The 1D Schrödinger equation with a spacetime white noise: the average wave function
TTHE 1D SCHRÖDINGER EQUATION WITH A SPACETIMEWHITE NOISE: THE AVERAGE WAVE FUNCTION
YU GU
Abstract.
For the 1D Schrödinger equation with a mollified spacetime whitenoise, we show that the average wave function converges to the Schrödingerequation with an effective potential after an appropriate renormalization.
Keywords: random Schrödinger equation, renormalization, path integral. Main result
Consider the Schrödinger equation driven by a weak stationary spacetime Gauss-ian potential V ( t, x ) :(1.1) i∂ t φ ( t, x ) +
12 ∆ φ ( t, x ) − √ εV ( t, x ) φ ( t, x ) = , t > , x ∈ R , on the diffusive scale ( t, x ) ↦ ( tε , xε ) ,(1.2) φ ε ( t, x ) ∶= φ ( tε , xε ) satisfies(1.3) i∂ t φ ε ( t, x ) +
12 ∆ φ ε ( t, x ) − ε / V ( tε , xε ) φ ε ( t, x ) = . With appropriate decorrelating assumptions on V , the rescaled large highly oscil-latory potential ε − / V ( t / ε , x / ε ) converges in distribution to a spacetime whitenoise, denoted by ˙ W ( t, x ) . To the best of our knowledge, the asymptotics of φ ε andmaking sense of the limit of (1.3), which formally reads i∂ t Φ ( t, x ) +
12 ∆Φ ( t, x ) − ˙ W ( t, x ) Φ ( t, x ) = , is an open problem. The goal of this short note is to take a first step by analyzing E [ φ ε ] as ε → Assumptions on the randomness.
We assume the spacetime white noise˙ W ( t, x ) is built on the probability space ( Ω , F , P ) , and V ( t, x ) = ˆ R % ( t − s, x − y ) ˙ W ( s, y ) dyds for some mollifier % ∈ C ∞ c with ´ % =
1. By the scaling property of ˙ W , we have1 ε / V ( tε , xε ) = ε / ˆ R % ( tε − s, xε − y ) ˙ W ( s, y ) dyds = ε / ˆ R ε % ( t − sε , x − yε ) ˙ W ( sε , yε ) dyds law = ˆ R ε % ( t − sε , x − yε ) ˙ W ( s, y ) dyds, a r X i v : . [ m a t h . P R ] M a y YU GU which converges in distribution to ˙ W independent of the choice of % . For simplicity,we choose % ( t, x ) = η ( t )√ π e − x , with η ∈ C ∞ c ( R ) and ´ η =
1. The covariance function of V is(1.4) R ( t, x ) = E [ V ( t, x ) V ( , )] = ˆ R % ( t + s, x + y ) % ( s, y ) dyds = R η ( t ) q ( x ) , with(1.5) R η ( t ) ∶= ˆ R η ( t + s ) η ( s ) ds, q ( x ) ∶= √ π e − x . We define ̃ R ( ω, ξ ) as the Fourier transform of R in ( t, x ) : ̃ R ( ω, ξ ) = ˆ R R ( t, x ) e − iωt − iξx dtdx. We use ̂ f to denote the Fourier transform of f in the x variable: ̂ f ( ξ ) = ˆ R f ( x ) e − iξx dx. Main result.
Assuming the initial data φ ε ( , x ) = φ ( x ) ∈ C ∞ c ( R ) , so we havea low frequency wave before rescaling: φ ( , x ) = φ ( εx ) . The following is the mainresult: Theorem 1.1.
There exists z , z ∈ C depending on the mollifier % , given by (2.13) and (2.18) respectively, such that for any t > , ξ ∈ R , (1.6) E [̂ φ ε ( t, ξ )] e z tε → ̂ φ ( ξ ) e − i ∣ ξ ∣ t + z t , as ε → . We make a few remarks.
Remark . The limit in (1.6) is the solution to i∂ t ¯ φ +
12 ∆ ¯ φ − iz ¯ φ = , ¯ φ ( , x ) = φ ( x ) , written in the Fourier domain:¯ φ ( t, x ) = π ˆ R ̂ φ ( ξ ) e − i ∣ ξ ∣ t + z t e iξx dξ. Remark . In the parabolic setting, a Wong-Zakai theorem is proved [3, 11, 13, 14]for ∂ t u ε =
12 ∆ u ε + ε / V ( tε , xε ) u ε , u ( , x ) = u ( x ) . The result says that there exists c , c > % such that(1.7) u ε ( t, x ) e − c tε − c t ⇒ U( t, x ) in distribution , where U solves the stochastic heat equation with a multiplicative spacetime whitenoise ∂ t U( t, x ) =
12 ∆ U( t, x ) + U( t, x ) ˙ W ( t, x ) , U( , x ) = u ( x ) , with the product U( t, x ) ˙ W ( t, x ) interpreted in the Itô’s sense. Writing the aboveequation in the mild formulation, it is easy to see that E [U] solves the unperturbedheat equation E [U( t, x )] = ˆ R √ πt e − ∣ x − y ∣ t u ( y ) dy, D SCHRÖDINGER EQUATION WITH SPACETIME WHITE NOISE 3 thus, a consequence of (1.7) is E [̂ u ε ( t, ξ )] e − c tε − c t → ̂ u ( ξ ) e − ∣ ξ ∣ t , which should be compared to (1.6) in the Schrödinger setting, with − c , c corre-sponding to z , z . Remark . Starting from the microscopic dynamics (1.1), if we consider a timescale that is shorter than the one in (1.2), and a low frequency initial data ( t, x ) ↦ ( tε , x √ ε ) , φ ( , x ) = φ (√ εx ) , a homogenization result was proved in [10]: for any t > , ξ ∈ R ,(1.8) ε d ̂ φ ( tε , √ εξ ) → ̂ φ ( ξ ) e − i ∣ ξ ∣ t − z t in probability. Here z is the same constant as in Theorem 1.1. If we instead considera high frequency initial data φ ( , x ) = φ ( x ) , which varies on the same scale as therandom media, a kinetic equation was derived on the time scale of tε in [2]:(1.9) E [∣̂ φ ( tε , ξ )∣ ] → W ( t, ξ ) , where W ( t, ξ ) = ´ R W ( t, x, ξ ) dx and W solves the radiative transfer equation(1.10) ∂ t W ( t, x, ξ ) + ξ ⋅ ∇ x W ( t, x, ξ ) = ˆ R ̃ R ( ∣ p ∣ − ∣ ξ ∣ , p − ξ )( W ( t, x, p ) − W ( t, x, ξ )) dp π . For similar results in the case of a spatial randomness, see [4, 7, 18]. The equation(1.10) shows that, in the high frequency regime where the wave and the random mediainteract fully, the momentum variable follows a jump process with the kernel givenby ̃ R ( ∣ p ∣ −∣ ξ ∣ , p − ξ ) . The real part of the constant z , in (1.6) and (1.8), describesthe total scattering cross-section, i.e., the jumping rate at the zero frequency:2Re [ z ] = ˆ R ̃ R ( ∣ p ∣ , p ) dp π . Thus, the renormalization in (1.6) can be viewed as a compensation of the exponentialattenuation of wave propagation on the time scale of tε . We emphasize that theaverage wave function (more precisely, the term E [̂ φ ε ( t, ξ )] E [̂ φ ∗ ε ( t, ξ )] ) only capturesthe ballistic component of wave. Remark . It is unclear at this stage what explicit information the conver-gence in (1.6) implies. On one hand, if we expect the family of random variables {̂ φ ε ( t, ξ )} ε ∈( , ) to converge in distribution to some random limit after any possiblerenormalization and assume the uniform integrability , then our result shows that e z t / ε is the only possible renormalization factor since the uniform integrabilityensures the mean E [̂ φ ε ( t, ξ )] also converges after the same rescaling. On the otherhand, without the uniform integrability it is a priori unclear whether the convergenceof ̂ φ ε ( t, ξ ) is related to the convergence of its first moment. In addition, based on thediscussion in Remark 1.4, we know that the wave field ̂ φ ( t, ξ ) decays exponentiallyon the time scale t / ε because Re [ z ] >
0, and the lost energy escapes to highfrequency regime through multiple scatterings. From this perspective, the physicalmeaning is unclear when we multiply the exponentially small solution by e z t / ε in(1.6) so that something “nontrivial” can still be observed. In the parabolic setting,the renormalization in (1.7) can be naturally viewed as a shift of the height functionby its average growing speed, after a Hopf-Cole transformationlog [ u ε ( t, x ) e − c tε − c t ] = log u ε ( t, x ) − c tε − c t. YU GU
For the Schrödinger equation, it is less clear to us what should be the right physicalquantity to look at. Another choice is to consider ̂ φ ( t, ξ ) for ξ ∼ O ( ) and t ∼ O ( ε − α ) with some α >
1. In light of (1.9) and the long time behavior of (1.10) analyzed in[15], we expect some diffusion equation to show up in the limit.
Remark . The convergences in (1.6), (1.8) and (1.9) hold in all dimensions d ⩾ √ ε , then in all dimensions: (i) on the time scale t / ε , depending onthe initial data, we have either (1.8) or (1.9); (ii) on the time scale t / ε , if we havea low frequency initial data, then (1.6) holds. The proof and the result does NOTdepend on the dimensions. Nevertheless, with a random potential of size √ ε , thechange of variables ( t, x ) ↦ ( tε , xε ) chosen in (1.3) only leads to a spacetime whitenoise in d = Remark . When the spacetime potential V ( t, x ) is replaced by a spatial potential V ( x ) , similar problems (including nonlinear ones) have been analyzed in [1, 5, 6, 9,12, 16, 19] in d = , ,
3. 2.
Proofs
The proof contains two steps. First, we derive a probabilistic representation of theaverage wave function E [̂ φ ( t, ξ )] with some auxiliary Brownian motion { B t } t ⩾ builton another probability space ( Σ , A , P B ) . Using this probabilistic representation, wepass to the limit using tools from stochastic analysis. Similar proofs have alreadyappeared in [9, 11].2.1. Probabilistic representation.
Assuming { B t } t ⩾ is a standard Brownianmotion starting from the origin, defined on ( Σ , A , P B ) . We denote the expectationwith respect to { B t } t ⩾ by E B . Lemma 2.1.
For the equation (2.1) i∂ t ψ +
12 ∆ ψ − V ( t, x ) ψ = , t > , x ∈ R , with ψ ( , x ) = ψ ( x ) , we have (2.2) E [ ̂ ψ ( t, ξ )] = ̂ ψ ( ξ ) E B [ e i √ iξB t e − ´ t ´ t R ( s − u, √ i ( B s − B u )) dsdu ] . On the formal level, (2.2) comes from an application of the Feynman-Kac formulato (2.1) then averaging with respect to V . We write (2.1) as ∂ t ψ = i ψ − iV ( t, x ) ψ = , and assume the following expression: ψ ( t, x ) = E B [ ψ ( x + √ iB t ) e − i ´ t V ( t − s,x +√ iB s ) ds ] . Averaging with respect to V and using the Gaussianity yields E [ ψ ( t, x )] = E B [ ψ ( x + √ iB t ) e − ´ t ´ t R ( s − u, √ i ( B s − B u )) dsdu ] , which, after taking the Fourier transform, gives (2.2). Proof.
We follow the proof of [9, Proposition 2.1], where a similar formula is derivedfor spatial random potentials. For the convenience of readers, we provide all thedetails here.Fix ( t, ξ ) , we define the function F ( z ) ∶= E B [ e izξB t − ´ t ´ t R ( s − u,z ( B s − B u )) dsdu ] , z ∈ ¯ D , D SCHRÖDINGER EQUATION WITH SPACETIME WHITE NOISE 5 with D ∶= { z ∈ C ∶ Re [ z ] > } . We also define the corresponding Taylor expansion F ( z ) = ∞ ∑ n = F ,n ( z ) , z ∈ ¯ D , with F ,n ( z ) ∶= (− ) n n ( π ) n n ! ˆ [ ,t ] n ˆ R n n ∏ j = ̂ R ( s j − u j , p j ) E B ⎡⎢⎢⎢⎣ e izξB t n ∏ j = e izp j ( B sj − B uj ) ⎤⎥⎥⎥⎦ dpdsdu. Recall that R ( t, x ) = R η ( t )√ π e − x / . In the definition of F , we have extended thedefinition so that R ( t, z ) = R η ( t )√ π e − z / for all z ∈ C . We also emphasize that ̂ R ( t, p ) is the Fourier transform of R ( t, x ) in the x − variable: ̂ R ( t, p ) = R η ( t ) e − p . It is straightforward to check that both F and F are analytic on D andcontinuous on ¯ D . Note that √ i ∈ ∂D . The goal is to show that(2.3) E [ ̂ ψ ( t, ξ )] = ̂ ψ ( ξ ) F (√ i ) . Since ( z, s, u ) ↦ R ( s − u, z ( B s − B u )) is bounded on ¯ D × R + , we have(2.4) F ( z ) = ∞ ∑ n = (− ) n n n ! E B [ e izξB t ( ˆ [ ,t ] R ( s − u, z ( B s − B u )) dsdu ) n ]= ∞ ∑ n = (− ) n n n ! E B ⎡⎢⎢⎢⎣ e izξB t ˆ [ ,t ] n n ∏ j = R ( s j − u j , z ( B s j − B u j )) dsdu ⎤⎥⎥⎥⎦= ∞ ∑ n = (− ) n n ( π ) n n ! E B ⎡⎢⎢⎢⎣ e izξB t ˆ [ ,t ] n ˆ R n n ∏ j = ̂ R ( s j − u j , p j ) e izp j ( B sj − B uj ) dpdsdu ⎤⎥⎥⎥⎦ . For z = x ∈ R , we can apply the Fubini theorem to see that F ( x ) = F ( x ) . Due tothe analyticity and continuity of F and F , we therefore have F ( z ) = F ( z ) for all z ∈ ¯ D . Hence, (2.3) is equivalent to(2.5) E [ ̂ ψ ( t, ξ )] = ̂ ψ ( ξ ) ∞ ∑ n = F ,n (√ i ) . For a fixed n , we rewrite F ,n (√ i ) = (− ) n n ( π ) n n ! ˆ [ ,t ] n ˆ R n n ∏ j = ̂ R ( s j − − s j , p j − ) δ ( p j − + p j )× E B [ e i √ iξB t e − ∑ nj = i √ ip j B sj ] dsdp. Let σ denote the permutations of { , . . . , n } . After a relabeling of the p -variableswe can write(2.6) F ,n (√ i ) = (− ) n n ( π ) n n ! ∑ σ ˆ [ ,t ] n < ˆ R n n ∏ j = ̂ R ( s σ ( j − ) − s σ ( j ) , p σ ( j − ) ) δ ( p σ ( j − ) + p σ ( j ) )× E B [ e i √ iξB t e − ∑ nj = i √ ip j B sj ] dsdp, where [ , t ] n < ∶= {( s , . . . , s n ) ∶ ⩽ s n ⩽ . . . ⩽ s ⩽ t } . Let F denote the pairingsformed over { , . . . , n } . It is straightforward to check that(2.7) F ,n (√ i ) = i n ( π ) n ∑ F ˆ [ ,t ] n < ˆ R n ∏ ( k,l )∈F ̂ R ( s k − s l , p k ) δ ( p k + p l )× E B [ e i √ iξB t e − ∑ nj = i √ ip j B sj ] dsdp. YU GU
The pre-factors in (2.6) and (2.7) differ by a factor of 2 n n ! since i − n = (− ) n , andthis comes from the mapping between the sets of permutations and pairings: for agiven pairing with n pairs, we have n ! ways of permutating the pairs, and insideeach pair, we have 2 options which leads to the additional factor of 2 n .The phase factor inside the integral in (2.7) can be computed explicitly:(2.8) E B [ e i √ iξB t e − ∑ nj = i √ ip j B sj ] = e − i ∣ ξ ∣ ( t − s )− i ∣ ξ − p ∣ ( s − s )− ... − i ∣ ξ − ... − p n ∣ s n . On the other hand, the equation (2.1) is written in the Fourier domain as ∂ t ̂ ψ = − i ∣ ξ ∣ ̂ ψ + ˆ R ̂ V ( t, dp ) πi ̂ ψ ( t, ξ − p ) , ̂ ψ ( , ξ ) = ̂ ψ ( ξ ) , where V ( t, x ) admits the spectral representation V ( t, x ) = ´ R ̂ V ( t,dp ) π e ipx . Using theabove formula, we can write the solution ̂ ψ ( t, ξ ) as an infinite series(2.9) ̂ ψ ( t, ξ ) = ∞ ∑ n = ˆ [ ,t ] n < ˆ R n n ∏ j = ̂ V ( s j , dp j ) πi e − i ∣ ξ ∣ ( t − s )− i ∣ ξ − p ∣ ( s − s )− ... − i ∣ ξ − ... − p n ∣ s n × ̂ ψ ( ξ − p − . . . − p n ) ds. Evaluating the expectation E [ ̂ ψ ( t, ξ )] in (2.9), using the Wick formula for computingthe Gaussian moment E [̂ V ( s , dp ) . . . ̂ V ( s n , dp n )] , and the fact that E [̂ V ( s i , dp i )̂ V ( s j , dp j )] = π ̂ R ( s i − s j , p i ) δ ( p i + p j ) dp i dp j , and comparing the result to (2.7)-(2.8), we conclude that (2.5) holds, which completesthe proof. (cid:3) Convergence of Brownian functionals.
By Lemma 2.1, the interestedquantity is written as E [̂ φ ε ( t, ξ )] = ̂ φ ( ξ ) E B [ e i √ iξB t e − ´ t ´ t R ε ( s − u, √ i ( B s − B u )) dsdu ] , with R ε defined as the covariance function of ε − / V ( t / ε , x / ε ) : R ε ( t, x ) = ε R ( tε , xε ) . After a change of variable and using the scaling property of the Brownian motion,we have ˆ t ˆ t R ε ( s − u, √ i ( B s − B u )) dsdu = ε ˆ t / ε ˆ t / ε R η ( s − u ) q (√ i ( B ε s − B ε u )/ ε ) dsdu law = ε ˆ t / ε ˆ t / ε R η ( s − u ) q (√ i ( B s − B u )) dsdu, where R η and q were defined in (1.5). Thus, by defining(2.10) X εt ∶= ε ˆ t / ε ˆ t / ε R η ( s − u ) q (√ i ( B s − B u )) dsdu, we have(2.11) E [̂ φ ε ( t, ξ )] = ̂ φ ( ξ ) E B [ e i √ iξεB t / ε − X εt ] . D SCHRÖDINGER EQUATION WITH SPACETIME WHITE NOISE 7
To pass to the limit of E [̂ φ ε ( t, ξ )] , it suffices to prove the weak convergence of therandom vector ( εB t / ε , X εt ) (for fixed t >
0) and a uniform integrability condition.The proof of Theorem 1.1 reduces to the following three lemmas.
Lemma 2.2. E B [ X εt ] = z tε + O ( ε ) with z defined in (2.13) . Lemma 2.3.
For fixed t > , as ε → , (2.12) ( εB t / ε , X εt − E B [ X εt ]) ⇒ ( N , N + iN ) in distribution, where N ∼ N ( , t ) and is independent of ( N , N ) ∼ N ( , t A ) , withthe × covariance matrix A defined in (2.15) . Lemma 2.4.
For any λ ∈ R , there exists a constant C > such that E B [∣ e λ ( X εt − E B [ X εt ]) ∣] ⩽ C uniformly in ε > .Remark . With some extra work as in [11, Proposition 2.3], the convergence in(2.12) can be upgraded to the process level. To keep the argument short, we onlyconsider the marginal distributions, which is enough for the proof of Theorem 1.1.
Proof of Lemma 2.2.
A straightforward calculation gives E B [ X εt ] = ε ˆ t / ε ds ˆ s R η ( s − u )√ π E B [ e − i ∣ B s − B u ∣ ] du = ε ˆ t / ε ds ˆ s R η ( u )√ π E B [ e − i ∣ B s − B s − u ∣ ] du. Since R η is compactly supported, it is clear that E B [ X εt ] = z tε + O ( ε ) , where(2.13) z = ˆ ∞ R η ( u )√ π E B [ e − i ∣ B u ∣ ] du = ˆ ∞ R η ( u )√ π ( + iu ) du. The proof is complete. (cid:3)
Proof of Lemma 2.3.
The proof is based on a martingale decomposition. Denotethe Brownian filtration by F r and the Malliavin derivative with respect to dB r by D r . An application of the Clark-Ocone formula [17, Proposition 1.3.14] leads to X εt − E B [ X εt ] = ˆ t / ε E B [ D r X εt ∣ F r ] dB r . Recall that X εt is defined in (2.10), by chain rule and the fact that(2.14) D r ( B s − B u ) = D r ˆ su dB r = [ u,s ] ( r ) , we have D r X εt = − iε ˆ t / ε ˆ s R η ( s − u )√ π e − i ∣ B s − B u ∣ ( B s − B u ) [ u,s ] ( r ) duds, r ∈ [ , t / ε ] . Taking the conditional expectation with respect to F r and computing the expectation E [ e − i X X ∣ B r − B u ] YU GU with X ∼ N ( B r − B u , s − r ) explicitly yields Y ε,tr ∶= ε − E B [ D r X εt ∣ F r ]= − i ˆ t / ε ˆ s R η ( s − u )√ π ( + i ( s − r )) / e − i ∣ Br − Bu ∣ ( + i ( s − r )) ( B r − B u ) [ u,s ] ( r ) duds. By the assumption, there exists M > R η ( s − u ) = s − u ⩾ M . Usingthe indicator function 1 [ u,s ] ( r ) in the above expression, we have for M ⩽ r ⩽ t / ε − M that Y ε,tr = Y r ∶ = − i ˆ r + Mr ˆ rr − M R η ( s − u )√ π ( + i ( s − r )) / e − i ∣ Br − Bu ∣ ( + i ( s − r )) ( B r − B u ) [ u,s ] ( r ) duds = − i ˆ M ˆ M R η ( s + u )√ π ( + is ) / e − i ∣ Br − Br − u ∣ ( + is ) ( B r − B r − u ) duds. The Y r defined above is only for r ∈ [ M, t / ε − M ] , but we can extend the definitionto r ∈ R by interpreting B as a two-sided Brownian motion. Thus, by the fact thatthe Brownian motion has stationary and independent increments, we know { Y r } r ∈ R is a stationary process with a finite range of dependence.It is easy to check that X εt − E B [ X εt ] − ε ˆ t / ε Y r dB r = ε ˆ t / ε ( Y ε,tr − Y r ) dB r → Y ,r = Re [ Y r ] and Y ,r = Im [ Y r ] , applying Ergodic theorem,we have ε ˆ t / ε Y j,r Y l,r dr → t E [ Y j,r Y l,r ] , j, l = , , and ε ˆ t / ε Y r ds → t E [ Y r ] = , almost surely. We apply the martingale central limit theorem [8, pp. 339] to derive ( εB t / ε , ε ˆ t / ε Y r dB r ) ⇒ ( B t , W t + iW t ) in C[ , ∞) , where B t is a standard Brownian motion, independent of the two-dimensional Brownian motion ( W t , W t ) with the covariance matrix A = ( A jl ) j,l = , given by(2.15) A jl = E [ Y j,r Y l,r ] . The proof is complete. (cid:3)
Proof of Lemma 2.4.
We write X εt − E B [ X εt ] = ε ˆ t / ε Z s ds, where Z s ∶= ˆ s R η ( u )√ π ( e − i ∣ B s − B s − u ∣ − E B [ e − i ∣ B s − B s − u ∣ ]) du. Again, assuming that R η ( u ) = ∣ u ∣ ⩾ M . Let N ε = [ tMε ] , we have X εt − E B [ X εt ] = ε N ε ∑ k = ˆ kM ( k − ) M Z s ds + ε ⎛⎝ ˆ M + ˆ t / ε N ε M ⎞⎠ Z s ds = ε N ε ∑ k = Z k + ε ⎛⎝ ˆ M + ˆ t / ε N ε M ⎞⎠ Z s ds D SCHRÖDINGER EQUATION WITH SPACETIME WHITE NOISE 9 where we defined Z k ∶= ´ kM ( k − ) M Z s ds for 2 ⩽ k ⩽ N ε . Since Z s is uniformly bounded,we have RRRRRRRRRRR ε ⎛⎝ ˆ M + ˆ t / ε N ε M ⎞⎠ Z s ds RRRRRRRRRRR ≲ ε. For the first part, we write ε N ε ∑ k = Z k = ⎛⎝ ∑ k ∈ A ε, + ∑ k ∈ A ε, ⎞⎠ εZ k , with A ε, = { ⩽ k ⩽ N ε ∶ k even } and A ε, = { ⩽ k ⩽ N ε ∶ k odd } . By theindependence of the increments of the Brownian motion, we know that { Z k } k ∈ A ε,j are i.i.d. for j = E B [∣ e λ ( X εt − E B [ X εt ]) ∣] ≲ E B [ e λε ∑ Nεk = Re [ Z k ] ]≲√ E B [ e λε ∑ k ∈ Aε, Re [ Z k ] ] E B [ e λε ∑ k ∈ Aε, Re [ Z k ] ] . By the fact that { Z k } are bounded random variables with zero mean, we have for j = , E B [ e λε ∑ k ∈ Aε,j Re [ Z k ] ] = ∏ k ∈ A ε,j E B [ e λε Re [ Z k ] ]⩽ N ε ∏ k = ( + λ ε E B [∣ Re [ Z k ]∣ ] + O ( ε )) ≲ . The proof is complete. (cid:3)
Proof of Theorem 1.1.
By (2.11), we have(2.17) E [̂ φ ε ( t, ξ )] e E B [ X εt ] = ̂ φ ( ξ ) E B [ e i √ iξεB t / ε −( X εt − E B [ X εt ]) ] . By Lemma 2.3, we know that, for fixed t > , ξ ∈ R , the random variable i √ iξεB t / ε − ( X εt − E B [ X εt ]) ⇒ i √ iξN − ( N + iN ) in distribution, where N ∼ N ( , t ) independent of ( N , N ) ∼ N ( , t A ) . SinceLemma 2.4 provides the uniform integrability: E B [∣ e i √ iξεB t / ε −( X εt − E B [ X εt ]) ∣ ] ⩽ √ E B [∣ e i √ iξεB t / ε ∣ ] E B [∣ e −( X εt − E B [ X εt ]) ∣ ] ≲ , sending ε → E [̂ φ ε ( t, ξ )] e z tε → ̂ φ ( ξ ) E B [ e i √ iξN −( N + iN ) ] = ̂ φ ( ξ ) e − i ∣ ξ ∣ t e ( A − A + i A ) t . Define(2.18) z = ( A − A + i A ) , the proof of Theorem 1.1 is complete. Acknowledgments.
We would like to thank Lenya Ryzhik for asking this questionand stimulating discussions. We also thank the two anonymous referees for a carefulreading of the manuscript and for pointing out several possible improvements as wellas a technical mistake in the original manuscript. The work is partially supportedby the NSF grant DMS-1613301/1807748 and the Center for Nonlinear Analysis atCMU.
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