Lt, x; B denote the localtime process of a Brownian motion B, we define the mutual intersection localtime of B and B 2 as lt := Lt, ; B B Proposition 2.2. For any fixed t >, Y ε,ε 2 t lt in L 2 Σ, as ε, ε 2. Proof. Instead of directly proving the convergence of Y ε,ε 2 t, let us first consider a modified process Ỹε,ε 2 t where B and B 2 are evaluated at the same time: Ỹ ε,ε 2 t := t t R ε,ε 2 s u, B s B 2 sdsdu, and show that Ỹε,ε 2 t converges to lt in L 2 Σ. To this end, set F ε,ε 2 s := t R ε,ε 2 s u, B s B 2 sdu, s, t.

6 6 YU GU, LI-CHENG TSAI In the preceding double-integral expression of Ỹε,ε 2 t, divide the range of integration over s into subintervals depending on its distance from and to t. We rewrite the expression as t Ỹ ε,ε 2 t = F ε,ε 2 s ε 2 +ε 2 2,t ε2 ε2 2 s +,ε 2 +ε 2 2 s + t ε 2 ε 2 2,t s ds. Recall that φt, =, t 2. This gives R ε,ε 2 s u, = for all s u ε 2 + ε 2 2. Consequently, for s ε 2 + ε 2 2, t ε 2 ε 2 2 we have F ε,ε 2 s = R ε,ε 2 s u, B s B 2 sdu. R Further setting Φx := R φt, xdx, Φ εx := ε Φε x, and Φ ε,ε 2 x := Φ ε x yφ ε2 ydy, we rewrite the last expression as F ε,ε 2 s = Φ ε,ε 2 B s B 2 s. On the other hand, we also have F ε,ε 2 s Φ ε,ε 2 B s B 2 s, for all s, t. This takes into account those values of s ε 2 + ε 2 2, t ε 2 ε 2 2, thereby giving Ỹ ε,ε 2 t = t ε 2 ε2 2 ε 2 +ε2 2 where r ε,ε 2 t is a remainder term satisfying r ε,ε 2 t t Φ ε,ε 2 B s B 2 sds + r ε,ε 2 t, 2.2 Φ ε,ε 2 B s B 2 s,ε 2 +ε 2 2 s + t ε 2 ε 2 2,t s ds. Now, for any interval a, b,, by the definition of the localtime, b Φ ε,ε 2 B s B 2 sds = Φ ε,ε 2 x Lb, x; B B 2 La, x; B B 2 dx a R = ΦxΦ y Lb, ε x + ε 2 y; B B 2 La, ε x + ε 2 y; B B 2 dxdy. R For almost every realization of B B 2, the function x Lt, x; B B 2 is continuous and compactly supported, and the function t Lt, x; B B 2 is increasing and continuous. Thus, from and the fact that Φdx =, we conclude that Ỹε,ε 2 t lt = Lt, ; B B 2 almost surely as ε, ε 2. Further, the same calculations via Fourier transform as in the proof Proposition 2. yields that sup E B e λỹε,ε 2 t E B e λlt,;b <. ε,ε 2 This property leverages the preceding almost sure convergence into a convergence in L 2 Σ: E B Ỹε,ε 2 t lt 2 as ε, ε Given 2.22, it remains to show that Y ε,ε 2 t Ỹε,ε 2 t in L 2 Σ. We will actually prove the following result: for any choice of Z, Z 2 {Y ε,ε 2 t, Ỹε,ε 2 t}, as ε, ε 2, E B Z Z 2 2π 2 e iξbs B2s e iξ B s B 2s dξdξ dsds. 2.23,t 2 R 2 E B Once this is done, expanding EY ε,ε 2 t Ỹε,ε 2 t 2 into four terms, and passing to the limit complete the proof. The proof for all cases of Z, Z 2 {Y ε,ε 2 t, Ỹε,ε 2 t} is the same, and we take Z = Z 2 = Y ε,ε 2 t as an example. As in the proof of Proposition 2., we express Y ε,ε 2 t via Fourier transform as Y ε,ε 2 t = =,t 2,t 2 2π Rε,ε 2 s u, ξe iξbs B2u dξ R 2πε 2 ε 2 2 s u w φ ε 2 R 2 dsdu, ε ξ φ w, ε ε 2 2 ξe iξbs B2u dξdw dsdu, 2

7 THE WONG ZAKAI THEOREM FOR SHE 7 where φ denotes the Fourier transform of φ in the x-variable. Squaring the last expression and taking expectation gives E B Y ε,ε 2 t 2 = G ε,ε 2 s, s dsds,,t 2 where G ε,ε 2 s, s :=,t 2 2πε 2 ε s u w φ ε 2 R 4, ε ξ φ w ε 2 2, ε 2 ξ φ s u w ε 2 E B e iξbs B2u e iξ B s B 2u dξdξ dwdw dudu., ε ξ φ w, ε ε 2 2 ξ 2 Fix s, s, t 2. Recall that φt, = whenever t 2, and note that, with s, s, t 2 being fixed, the conditions t s, t s, s, s 2 ε2 + ε 2 2 holds for all small enough ε, ε 2. Consequently, in the last expression of G ε,ε 2 s, s, for all ε, ε 2 small enough, the integration domain of u, u can be and is replaced, t 2 R 2. A change of variables in this case yields G ε,ε 2 s, s = 2π 2 φu, ε ξ φ w, ε 2 ξ φu, ε ξ φ w, ε 2 ξ R 6 E B e iξbs B2s ε2 u ε2 2 w e iξ B s B 2s ε 2 u ε 2 2 w dξdξ dwdw dudu. By the dominated convergence theorem and the fact that φu, du = φdudx =, we obtain G ε,ε 2 s, s 2π R 2 E B e iξbs B2s e iξ B s B 2s dξdξ, pointwisely in, t To achieve 2.23, we need to upgrade the pointwise convergence of 2.24 to convergence in L, t 2. To this end, with φ, ξ φ,, we bound G ε,ε 2 s, s 2πε 2 ε s u w φ, φ w, φ s u w, φ w, ε 2 R 6 ε 2 2 ε 2 ε 2 2 E B e iξbs e iξ B s dξdξ dwdw dudu. After integrating in w, w, u, u on the RHS of the last integral, we have G ε,ε 2 s, s 2π R 2 E B e iξbs e iξ B s dξdξ C 2 s s s s L, t 2. Given this, the dominated convergence theorem upgrades 2.24 into a convergence in L, t 2. gives 2.23 and hence completes the proof. This We next turn to the distributional limit of X,ε and X 2,ε defined in 2.3. Hereafter, we use to denote the weak convergence of probability laws in a designated space, and endow the space C, with the topology of uniform convergence over compact subsets of,. Proposition 2.3. As ε, B, B 2, l, X,ε, X 2,ε = B, B 2, l, σ W, σ W 2 in C, where W, W 2 are standard Brownian motions independent of B, B 2, and σ, is given in 2.. Proof. The proof consists of two steps. Step : Instead of showing 2.25 directly, let us first establish B, B 2, X,ε, X 2,ε = B, B 2, σ W, σ W 2 in C, To this end, we appeal to microscopic coordinates. That is, with Xε,j mi B j, we have given in 2.4 with B replaced by B t, B 2 t, X,ε t, X 2,ε t t law = εb ε 2 t, εb 2 ε 2 t, X mi,εt, X mi 2,εt t. 2.27

8 8 YU GU, LI-CHENG TSAI We begin by writing Xj,ε mi in terms of stochastic integrals. Let D j,r denote the Malliavin derivative with respect to db j r on Σ, F B, P B, and let F j r denote the canonical filtration of B j. The Clark Ocone formula states that with Xj,ε mi t having zero mean X mi j,εt = ε A direct calculation yields so ε 2 t D j,r X mi j,εt = ε Z j,ε r, t = Z j,ε r, tdb j r, ε 2 t s ε 2 t r r Z j,ε r, tr, t := ε ED j,r X mi j,εt F j r. x Rs u, B j s B j ududs {u r<s}, E B x Rs u, B j s B j u F j r duds The function x Rs, x vanishes for all s because R does, so by defining r+ r Z j r := {r } E B x Rs u, B j s B j ududs F j r duds, 2.29 r r we have Z j,ε r, t = Z j r, for all r, ε 2 t. The latter is preferred for our purpose, because it does not depend on t. In particular, the analogous integrated process: t X mi j,εt := ε ε 2 t Z j rdb j r is a martingale unlike Xj,ε mit, which is not due to the t-dependence of Z j,εr, t. Also, for each t R +, the L 2 Σ-distance between vanishes as ε : E B X mi j,ε and Xmi j,ε ε 2 t mi X j,εt Xj,εt mi 2 = ε 2 Given this, let us focus on the modified process Now, consider the C,, R 4 -valued process mi X j,ε E B Z j,ε r, t Z j r 2 dr Cε instead of Xmi j,ε. M ε t = M j,ε t 4 := εb ε 2 t, εb 2 ε 2 mi mi t, X,εt, X 2,εt. It is a continuous martingale, with cross-variance, for i, j =, 2, 3, 4,, 4, 2, 3, ε M i,ε, M j,ε t = 2 ε 2 t Z i sds, for i, j =, 3, 2, 4, t, for i, j =,, 2, 2, ε 2 ε 2 t Z 2 i sds, for i, j = 3, 3, 4, 4. Further, straightforward calculations from the expression 2.28 gives, with x R being an odd function in x, E B Z i s = and E B Z i 2s = σ 2, for all s, where σ 2 := E B Z 2 i 2 2 = E B E B x Rs u, B j s B j ududs F j. 2.3 Calculating the conditional expectation in 2.29 gives Z j r = {r } r+ r r r Rs, r, u, B j r B j ududs, where Rs, r, u, x := R qs r, x y yrs u, ydy, and qt, x := 2πt e x2 2t denotes the standard heat kernel. From this expression, it is readily check that Z i s s is bounded, stationary, and has finite range correlation similar to 2.9. Consequently, Ergodic theorem applied to Z i s s and { Z 2 i s} s gives ε 2 t ε 2 Z i sds, ε 2 t ε 2 Z i 2 sds σ 2 t,

9 THE WONG ZAKAI THEOREM FOR SHE 9 almost surely as ε, for any fixed t R +. Given these properties, the martingale central limit theorem EK86, Thm.4, p339 yields that M B, B 2, σ W, σ W 2, in C, 4. This together with 2.27 and 2.3 gives B t, B 2 t, X,ε t, X 2,ε t = B t, B 2 t, σ W t, σ W 2 t, in fdd Given 2.32, it now suffices to establish the tightness of B, B 2, X,ε, X 2,ε in C, 4, and show that σ 2 = σ 2. The first step is to appeal to microscopic coordinates. Using we write X j,ε t law = X mi j,εt = ε ε 2 t X j sds + r ε t. Given the properties of X j, a classical functional central limit theorem, see, e.g., Bil99, pp 78 79, asserts that X j,ε = σ W j, in C, This in particular implies the tightness of B, B 2, X,ε, X 2,ε. Further, comparing , we see that σ 2 = σ 2 must holds. Alternatively, it is possibly to show σ 2 = σ 2 by calculations from the expressions 2. and 2.3. We thus conclude Step 2: Having established 2.26, our next goal is to extend the convergence result to include the localtime process l. First, Tanaka s formula gives 2lt = B t B 2 t t sgnb s B 2 sdb B 2 s. Had it been the case that the RHS were a continuous function of B B 2, the desired result 2.25 would follow immediately from We show in Lemma 2.4 that, in fact, the stochastic integral t sgnb s B 2 sdb B 2 s is well-approximated by a sequence of continuous functions of B B 2. That is, there exists a sequence {f n } n CC, ; C, such that sgnb s B 2 sdb B 2 s f n B B 2 in probability, as n C, Now, fix arbitrary bounded and continuous g : C, 3 R and h : C, 2 R, and consider test functions of the type g h C, 5 R. It is known that the linear span of functions of this type is dense in CC, 5 ; R. Hence proving 2.25 amounts to proving E B gb, B 2, lhx,ε, X 2,ε E B gb, B 2, lhσ W, σ W 2, as ε Set 2l n t := B t B 2 t f n Bt. Since f n is continuous, for each fixed n, from 2.26 we have E B gb, B 2, l n hx,ε, X 2,ε E B gb, B 2, l n hσ W, σ W 2 as ε. On the other hand, with g being bounded and continuous, by 2.34, we have E B gb, B 2, l n gb, B 2, l, as n. From these the desired result 2.35 follows. The proof is complete. Lemma 2.4. The claim 2.34 holds for a sequence {f n } n CC, ; C,. Proof. Set Bt := B t B 2 t and Ut := t sgnbsdbs to simplify the notation. We begin by constructing the continuous function f n. Set ζx := x{ x } + sgnx{ x > }, ζ n x := ζx/n /4, and define f n yt := t k= ζ n y k n k n, k+ n sdys := k= ζ n y k n y k+ n t y k n t.

10 YU GU, LI-CHENG TSAI Indeed, f n is continuous for each fixed n. Fix an arbitrary T >. Using Doob s L 2 -martingale inequality and Itô isometry, we calculate E B sup Ut f n Bt 2 T C E B ζ n B k 2 n k n, k+ t sgnbt dt n,t = C k= I k,n T k= E B ζ n B k n sgnbt 2 dt, 2.36 where I k,n T := k n, k+ n, T. Set V k,n := sup s, n Bs + k n B k n. On the interval t I k,nt, we have ζb k n = sgnbt whenever B k n > n 4 and V k,n < n 4. Hence E B sup Ut f n Bt 2 nt C P B Vk,n n 4 + PB B k n n T 4 n.,t k= law By the scaling property of Brownian motion and the reflection principle, we have that V k,n = 2/n Z and that B k n law = 2k/nZ, where Z is a standard Gaussian. This gives E B sup Ut f n Bt 2 nt C,T k= P B Z 2 2 n 4 + PB Z n /4 2k /2 T n, with n/4 := when k =. It is now readily verified that the last expression tends to as n. From 2k /2 this the desired result follows: sup,t Ut f n Bt in probability, as n, for each fixed T. 3. Proof of Theorem. Let us first establish the boundedness of moments of u ε t, x. Set λ n := nn+ 2. With the initial condition u being bounded, applying Hölder s inequality with exponents λ n,..., λ n in the formula 2.2, we have n E u ε t, x n C E B exp λn Xj,ε t 2 σ2 t + r ε t E B exp λn Y i,j,ε t /λn C i<j n E B exp λn X ε t n/λ n E B exp λn Y i,j,ε t nn /2λ n Using the exponential moment bounds from Proposition 2., we obtain sup ε, E u ε t, x n <. That is, moments of u ε t, x are bounded uniformly in ε. This reduces proving.4 for all n to proving.4 for just one n. We henceforward consider n = 2. Let us first identify the limit of Eu ε t, xu ε2 t, x. Recall from 2.9 that lt denotes the mutual intersection localtime of B, B 2, and that U denotes the solution of the SHE.. Proposition 3.. We have lim Eu ε t, xu ε2 t, x = E B u x + B tu x + B 2 t explt. ε,ε 2 Proof. The starting point of the proof is the formula 2.4: Eu ε t, xu ε2 t, x = E B 2 By virtue of Propositions , we have u x + B j t exp Y ε,ε 2 t + 2. X j,εj t 2 σ2 t + r εj t. 3. B t, B 2 t, Y ε,ε 2 t, X,ε t, X 2,ε2 t B t, B 2 t, lt, σ W t, σ W 2 t in distribution. The proof is complete by invoking Propositions 2.. Now, with Eu ε t, x u ε2 t, x 2 = Eu ε t, xu ε t, x 2Eu ε t, xu ε2 t, x + Eu ε2 t, xu ε2 t, x, Proposition 3. has an immediate corollary:

11 THE WONG ZAKAI THEOREM FOR SHE Corollary 3.2. The sequence {u ε t, x} ε, is Cauchy in L 2 Ω. Given this result, it suffices to identify the unique limit of u ε t, x in L 2 Ω. We achieve this by Wiener chaos expansion. Fix t, x R + R hereafter, and denote R< k := {s,..., s k R k : s <... < s k }. Given any f L 2 R< k R k, we consider the k-th order multiple stochastic integral n I k f := fs,..., s k, x,..., x k ξs i, y i ds i dy i. R k < Rk Let qs, y = 2πs e y2 /2s denotes the standard heat kernel, the solution U of the SHE permits the chaos expansion we omit the dependence on t, x Ut, x = I k f k, k= f k := {<s<...<s k <t} R u y i= k qs i+ s i, y i+ y i dy, 3.2 under the convention y k+ := x, s :=, and s k+ := t. For u ε, a similar expansion also exists: the Stroock formula Str87, Eqn 7, p3 states that u ε t, x = I k f k,ε, f k,ε := ED k u ε t, x. 3.3 k= Here D denotes the Malliavin derivative with respect to ξ on Ω, F, P. To calculate the chaos coefficient f k,ε, we set Ψ ε,b r, y := t i= φ ε t s r, x + Bs yds, 3.4 and rewrite the Feynman Kac formula 2. as u ε t, x = E B u x + Bt exp Ψ ε,b r, yξr, ydydr c ε t. R 2 From this expression we calculate f k,ε r,..., r k, y,..., y k = E B u t x + Bt exp s k R ε s u, Bs Bududs c ε t Ψ ε,b r i, y i. Denoting the L 2 Ω limit of u ε t, x by U t, x, and the chaos expansion of U t, x is written as U t, x = I k f k. 3.6 k= The following lemma completes the proof of Theorem.. Lemma 3.3. U t, x = Ut, x in L 2 Ω. Proof. Given the chaos expansions in 3.2 and 3.6, it suffices to show f k = f k. Since u ε t, x U t, x in L 2 Ω, using the orthogonality of the chaos, i.e., E u ε U 2 = f k,ε f k 2 dsdy, R k < Rk k= we see that, for each k, f k,ε f k in L 2 R k < R k. Fix arbitrary g Cc R 2n, we consider f k,ε, g := f k,ε r,..., r k, y,..., y k gr,..., r k, y,..., y k drdy. 3.7 R k < Rk To prove f k = f k, it suffices to show f k,ε, g f k, g as ε. i= 3.5

12 2 YU GU, LI-CHENG TSAI The formula 3.5 yields f k,ε, g = u x + Bte t s Rεs u,bs Bududs cεt R k < Rk E B k,t k i= φ ε t s i r i, x + Bs i y i ds gr,..., r k, y,..., y k drdy. With φ ε, := ε 3 φε 2, ε, we perform a change of variables r i ε 2 r i + t s i, y i εy i + x + Bs i to rewrite the last expression as k f k,ε, g = Aε A r, s φ r i, y ie B u x + Bte t s Rεs u,bs Bududs cεt R 3k i= g ε 2 r + t s,..., ε 2 r k + t s k, εy + x + Bs,..., εy k + x + Bs k dr dy ds, where A ε := {ε 2 r + t s <... < ε 2 r k + t s k} and A := {s,..., s k, t k } translate the constraints on the old variables into the new ones. In order to pass to the limit, we note that, by Proposition 2.3, for any fixed s,..., s k, t k, t s R ε s u, Bs Bududs c ε t, Bs,..., Bs k, Bt σ W t 2 σ2 t, Bs,..., Bs k, Bt. In addition, for any fixed r,..., r k Rk, Aε A r, s {<sk <...<s <t}. By applying Proposition 2. and the dominated convergence theorem, we arrive at k f k,ε, g φ r i, y i {<sk <...<s <t} R 3k i= E B u x + Bte σ Wt 2 σ2 t gt s,..., t s k, x + Bs,..., x + Bs k dr dy ds. In the last expression, integrate over r i, y i using φdrdy =, and perform a change of variables t s i s i. We see that it equals u x + Btgs,..., s k, x + Bt s,..., x + Bt s k ds = f k, g. R k <,tk E B The proof is complete. References BBF7 I. Bailleul, F. Bernicot, and D. Frey. Spacetime paraproducts for paracontrolled calculus, 3d PAM and multiplicative Burgers equations. In Annales Scientifiques de l École Normale Supérieure, 27. BC95 L. Bertini and N. Cancrini. The stochastic heat equation: Feynman Kac formula and intermittence. J Stat Phys, 785:377 4, 995. Bil99 P. Billingsley. Convergence of probability measures. John Wiley & Sons, 999. CS7 A. Chandra and H. Shen. Moment bounds for spdes with non-gaussian fields and application to the Wong-Zakai problem. Electronic Journal of Probability, 22, 27. CSZ7 F. Caravenna, R. Sun, and N. Zygouras. Universality in marginally relevant disordered systems. Ann Appl Probab, 275:35 32, 27. EK86 S. N. Ethier and T. G. Kurtz. Markov processes: characterization and convergence, volume 282. John Wiley & Sons, 986. Fen2 Z. S. Feng. Diffusivity of rescaled random polymer in random environment in dimensions and 2. arxiv:2.625, 22. Fen6 Z. S. Feng. Rescaled Directed Random Polymer in Random Environment in Dimension + 2. PhD thesis, University of Toronto Canada, 26. GIP5 M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singular PDEs. In Forum of Mathematics, Pi, volume 3. Cambridge University Press, 25. GRZ7 Y. Gu, L. Ryzhik, and O. Zeitouni. The Edwards Wilkinson limit of the random heat equation in dimensions three and higher. arxiv:7.344, 27. Hai3 M. Hairer. Solving the KPZ equation. Ann of Math, 87: , 23. Hai4 M. Hairer. A theory of regularity structures. Invent Math, 982:269 54, 24. HL5 M. Hairer and C. Labbé. Multiplicative stochastic heat equations on the whole space. arxiv:54.762, to appear in J Eur Math Soc,

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