MOMENTS OF 2D PARABOLIC ANDERSON MODEL. 1. Introduction

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1 MOMENTS OF D PARABOLIC ANDERSON MODEL YU GU, WEIJUN XU Abstract. In this not, w us th Fynman-Kac formula to driv a momnt rprsntation for th D parabolic Andrson modl in small tim, which is rlatd to th intrsction local tim of planar Brownian motions. Kywords: Fynman-Kac formula, rnormalization, intrsction local tim. 1. Introduction Th aim of this not is to study th xistnc of momnts of th solution to th parabolic Andrson modl (PAM in two spatial dimnsions, formally givn by (1.1 t u = 1 u + u ξ, (t, x R+ R, whr ξ is th two dimnsional spatial whit nois, that is, a gnralizd Gaussian procss with covarianc Eξ(xξ(y] = δ(x y. Th quation is wll-posd in dimnsion 1, but th product btwn u and ξ bcoms ill-dfind as soon as d. For d =, th solution u is dfind in 7, 8, 1] as th limit of a squnc of th rgularizd and rnormalizd quations. Mor prcisly, fix a symmtric mollifir ρ : R R + with ρ(x = ρ( x and ρ = 1. Lt and considr th quation ρ ε (x = ε ρ(x/ε, ξ ε = ξ ρ ε, (1. t u ε = 1 u ε + (ξ ε C ε u ε, for som larg constant C ε. Thn, for (1.3 C ε = 1 log ε + O(1, π th squnc of solutions {u ε } convrgs to som limit u (indpndnt of th mollification in probability, and w call this limit u th solution to D PAM. In d = 3, th mollifir ρ ε (x = ε 3 ρ(x/ε, and th rnormalization constant taks th form C ε = c 1 ε 1 + c log ε + O(1 9]. So far, most of th rsults mntiond abov focusd on th xistnc of th solution and th convrgnc of th rgularizd PDE aftr rnormalization. Th statistical proprtis of u rmains a challng; s 1,, 4] for som rlvant discussions. Th goal of this not is to show that th n-th momnt of th solution u to D PAM xists for small tim, and w prsnt a Fynman-Kac formula for Eu n ] in th tim intrval it xists. Th following is our main rsult. 1

2 YU GU, WEIJUN XU Thorm 1.1. Thr xists a univrsal constant δ > such that for vry n 1, th n-th momnt of u xists for t (, δ with Eu(t, x n ] givn by (1.1. n 1.1. Huristic argumnt. W first giv a huristic drivation of Eu(t, x] n by writing down a rprsntation for Eu ε (t, x] n and passing to th limit formally. Suppos u ε (, x = u (x for som continuous function u with u 1, w writ th solution to (1. by th Fynman-Kac formula ( t ] (1.4 u ε (t, x = E B u (x + B t xp ξ ε (x + B s ds C ε t. Hr, B t is a standard planar Brownian motion starting from th origin and indpndnt of th whit nois ξ, and C ε is th constant dfind in (1.3. W us E B to dnot th xpctation with rspct to B. W now procd to calculating th n-th momnt of u ε (t, x. First of all, th covarianc function of ξ ε satisfis ( x y Eξ ε (xξ ε (y] = R ε (x y := ε R, ε whr R = ρ ρ, and ρ is th mollifir usd to rgulariz th nois ξ. Nxt, on raiss th xprssion (1.4 to th n-th powr, and tak a furthr xpctation with rspct to ξ ε. Sinc B is indpndnt of ξ ε, on can intrchang this xpctation with th on with rspct to th Brownian motions, and gt ] n (1.5 Eu ε (t, x n ] = E B xp (In(t ε nc ε t u (x + Bt k. Hr, {B k },...,n ar indpndnt Brownian motions, and E B dnots th xpctation with rspct to ths B k s. Also, I ε n(t is givn by (1.6 I ε n(t = n t s R ε (B k s B k ududs + t t 1 i<j n R ε (B i s B j udsdu, whr R ε (x = ε R(x/ε convrgs to th Dirac function as ε. It is wll know (s for xampl 3, Chaptr ] that ach trm in th scond trm abov (whn i j convrgs to th mutual intrsction local tim of Brownian motion, formally writtn as δ(b i,t] s Budsdu. j Th first trm abov (whn on has th sam Brownian motion in th argumnt of R ε unfortunatly dos not convrg as ε, but it dos whn on subtracts its man (s 14, 15, 1]. Thus, w dfin (1.7 ν ε (t = and for vry t, w hav t s E B R ε (B s B u ]duds, (1.8 I ε n(t nν ε (t X n (t

3 MOMENTS OF PAM 3 in probability, whr X n (t is a linar combination of slf- and mutual-intrsction local tims of planar Brownian motions, formally writtn as n t s ( X n (t = δ(bs k Bu k E B δ(bs k Bu] k duds ( i<j n t t δ(b i s B j udsdu. Not that w do not hav th factor 1 in front of th first trm sinc th intgration is on th simplx rathr than th squar, t]. It is wll known from 1] that X n (t has xponntial momnts for small nough t (dpnding on n. In ordr for th xprssion (1.5 to convrg, on nds th divrgnt constant C ε t coincids with ν ε (t. A simpl calculations shows that this is indd th cas up to an O(1 corrction. Lmma 1.. Giv a choic of th constant C ε in (1.3 (that is, fix th choic of th O(1 part, thr xists constants µ 1 and µ such that ν ε (t C ε t t(µ 1 +µ log t as ε. By (1.8 and Lmma 1., w hav I ε n(t nc ε t =I ε n(t nν ε (t + n(ν ε (t C ε t X n (t + nt(µ 1 + µ log t in probability. If th squncs {u ε (t, x n } and { Iε n(t nc εt } ar both uniformly intgrabl, thn w can pass both sids of (1.5 to th limit, and obtain (1.1 Eu(t, x n ] = E B xp ( X n (t + nt(µ 1 + µ log t n ] u (x + Bt k. Th rst of th not is to show th uniform intgrability of {u ε (t, x n } and { Iε n (t ncεt } for small tim t, so (1.1 dos hold. 1.. Discussions. Rmark 1.3. Th sam argumnt lads to a similar rsult in d = 1, whr w choos C ε = and do not hav th small tim constraint. Th rnormalizd slf-intrsction local tim can b writtn as t s (δ(b s B u E B δ(b s B u ]duds = 1 L t (x dx 1 E B L t (x ]dx, R R with L t (x dnoting th local tim of 1D Brownian motion up to t. Rmark 1.4. For n = 1, th momnt formula rads Eu(t, x] = E B u (x + B t γ(,t] < +t(µ 1+µ log t ], s with γ(, t] < = t (δ(b s B u E B δ(b s B u ]duds rprsnting th slfintrsction local tim of B. It was provd in 13] that thr xists t > such that E B γ(,t] < ] < t < t, = t > t.

4 4 YU GU, WEIJUN XU Thus, it is natural to xpct that th momnts of u dos not xist for larg t, although w do not hav a rigorous proof for it. Rmark 1.5. In 1], th authors dfind th D Andrson Hamiltonian H = +ξ on th torus T = R /Z using para-controlld calculus. An intrsting application is th xponntial tail bounds for th ground stat ignvalu Λ 1. It was provd in 1, Proposition 5.4] that thr xists C 1, C > such that C 1x PΛ 1 x] C x as x. Using th orthonormal ignvctors of H, dnotd by { n }, w writ th solution to PAM as u(t, x = Λnt u, n n (x, n=1 thrfor, E u(t, x ]dx E Λ1t ] T u (t, x dx. T By th xponntial tail bounds on Λ 1, it is clar th r.h.s. of th abov display is only finit for small t, which is consistnt with our rsult. Rmark 1.6. In th forthcoming articl 5], th authors considr th D PAM with a small nois (1.11 t u = u + βu ξ, (t, x R + R. Thy obtain an xplicit chaos xpansion of crtain polymr masur associatd with (1.11 for β 1. In particular, this implis that th scond momnt of u xists for t, 1], x R and β sufficintly small. Th rstriction of β 1 is quivalnt with our small tim rstriction. Indd, dfin u β (t, x := u(t/β, x/β, on ss that u β satisfis (1.1, hnc for u β (t, x to b squar intgrabl, w nd t/β 1, i.., t β 1. Rmark 1.7. A simpl calculation shows that th momnts of th approximations to 3D PAM xplod as ε, and indicats that th solution to 3D PAM may not hav a momnt. To s this, w considr th constant initial condition u 1, so ( t s ] Eu ε (t, x] = Cεt E B xp R ε (B s B u duds, whr R ε (x = ε 3 R(x/ε. Sinc R(x is continuous and R( >, w assum for simplicity that R(x > δ > for x. Thus, by considring th vnt that B s < ε for all s, t], w hav ( t E B xp s ] R ε (B s B u duds xp ( δt ε 3 P sup B s < ε ]. s,t] Th probability Psup s,t] B s < ε] c tε for som c > dpnding on th dimnsion. Whn d = 3, th rnormalization constant C ε = c 1 ε 1 +c log ε +O(1.

5 MOMENTS OF PAM 5 It implis that for any t >, x R 3, w hav lim ε Eu ε (t, x] =. Th sam discussion applis to d =, whr ( δt Eu ε (t, x] xp ε c t ε C εt. If t > c /δ, w also hav lim ε Eu ε (t, x] =. Rmark 1.8. Whn d =, th small tim constraint for th xistnc of momnts in our contxt also appars in 11, Thorm 4.1], whr th usual product u ξ is rplacd by th Wick product u ξ. Rmark 1.9. In th forthcoming articl 6], a similar rsult is drivd for th random Schrödingr quation i t φ + 1 φ φ ξ =.. Proof of Lmma 1. and Thorm 1.1 W dnot, t] n < = { s 1 <... < s n t}, and writ a b if a Cb with som constant C indpndnt of ε. Proof of Lmma 1.. By scaling proprty of Brownian motion, w hav ( R ε (B s B u = ε Bs B u law R = ε R ( B s/ε B u/ε. ε A chang of variabl (u/ε, s/ε (u, s thn yilds ν ε (t = ε t/ε s E B R(B s B u ]duds. Now, B s B u has th standard normal dnsity ( π(s u 1 (s u. W thn do anothr chang of variabl s u v, intgrat s out, and rscal v vε. This lads us to ν ε (t = t ( t R(x v 1 ε x v dv dx 1 t ( R(x ε x v dx dv π R π R := (i (ii. Sinc R intgrats to 1, it is clar that (ii t π as ε. As for (i, a substitution of variabl ε x v by parts yilds (i = t π = t π ( R(x R t ( R(x R ε x ε x t λ 1 λ dλ dx λ log λdλ ε x t x λ and thn an intgration ( ε x log dx. t It is clar from th abov xprssion that as ε, th only divrgnt part of (i is from th trm log(ε, and a dirct calculation shows ν ε (t t π log ε µ 1t + µ t log t

6 6 YU GU, WEIJUN XU for som constant µ 1, µ. Proof of Thorm 1.1. Fix (t, x and n, and rcall that ] n (.1 Eu ε (t, x n ] = E B xp(i n(t ε nc ε t u (x + Bt k, whr E B is th xpctation with rspct to indpndnt planar Brownian motions B k s, and In ε is givn by th xprssion (1.6. Not that u ε (t, x n u(t, x n in probability, and that by (1.8 and Lmma 1., w hav I ε n(t nc ε t X n (t + nt(µ 1 + µ log t in probability. Thus, in viw of (.1, it suffics to show th uniform intgrability of u ε (t, x n and xp(in(t ε nc ε t n u (x + Bt k. This allows us to pass both sids of (.1 to th limit and conclud Thorm 1.1. To prov th uniform intgrability, w bound th scond momnt of ths two objcts: and E B I ε n(t ncεt E u ε (t, x n] E B I ε n (t ncεt] E B I ε n (t nνε(t], n u (x + B k t ] E B I ε n(t nc εt ] E B (I ε n(t nν ε(t ], whr w hav usd u 1. Thus, it suffics to show that for vry n and θ, thr xists t small nough such that E B θ(iε n(t nν ε(t ] is uniformly boundd in ε for all t < t. To s this, using Höldr s inquality, w gt E B θ(i ε n (t nνε(t] n E B θnβk ε (,t] < E Bβ k ε (,t] < ] ] 1 N whr N = n(n+1, and w hav usd th notations β k ε (, t] < = t s R ε (B k s B k ududs, α i,j ε (, t] = 1 i<j n t t 1 (E B θnαi,j ε (,t] N, R ε (B i s B j udsdu. By chang of variabls and th scaling proprty of th Brownian motion, w hav β k ε (, t] < law = tβ k ε/ t (, 1] <, α i,j ε (, t] law = tα i,j ε/ t (, 1]. Thn, Lmma A.1 implis that thr xists λ, C > such that t < This complts th proof. λ θn sup E B θ(i ε(t nνε(t] n C. ε>

7 MOMENTS OF PAM 7 Appndix A. Exponntial momnts of intrsction local tim of planar Brownian motions Rcall that R ε (x = ε R( x, w dfin ε α ε (A = R ε (Bs 1 Budsdu, β ε (A = A for any st A R +, and A R ε (B s B u dsdu X ε = β ε (, 1] < E B β ε (, 1] <], Y ε = α ε (, 1]. Lmma A.1. Thr xists univrsal constants λ, C > such that ( EB λxε ] + E B λyε ] C. sup ε> Th abov rsult is standard. Th cas ε =, i.., th xponntial intgrability of intrsction local tim, was addrssd in th classical work 13]. W could not find a dirct rfrnc for ε >, though th proof follows ssntially in th sam lin as th cas of ε =. For th convninc of th radr, w prsnt th dtails hr. Proof. W considr Y ε first. Sinc R = ρ ρ, w can writ Y ε = ρ ε (Bs 1 xρ ε (Bu xdxdsdu, R,1] with ρ ε (x = ε ρ(x/ε. For any n N, ( n ] E B Yε n ] = ρ ε (Bs 1 k x k ρ ε (Bu k x k dsdu dx = R n R n (,1] n E B,1] n E B n ] ρ ε (B sk x k ds dx. By 3, (..11], w hav ( n E B Yε n ] = ρ ε (z k x k R n R n σ,1] n < n p sk s k 1 (z σ(k z σ(k 1 dsdz dx, whr p t (x is th dnsity of N(, t,, t] n < = { s 1 <... < s n t}, and σ dnots th summation ovr all prmutations ovr {1,..., n}. If w dnot h(z 1,..., z n = n n p sk s k 1 (z σ(k z σ(k 1 ds, Q ε (z 1,..., z n = ρ ε (z k, σ,1] n < thn E B Yε n ] quals to (A.1 ( Q ε h(x 1,..., x n dx Q ε (x 1,..., x n dx h(x 1,..., x n dx R n R n R n = h(x 1,..., x n dx = E B α(, 1] n ], R n

8 8 YU GU, WEIJUN XU whr α(, 1] is th mutual-intrsction local tim formally writtn as α(, 1] = 1 1 δ(b 1 s B udsdu, and w usd th L Gall s momnt formula in th scond lin of (A.1. By 13], w hav E B xp(µα(, 1] ] < C for som µ >, hnc w only nd to choos λ = µ to gt E B λyε ] = n= λ n E B Y n ε ] n! n= λ n n! E B α(, 1] n ] = E B µα(,1] ] <. Nxt, w considr X ε. W dfin th triangl approximation of {(u, s : u < s 1}: l A k l + 1 l + 1 l + l =,,, l =, 1,..., k 1, k =, 1,.... k+1 k+1 k+1 k+1 W will us th following thr proprtis: (i Fix any k, {β ε (A k l } l=,..., k 1 ar i.i.d. random variabls. (ii β ε (A k l law = (k+1 β ε (k+1/(, 1] 1, ] law = (k+1 α ε (k+1/(, 1] (iii sup ε> E B λαε(,1] ] C for som λ, C >. By (iii and a Taylor xpansion, thr xists C > such that for sufficintly small λ (A. sup E B λ(αε(,1] E B α ε(,1] ] ] Cλ. ε> W fix th constants λ, C from now on, and writ X ε = k 1 (β ε (A k l E B β ε (A k l ]. k= l= Fix a (, 1 and dfin a squnc of constants b 1 = λ, N b N = λ (1 a(j 1, N =, 3,..., j=

9 MOMENTS OF PAM 9 w hav E B xp b N E B xp N k 1 k= l= N 1 k 1 b N 1 (β ε (A k l E B β ε (A k l (β ε (A k l E B β ε (A k l k= l= 1 a(n 1 N 1 E B xp a(n 1 b N (β ε (A N l E B β ε (A N l E B xp N 1 k 1 b N 1 l= (β ε (A k l E B β ε (A k l k= l= a(n 1 ( E B xp a(n 1 b N (β ε (A N E B β ε (A N ] N a(n 1. Sinc β ε (A N law = (N+1 α ε (N+1/(, 1], w hav E B xp a(n 1 b N (β ε (A N E B β ε (A N ] =E B xp a(n 1 b N (N+1 (α ε (N+1/(, 1] E B α ε (N+1/(, 1] ]. Using th fact that a(n 1 b N (N+1 < λ and (A., w driv for all ε > that E B xp a(n 1 b N (β ε (A N E B β ε (A N ] Cb N N+a(N 1, so thr xists C > such that N k 1 E B xp b N (β ε (A k l E B β ε (A k l E B xp k= l= (β ε (A k l E B β ε (A k l C (a 1N. N 1 k 1 b N 1 k= l= Itrating th abov inquality, w gt N k 1 E B xp b N (β ε (A k l E B β ε (A k l xp(c (1 a 1 1 k= l= Sinc b N b for som b >, w hav which complts th proof. E B xp(b X ε ] xp(c (1 a 1 1, Acknowldgmnts. W thank Dirk Erhard and Nikolaos Zygouras for stimulating discussions and for showing us th argumnt in Rmark 1.7. YG is partially supportd by th NSF through DMS WX is supportd by EPSRC through th rsarch fllowship EP/N1568/1.

10 1 YU GU, WEIJUN XU Rfrncs 1] R. Allz and K. Chouk, Th continuous Andrson hamiltonian in dimnsion two, arxiv prprint arxiv: , (15. ] G. Cannizzaro and K. Chouk, Multidimnsional SDEs with singular drift and univrsal construction of th polymr masur with whit nois potntial, arxiv prprint arxiv: , (15. 3] X. Chn, Random walk intrsctions: Larg dviations and rlatd topics, no. 157, Amrican Mathmatical Soc., 1. 4] K. Chouk, J. Gairing, and N. Prkowski, An invarianc principl for th twodimnsional parabolic Andrson modl with small potntial, arxiv prprint arxiv: , (16. 5] D. Erhard and N. Zygouras, privat communication. 6] Y. Gu, T. Komorowski, and L. Ryzhik, Th SPDE limit for th random Schrödingr quation: th avrag wav function, in prparation, 17. 7] M. Gubinlli, P. Imkllr, and N. Prkowski, Paracontrolld distributions and singular pds, in Forum of Mathmatics, Pi, vol. 3, Cambridg Univ Prss, 15, p. 6. 8] M. Hairr, A thory of rgularity structurs, Invntions mathmatica, 198 (14, pp ] M. Hairr and C. Labbé, Multiplicativ stochastic hat quations on th whol spac, arxiv prprint arxiv: , (15. 1], A simpl construction of th continuum parabolic andrson modl on R, Elctronic Communications in Probability, (15. 11] Y. Hu, Chaos xpansion of hat quations with whit nois potntials, Potntial Analysis, 16 (, pp ] J.-F. L Gall, Som proprtis of planar brownian motion, in Ecol d Eté d Probabilités d Saint-Flour XX-199, Springr, 199, pp ], Exponntial momnts for th rnormalizd slf-intrsction local tim of planar brownian motion, in Séminair d Probabilités XXVIII, Springr, 1994, pp ] S. Varadhan, Appndix to uclidan quantum fild thory by k. symanzik, Local Quantum Thory. Acadmic Prss, Rading, MA, 1 ( ] M. Yor, Prcisions sur l xistnc t la continuit ds tmps locaux d intrsction du mouvmnt brownin dans R, in Séminair d Probabilités XX 1984/85, Springr, 1986, pp (Yu Gu Dpartmnt of Mathmatics, Building 38, Stanford Univrsity, Stanford, CA, 9435, USA (Wijun Xu Mathmatics Institut, Univrsity of Warwick, Covntry, CV4 7AL, UK