aa r X i v : . [ m a t h . P R ] A p r CHAOS EXPANSION OF 2D PARABOLIC ANDERSON MODEL YU GU JINGYU
HUANGAbstract.
We prove a chaos expansion for the 2D parabolic Anderson Modelin small time, with the expansion coefficients expressed in terms of the densityfunction of the annealed polymer in a white noise environment.Keywords: parabolic Anderson model, chaos expansion, renormalized self-intersection local time . AMS 2010 subject classification.
Primary 60H15, 60H07; Secondary 35R60. Introduction and main result
Consider the continuous parabolic Anderson model in d = 2 formally writtenas(1.1) ∂ t u ( t, x ) = 12 ∆ u ( t, x ) + u · ( ˙ W ( x ) − ∞ ) , t ≥ , x ∈ R , where ˙ W ( x ) is a spatial white noise defined on the probability space (Ω , F , P ) formally satisfying E [ ˙ W ( x ) ˙ W ( y )] = δ ( x − y ) . The equation (1.1) was analyzed in [7, 8, 9] by different approaches includingthe theory of regularity structures, para-controlled calculus, and the method ofcorrectors and two-scale expansions. The main results in these references showedthat a smoothed version of (1.1) converges to some limit that is independent ofthe mollification.More precisely, let ϕ : R → R + be a smooth and compactly supported func-tion on R satisfying ϕ ( x ) = ϕ ( − x ) and R ϕ = 1 . Define ϕ ǫ ( · ) = ǫ − ϕ ( · /ǫ ) and(1.2) ˙ W ǫ ( x ) = Z R ϕ ǫ ( x − y ) dW ( y ) as the mollification of ˙ W . The covariance function of ˙ W ǫ is(1.3) R ǫ ( x − y ) := E [ ˙ W ǫ ( x ) ˙ W ǫ ( y )] = ϕ ǫ ⋆ ϕ ǫ ( x − y ) . Let u ǫ be the solution to the equation with smooth coefficients(1.4) ∂ t u ǫ ( t, x ) = 12 ∆ u ǫ ( t, x ) + u ǫ ( ˙ W ǫ ( x ) − C ǫ ) , with the diverging constant(1.5) C ǫ = 1 π log ǫ − . Then u ǫ converges in some weighted Hölder space to a limit u that is defined tobe the solution to (1.1), see [9, Theorem 4.1]. GU JINGYU
HUANG
While the solution to (1.1) is well-defined, its statistical property remains achallenge. We refer to [1, 2, 5, 6, 14] for some relevant discussions. The goalof this note is to provide a Wiener chaos expansion of the solution u , in theshort time regime. We assume u ǫ (0 , x ) = u ( x ) for some bounded function u .Theorem 1.1 below shows that for small t , u ǫ ( t, x ) → u ( t, x ) in L (Ω) as ǫ → ,and u ( t, x ) is written explicitly as a Wiener chaos expansion in terms of theprobability density of a polymer in a white noise environment, see (1.17). Wehope that the explicit chaos expansion will provide another way of proving theconvergence to (1.1), e.g. from a discrete system using the general criteria provedin [3, 14]. The tool we use is a combination of the Feynman-Kac representationand Malliavin calculus. By writing u ǫ ( t, x ) in terms of a chaos expansion, itsuffices to pass to the limit in each chaos.1.1. Elements of Malliavin calculus.
We give a brief introduction to Malli-avin calculus and refer to [15] for more details. For any function φ ∈ L ( R ) , wedefine W ( φ ) = R φ dW . Let F be a smooth and cylindrical random variable ofthe form F = f ( W ( φ ) , . . . , W ( φ n )) , with φ i ∈ L ( R ) , f ∈ C ∞ p ( R n ) (namely f and all its partial derivatives havepolynomial growth), then the Malliavin derivative of F , denoted by DF , is the L ( R ) − valued random variable defined by(1.6) DF = n X j =1 ∂f∂x j ( W ( φ ) , . . . , W ( φ n )) φ j . For each positive integer k , D k F is defined to be the k -th iterated derivative of F , which is a random variable taking values in L ( R ) ⊗ k , the k -th tensor productof L ( R ) . The operator D k is closable from L (Ω) into L (Ω; L ( R ) ⊗ k ) and wedefine the Sobolev space D k, as the closure of the space of smooth and cylindricalrandom variables under the norm(1.7) k D k F k k, = vuut E " F + k X j =1 k D j F k L ( R ) ⊗ j . Define D ∞ , = T ∞ k =1 D k, , and L ( R ) ⊙ k as the k -th symmetric tensor product of L ( R ) .For any integer n ≥ , we denote by H n the n -th Wiener chaos of W . Werecall that H is simply R , and for n ≥ , H n is the closed linear subspace of L (Ω) generated by the random variables { H n ( W ( h )) : h ∈ L ( R ) , k h k L ( R ) = 1 } , where H n is the n -th order Hermite polynomials. For any n ≥ , the mapping I n ( h ⊗ n ) := H n ( W ( h )) can be extended to a linear isometry between L ( R ) ⊙ n and H n , with the iso-metric relation(1.8) E [ I n ( h ⊗ n ) ] = n ! k h ⊗ n k L ( R ) ⊗ n . HAOS EXPANSION OF 2D PARABOLIC ANDERSON MODEL 3
Consider now a random variable F ∈ L (Ω) , it can be written as(1.9) F = E [ F ] + ∞ X n =1 I n ( f n ) , where the series converges in L (Ω) , and the coefficients f n ∈ L ( R ) ⊙ n aredetermined by F . This identity is called the Wiener-chaos expansion of F .When the above F ∈ D ∞ , , the n − th coefficient f n in the Wiener chaos expan-sion of F can be explicitly written as [16, Page 3, equation (7)](1.10) f n = E [ D n F ] n ! . Brownian self-intersection local time and polymer in white noise.
The self-intersection local time of the planar Brownian motion is a classical sub-ject in probability theory [4, 12, 13, 17, 18]. In the following, we discuss itsconnections to the parabolic Anderson model.Using the Feynman-Kac formula, we write the solution to (1.4) as(1.11) u ǫ ( t, x ) = E B (cid:20) u ( x + B t ) exp (cid:18)Z t ˙ W ǫ ( x + B s ) ds − C ǫ t (cid:19)(cid:21) , where B is a standard Brownian motion starting from the origin which is inde-pendent from ˙ W , and E B denotes the expectation with respect to B . Takingexpectation with respect to ˙ W ǫ and using the fact that the exponent inside theexpectation in (1.11) is of Gaussian distribution for each realization of the Brow-nian motion, we obtain E [ u ǫ ( t, x )] = E B (cid:20) u ( x + B t ) exp (cid:18)Z t Z s R ǫ ( B s − B u ) duds − C ǫ t (cid:19)(cid:21) , where we recall that R ǫ is the covariance function of ˙ W ǫ . It is well-known that(1.12) γ ǫ ( t, B ) := Z t Z s R ǫ ( B s − B u ) duds − Z t Z s E B [ R ǫ ( B s − B u )] duds → γ ( t, B ) almost surely, and γ ( t, B ) is the so-called renormalized self-intersection local timeof the planar Brownian motion formally written as(1.13) γ ( t, B ) = Z t Z s δ ( B s − B u ) duds − Z t Z s E B [ δ ( B s − B u )] duds. In addition, there exists some critical t c > such that(1.14) E B [exp( γ ( t, B ))] (cid:26) < ∞ t < t c , = ∞ t > t c . The renormalization constant in (1.5) matches the expectation in (1.12) up to an O (1) correction, and a calculation as in [6, Lemma 1.1] shows that there existsconstants µ , µ such that(1.15) Z t Z s E B [ R ǫ ( B s − B u )] duds − C ǫ t → t ( µ + µ log t ) YU GU JINGYU
HUANG as ǫ → . For small t , it was shown in [6] that(1.16) E [ u ( t, x )] = lim ǫ → E [ u ǫ ( t, x )] = e t ( µ + µ log t ) E B [ u ( x + B t ) e γ ( t,B ) ] . This motivates us to define F ( t ) := log E B [ e γ ( t,B ) ] , so we can write E [ u ( t, x )] = e t ( µ + µ log t )+ F ( t ) ˆ E t,B [ u ( x + B t )] , where ˆ E t,B denotes the expectation with respect to the Wiener measure tilted bythe factor e γ ( t,B ) , i.e., ˆ E t,B [ X ] = E B [ Xe γ ( t,B ) ] E B [ e γ ( t,B ) ] = E B [ Xe γ ( t,B ) ] e − F ( t ) for any bounded X . By the formal expression in (1.13), we can view ˆ E t,B as theexpectation with respect to the annealed measure of a polymer in a white noiseenvironment. By (1.14), it is clear that the measure is absolutely continuouswith respect to the Wiener measure for small t . Applying the Radon-Nikodymtheorem, for any n ∈ Z + and < s < . . . < s n ≤ t < t c , there exists anon-negative measurable function, denoted by F s ,...,s n : R n → R , such that ˆ E t,B [1 A ( B s , . . . , B s n )] = Z A F s ,...,s n ( x , . . . , x n ) dx for all A ⊂ R n . In other words, F s ,...,s n is the joint spatial density function ofthe polymer path at s < . . . < s n . We note that F actually depends on t sincethe tilted measure depends on t . For our purpose, we use the simplified notationsince t is fixed. It is an elementary exercise to show that F s ,...,s n ( x , . . . , x n ) isjointly measurable in ( s , . . . , s n , x , . . . , x n ) . For the convenience of the reader,we present a proof in the appendix.Denote [0 , t ] n< := { ≤ s < . . . < s n ≤ t } , the following is our main result. Theorem 1.1.
There exists t > such that for each t ∈ (0 , t ) , x ∈ R , therandom variable u ǫ ( t, x ) converges in L (Ω) to (1.17) u ( t, x ) = ∞ X n =0 I n ( f n ( · ; t, x )) . The coefficient f n ( · ; t, x ) is given by (1.18) f n ( y , . . . , y n ; t, x )= e t ( µ + µ log t )+ F ( t ) Z R Z [0 ,t ] n< u ( x + z ) F s ,...,s n ,t ( y − x, . . . , y n − x, z ) dsdz. Remark 1.2.
The small time constraint in Theorem 1.1 seems necessary. It wasshown in [6] that E [ u ( t, x ) ] is finite and admits a Feynman-Kac representationfor small t , and we expect that E [ u ( t, x ) ] = ∞ when t is large, in light of (1.14) . HAOS EXPANSION OF 2D PARABOLIC ANDERSON MODEL 5
Remark 1.3.
Since the formal product u · ˙ W in (1.1) comes from the classicalphysical product u ǫ ˙ W ǫ in (1.4) , we may interpret it in the Stratonovich’s sense.If it is replaced by the Wick product: (1.19) ∂ t u ( t, x ) = 12 ∆ u ( t, x ) + u ( t, x ) ⋄ ˙ W ( x ) , a different chaos expansion was proved in [10] . Compared with (1.17) , the onlydifference is the lack of the weight e γ ( t,B ) in the definition of F . This reduces thepolymer measure to the original Wiener measure, in which case we have (1.20) F s ,...,s n ( x , . . . , x n ) = q s ( x ) q s − s ( x − x ) . . . q s n − s n − ( x n − x n − ) , where q t ( x ) := (2 πt ) − e −| x | / t is the standard heat kernel. With µ = µ = F = 0 ,the expansion coefficient is given by (1.21) f n ( y , . . . , y n ; t, x )= Z R Z [0 ,t ] n< u ( x + z ) F s ,...,s n ,t ( y − x, . . . , y n − x, z ) dsdz = Z [0 ,t ] n Z R {
The same proof works in the one dimensional case, where thesmall time constraint can be removed, and there is no need to renormalize. Asimilar expansion coefficient as (1.18) holds. Proof of the main result
For fixed t > , x ∈ R , ǫ > and each realization of the Brownian motion, wewrite the exponent in (1.11) as Z t ˙ W ǫ ( x + B s ) ds = Z t Z R ϕ ǫ ( x + B s − y ) dW ( y ) ds = Z R (cid:18)Z t ϕ ǫ ( x + B s − y ) ds (cid:19) dW ( y )= Z R Φ ǫt,x,B ( y ) dW ( y ) , with Φ ǫt,x,B ( y ) := Z t ϕ ǫ ( x + B s − y ) ds. YU GU JINGYU
HUANG
Then it is easy to see that u ǫ ( t, x ) ∈ D ∞ , , and(2.1) D n u ǫ ( t, x ) = E B (cid:20) u ( x + B t ) D n exp (cid:18)Z R Φ ǫt,x,B ( y ) dW ( y ) − C ǫ t (cid:19)(cid:21) = E B (cid:20) u ( x + B t ) exp (cid:18)Z R Φ ǫt,x,B ( y ) dW ( y ) − C ǫ t (cid:19) (Φ ǫt,x,B ( · )) ⊗ n (cid:21) . By the Stroock’s formula (1.10), we can write the Wiener chaos expansion of u ǫ ( t, x ) as(2.2) u ǫ ( t, x ) = ∞ X n =0 I n ( f ǫ,n ( · ; t, x )) , with(2.3) f ǫ,n ( · ; t, x ) = 1 n ! E [ D n u ǫ ( t, x )]= 1 n ! E B (cid:20) u ( x + B t ) exp (cid:18)Z t Z s R ǫ ( B s − B u ) duds − C ǫ t (cid:19) (cid:0) Φ ǫt,x,B ( · ) (cid:1) ⊗ n (cid:21) . By (1.15), we define(2.4) r ǫ := Z t Z s E B [ R ǫ ( B s − B u )] duds − C ǫ t − t ( µ + µ log t ) , which goes to zero as ǫ → , and rewrite(2.5) f ǫ,n ( · ; t, x ) = e t ( µ + µ log t )+ r ǫ n ! E B h u ( x + B t ) exp( γ ǫ ( t, B )) (cid:0) Φ ǫt,x,B ( · ) (cid:1) ⊗ n i . To prove Theorem 1.1, it suffices to show that as ǫ → ,(2.6) ∞ X n =0 n ! k f ǫ,n ( · ; t, x ) − f n ( · ; t, x ) k L ( R ) ⊗ n → . Define(2.7) ˜ f ǫ,n ( · ; t, x ) := e t ( µ + µ log t )+ r ǫ n ! E B h u ( x + B t ) exp( γ ( t, B )) (cid:0) Φ ǫt,x,B ( · ) (cid:1) ⊗ n i . Since k f ǫ,n ( · ; t, x ) − f n ( · ; t, x ) k L ( R ) ⊗ n ≤ k f ǫ,n ( · ; t, x ) − ˜ f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n + 2 k ˜ f ǫ,n ( · ; t, x ) − f n ( · ; t, x ) k L ( R ) ⊗ n , the proof of (2.6) reduces to the following three lemmas. Lemma 2.1.
There exists t , C > independent of ǫ, n such that if t < t , k f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n + k ˜ f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n + k f n ( · ; t, x ) k L ( R ) ⊗ n ≤ ( Ct ) n n ! . Lemma 2.2.
There exists t > such that if t < t , k f ǫ,n ( · ; t, x ) − ˜ f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n → , as ǫ → . HAOS EXPANSION OF 2D PARABOLIC ANDERSON MODEL 7
Lemma 2.3.
There exists t > such that if t < t , k ˜ f ǫ,n ( · ; t, x ) − f n ( · ; t, x ) k L ( R ) ⊗ n → , as ǫ → . In the following, we use the notation a . b when a ≤ Cb for some constant C > independent of ǫ, n . Proof of Lemma 2.1 . The proof of f ǫ,n and ˜ f ǫ,n is the same. Take f ǫ,n for example: k f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n . n !) Z R n E B ,B " Y j =1 e γ ǫ ( t,B j ) n Y k =1 Φ ǫt,x,B j ( y k ) ! dy, where B , B stand for independent Brownian motions. Performing the integralin the y variable, the r.h.s. of the above display is bounded by n !) E B ,B (cid:20) e γ ǫ ( t,B )+ γ ǫ ( t,B ) (cid:18)Z [0 ,t ] R ǫ ( B s − B u ) dsdu (cid:19) n (cid:21) . Now we use Cauchy-Schwarz inequality and (A.1)-(A.2) to derive E B ,B (cid:20) e γ ǫ ( t,B )+ γ ǫ ( t,B ) (cid:18)Z [0 ,t ] R ǫ ( B s − B u ) dsdu (cid:19) n (cid:21) ≤ vuut E B ,B [ e γ ǫ ( t,B )+2 γ ǫ ( t,B ) ] E B ,B "(cid:18)Z [0 ,t ] R ǫ ( B s − B u ) dsdu (cid:19) n . ( Ct ) n p (2 n )! . An application of Stirling’s approximation yields the desired result. By Lemma 2.3,the same estimate holds for f n . The proof is complete. Proof of Lemma 2.2 . By the same discussion as in the proof of Lemma 2.1, wehave k f ǫ,n ( · ; t, x ) − ˜ f ǫ,n ( · ; t, x ) k L ( R ) ⊗ n ≤ n !) E B ,B (cid:20)(cid:12)(cid:12)(cid:12) ( e γ ǫ ( t,B ) − e γ ( t,B ) )( e γ ǫ ( t,B ) − e γ ( t,B ) ) (cid:12)(cid:12)(cid:12) (cid:18)Z t Z t R ǫ ( B s − B u ) dsdu (cid:19) n (cid:21) . By the fact that γ ǫ → γ a.s. as ǫ → and (A.3), we know that the random vari-able inside the above expectation converges to zero in probability. The uniformintegrability is guaranteed by (A.1) and (A.2). Thus, the r.h.s. of the abovedisplay goes to zero as ǫ → . Proof of Lemma 2.3 . First, we claim that ˜ f ǫ,n ( · ; t, x ) is a Cauchy sequence in L ( R ) ⊗ n . It suffices to prove the convergence of(2.8) lim ǫ ,ǫ → h ˜ f ǫ ,n ( · ; t, x ) , ˜ f ǫ ,n ( · ; t, x ) i L ( R ) ⊗ n . By applying Lemma A.1, we have E B ,B " Y j =1 u ( x + B jt ) e γ ( t,B j ) (cid:18)Z t Z t R ǫ ,ǫ ( B s − B u ) dsdu (cid:19) n YU GU JINGYU
HUANG converges as ǫ , ǫ → , where R ǫ ,ǫ := ϕ ǫ ⋆ ϕ ǫ . This proves (2.8).Next, we show that ˜ f ǫ,n ( · ; t, x ) → f n ( · ; t, x ) in L ( R ) ⊗ n which implies that f n ( · ; t, x ) ∈ L ( R ) ⊗ n and completes the proof. We have(2.9) ˜ f ǫ,n ( y , . . . , y n ; t, x )= e t ( µ + µ log t )+ r ǫ n ! E B " u ( x + B t ) e γ ( t,B ) n Y k =1 Φ ǫt,x,B ( y k ) = e t ( µ + µ log t )+ F t + r ǫ n ! ˆ E t,B " u ( x + B t ) n Y k =1 Φ ǫt,x,B ( y k ) = e t ( µ + µ log t )+ F t + r ǫ ϕ ⊗ nǫ ⋆ G ( y − x, . . . , y n − x ) , with G ( z , . . . , z n ) := Z R Z [0 ,t ] n< u ( x + z n +1 ) F s ,...,s n ,t ( z , . . . , z n , z n +1 ) dsdz n +1 ∈ L ( R ) ⊗ n . Since ϕ ⊗ nǫ is an approximation to identity, by the classical convolution theorem, ϕ ⊗ nǫ ⋆ G → G in L ( R ) ⊗ n . Thus, ˜ f ǫ,n ( y , . . . , y n ; t, x ) → e t ( µ + µ log t )+ F t G ( y − x, . . . , y n − x )= f n ( y , . . . , y n ; t, x ) in L ( R ) ⊗ n . Appendix A. Technical lemmas
A.1.
Measurability of F . We show that F s ,...,s n ( x , . . . , x n ) is jointly measur-able in the ( s, x ) variable. Fix any < s < . . . < s n ≤ t , consider F ǫs ,...,s n ( x , . . . , x n ) := ˆ E t,B " n Y j =1 ϕ ǫ ( B s j − x j ) = Z R n n Y j =1 ϕ ǫ ( y j − x j ) F s ,...,s n ( y , . . . , y n ) dy. The last integral converges in L ( R n ) to F s ,...,s n . It is clear that F ǫs ,...,s n ( x , . . . , x n ) is continuous in both s and x variable, hence it is measurable. If we can show F ǫs ,...,s n ( x , . . . , x n ) converges in L ([0 , t ] n< × R n ) to some g s ,...,s n ( x , . . . , x n ) ,then g = F almost everywhere in [0 , t ] n< × R n , which implies F is measurable.For fixed s , . . . , s n , we have Z R n | F ǫs ,...,s n ( x , . . . , x n ) − F δs ,...,s n ( x , . . . , x n ) | dx → HAOS EXPANSION OF 2D PARABOLIC ANDERSON MODEL 9 as ǫ, δ → . In addition, Z R n | F ǫs ,...,s n ( x , . . . , x n ) − F δs ,...,s n ( x , . . . , x n ) | dx ≤ Z R n (cid:0) F ǫs ,...,s n ( x , . . . , x n ) + F δs ,...,s n ( x , . . . , x n ) (cid:1) dx = 2 . Thus, by the dominated convergence theorem, we have Z [0 ,t ] n< Z R n | F ǫs ,...,s n ( x , . . . , x n ) − F δs ,...,s n ( x , . . . , x n ) | dxds → as ǫ, δ → . This completes the proof.A.2. Estimates on intersection local time.
We collect some standard esti-mates on the intersection local time of planar Brownian motion. Recall that R ǫ ,ǫ = ϕ ǫ ⋆ ϕ ǫ , and assume that the Brownian motion is built on the proba-bility space (Σ , A , P B ) . Lemma A.1.
For any λ > , there exist constants C, t > such that (A.1) sup ǫ ∈ (0 , ,t ∈ [0 ,t ] E B [ e λγ ǫ ( t,B ) ] ≤ C , and for all n ∈ N , (A.2) sup ǫ ,ǫ ∈ (0 , E B ,B (cid:20)(cid:18)Z [0 ,t ] R ǫ ,ǫ ( B s − B u ) dsdu (cid:19) n (cid:21) ≤ n !( Ct ) n . In addition, (A.3) Z [0 ,t ] R ǫ ,ǫ ( B s − B u ) dsdu → Z [0 ,t ] δ ( B s − B u ) dsdu in L (Σ) as ǫ , ǫ → , where the r.h.s. is the so-called mutual intersection localtime of planar Brownian motions.Proof The uniform exponential integrability (A.1) is shown in [6, Lemma A.1].It also contains a moment estimate of the form sup ǫ ∈ (0 , E B ,B (cid:20)(cid:18)Z [0 ,t ] R ǫ ( B s − B u ) dsdu (cid:19) n (cid:21) ≤ E B ,B (cid:20)(cid:18)Z [0 ,t ] δ ( B s − B u ) dsdu (cid:19) n (cid:21) ≤ n !( Ct ) n . The same proof leads to (A.2).Since Z [0 ,t ] R ǫ ( B s − B u ) dsdu → Z [0 ,t ] δ ( B s − B u ) dsdu in L (Σ) , to prove (A.3), it suffices to show that as ǫ , ǫ → , Z [0 ,t ] R ǫ ,ǫ ( B s − B u ) dsdu − Z [0 ,t ] R ǫ ( B s − B u ) dsdu → GU JINGYU
HUANG in L (Σ) , which reduces to the convergence of E B ,B (cid:20)Z [0 ,t ] R ǫ ,ǫ ( B s − B u ) R ǫ ,ǫ ( B s − B u ) dsdu (cid:21) as ǫ j → , j = 1 , , , . We write R ǫ i ,ǫ j in the Fourier domain so that the aboveexpectation equals to π ) Z [0 ,t ] Z R ˆ ϕ ( ǫ ξ ) ˆ ϕ ( ǫ ξ ) ˆ ϕ ( ǫ η ) ˆ ϕ ( ǫ η ) E B ,B [ e iξ · ( B s − B u ) e iη · ( B s − B u ) ] dξdηdsdu . It suffices to use the bound Z [0 ,t ] Z R E B ,B [ e iξ · ( B s − B u ) e iη · ( B s − B u ) ] dξdηdsdu < ∞ and the dominated convergence theorem to complete the proof. Acknowledgments.
We would like to thank the anonymous referee for a verycareful reading of the manuscript and many useful suggestions that helped im-prove the presentation. YG is partially supported by the NSF through DMS-1613301.
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