Semi-discrete semi-linear parabolic SPDEs

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1 Semi-dicree emi-linear parabolic SPDE Nico Georgiou Univeriy of Suex Davar Khohnevian Univeriy of Uah Mahew Joeph Univeriy of Sheffield Shang-Yuan Shiu Naional Cenral Univeriy La Updae: November 2, 213 Abrac Conider he emi-dicree emi-linear Iô ochaic hea equaion, u x = L u x + σu x B x, ared a a non-random bounded iniial profile u : Z d R +. Here: {Bx} x Z d i an field of i.i.d. Brownian moion; L denoe he generaor of a coninuou-ime random walk on Z d ; and σ : R R i Lipchiz coninuou and non-random wih σ =. The main finding of hi paper are: i The kh momen Lyapunov exponen of u grow exacly a k 2 ; ii The following random Radon Nikodým heorem hold: lim τ u +τ x u x B +τ x B x = σux in probabiliy; iii Under ome non-degeneracy condiion, here ofen exi a cale funcion S : R,, uch ha he finie-dimenional diribuion of x {Su +τ x Su x}/ τ converge o hoe of whie noie a τ ; and iv When he underlying walk i ranien and he noie level i ufficienly low, he oluion can be a.. uniformly diipaive provided ha u l 1 Z d. Keyword: The ochaic hea equaion; ineracing diffuion. Primary 6J6, 6K35, 6K37; Sec- AMS 2 ubjec claificaion: ondary: 47B8, 6H25. Reearch uppored in par by he NSF gran DMS N.G.; M.J. and DMS M.J.; D.K., he NSC gran M-8-1-MY2 S.-Y.S., and he NCU gran 12G67-3 S.-Y.S.. 1

2 1 Inroducion Conider he following emi-dicree ochaic hea equaion, du x d = L u x + σu x db x, SHE d where {Bx} x Z d i a field of independen andard linear Brownian moion, L denoe he generaor of a coninuou-ime random walk X := {X } := { N j=1 Z j} on Z d where N i a Poion proce wih jump-rae one and he Z j are i.i.d. random variable aking value on Z d, and σ : R R i a Lipchiz-coninuou non-random funcion wih σ =. 1.1 I i well-known ha if he iniial ae u : Z d R i non-random and bounded, hen SHE ha an a..-unique oluion in he ene of K. Iô; ee for example Shiga and Shimizu [36]. We will concenrae only on he cae ha u x for all x Z d, and up x Z d u x >, 1.2 hough ome of our heory work for more general iniial funcion, a well. Semi-dicree ochaic parial differenial equaion uch a SHE have been udied a grea lengh [12, 13, 17 21, 25, 28, 31, 35, 36], mo commonly in he conex of well-eablihed model of aiical mechanic or populaion geneic. The purpoe of hi aricle i o highligh ome uble local and global feaure of he oluion o SHE. For our fir reul, conider he [maximal] kh momen Lyapunov exponen γ k u := lim inf γ k u := lim up up 1 x Z log E u x k, d 1 up x Z log E u x k. d 1.3 In he very imporan pecial cae ha σx x and L := he generaor of a imple ymmeric walk on Z d hi i he ocalled parabolic Anderon model i i he frequenly he cae ha γ k u = γ k u < for all k More ignificanly, i i frequenly he cae ha γ k u > for all k 2 if and only d {1, 2}; ee he memoir of Carmona and Molchanov [13] for hee reul in he cae ha u i a conan, for inance. In he preen non-linear eing, one doe no expec he equaliy of he Lyapunov exponen γ k u and γ k u. Sill, our fir reul how ha, under ome inermiency condiion, he Lyapunov exponen are alway poiive 2

3 and finie, and ha he kh momen Lyapunov exponen grow a k 2, a k. Thi i conra wih coninuou SPDE where he Lyapunov exponen ypically grow exacly a k 3 a k [4 7, 24]. Wih he preceding aim in mind, le u define Lip σ := up <x y< σx σy x y, l σ := inf x R σx x. 1.5 Noe, in paricular, ha l σ x σx Lip σ x x R by 1.1; he upper bound i alway finie, and he lower bound i > for x iff l σ >. The following reul make he previou aerion more precie. For he ake of compleene, we include alo a careful exience-uniquene and poiiviy aemen, ince hoe aerion are free byproduc of he proof of he main par of he heorem, which involve he numerical [upper and lower] bound on he growh of he Lyapunov exponen. Theorem 1.1. The non-linear ochaic hea equaion SHE ha a oluion u ha i coninuou in he variable, and i unique among all predicable random field ha aify up [,T ] up x Z d E u x 2 < for all T >. Moreover, γ k u 8Lip 2 σk 2 for all ineger k Furhermore, u x for all and x Z d a.., provided ha u x for all x Z d. Finally, if l σ > and σx > for all x >, hen for all ε, 1, γ k u 1 εl 2 σk 2 for every ineger k ε 1 + εl 2 σ Sandard momen mehod which we will have o reproduce here a well how ha u x i almo urely a Hölder-coninuou random funcion for every Hölder exponen < 1 /2. The following prove ha he Hölder exponen 1/2 i harp. Theorem 1.2 A Radon Nikodým propery. For every and x Z d, In addiion, lim up τ almo urely. lim τ u +τ x u x B +τ x B x = σ u x almo urely. 1.8 u +τ x u x 2τ log log1/τ = lim inf τ u +τ x u x 2τ log log1/τ = σ u x, 1.9 Remark 1.3. Local ieraed-logarihm law, uch a 1.9 are well known in he conex of finie-dimenional diffuion; ee for inance Anderon [3, Theorem 4.1]. The ime-change mehod employed in he finie-dimenional eing will, however, no work effecively in he preen infinie-dimenional conex. Here, we obain 1.9 a a ready conequence of he proof of he random Radon Nikodým propery

4 Remark 1.4. Le u fix an x Z d and a >, and le u conider Rτ := [u +τ x u x]/[b +τ x B x]; hi i a well-defined random variable for every τ >, ince B +τ x B x wih probabiliy one for every τ >. However, {Rτ} τ> i no a well-defined ochaic proce ince here exi random ime τ > uch ha B +τ x B x = a.. Thu, one doe no expec ha he mode of convergence in 1.8 can be improved o almo-ure convergence. Thi aemen can be renghened furher ill, bu we will no do o here. According o 1.8, he oluion o he ochaic hea equaion behave a he non-ineracing yem du x σu xdb x of diffuion, locally o fir order. Thi migh eem o ugge he [fale] aerion ha x u x ough o be a equence of independen random variable. Tha i no he cae, a can be een by looking more cloely a he ime incremen of u x. In fac, our argumen can be exended o how ha he paial correlaion rucure of u appear a econd-order approximaion level in he ene of he following hree erm ochaic Taylor expanion in he cale τ 1 /2 : u +τ x u x + τ 1 /2 σu xz 1 + τz 2 + τ 3 /2 Uτ a τ, 1.1 where: i denoe approximaion in he ene of diribuion; ii Z 1 i a andard normal variable independen of u x; iii Z 2 i a non-rivial random variable ha depend on he enire random field {u y} [,],; and iv Uτ = O P 1 a τ. The laer mean ha lim m lim up τ P{ Uτ m} =. In paricular, 1.1 ell u ha he emporally-local ineracion in he random field x u x are econd order in naure. Raher han prove hee refined aerion, we nex urn our aenion o a differen local propery of he oluion o SHE and how ha, afer a cale change, he local-in-ime behavior of he oluion o SHE i ha of paial whie noie. Namely, we offer he following: Theorem 1.5. Suppoe σz > for all z R \ {}, and define Sz := z z dw σw z, 1.11 where z R\{} i a fixed number. Then, Su x < a.. for all > and x Z d. Furhermore, if we chooe and fix m diinc poin x 1,..., x m Z d, hen for all > and q 1,..., q m R, m { } m lim S u+τ x j S u x j q j τ = Φq j, 1.12 τ P j=1 where Φq := 2π 1/2 q exp w2 /2 dw denoe he andard Gauian cumulaive diribuion funcion. The preceding manife ielf in amuing way for differen choice of he non-lineariy coefficien σ. Le u menion he following parabolic Anderon model, which ha been a moivaing example for u. j=1 4

5 Example 1.6. Conider he emi-dicree parabolic Anderon model, which i SHE wih σx = c x [for ome fixed conan c > ]. In ha cae, he oluion o SHE i poiive [if u i] and he cale funcion S i Sz = c 1 lnz/z for z, z >. A uch, σu x = cu x in SHE, and we find he following log-normal limi law: For every > and x 1,..., x m Z d fixed, [u+τ ] 1/ τ x 1,..., u x 1 [ ] 1/ τ u+τ x m = e cn1,..., e cnm, 1.13 u x m a τ, where N 1,..., N m are i.i.d. andard normal variable, and denoe convergence in diribuion. Our final main reul i a aemen abou he large-ime behavior of he oluion u o SHE. We inend o prove a rigorou verion of he following aerion: If he random walk X i ranien and Lip σ i ufficienly mall o ha SHE i no very noiy hen a decay condiion uch a u l 1 Z d on he iniial profile i enough o enure ha up x Z d u x almo urely a. Thi i new even for he parabolic Anderon model, where σx x and L := he generaor of he imple walk on Z d. In fac, hi reul give a parial [hough rong] negaive anwer o an open problem of Carmona and Molchanov [13, p. 122] and rule ou he exience of [he analogue of] a nonrivial Anderon mobiliy edge in he preen non-aionary eing, when u l 1 Z d. We are aware only of one uch non-exience heorem, hi ime for he original aionary Anderon model on ree graph ; ee he recen paper by Aizenman and Warzel [2]. Recall ha X := {X } i a coninuou-ime random walk on Z d wih generaor L. Le X denoe an independen copy of X, and define Υ := P{X = X } d = E 1 {} X X d We can hink of Υ a he expeced value of he oal occupaion ime of {}, a viewed by he ymmerized random walk X X. Alhough Υ i alway well defined, i i finie if and only if he ymmerized random walk X X i ranien [14]. We are ready o ae our final reul. Theorem 1.7. Suppoe ha and ha here exi α 1, uch ha Lip σ < [Υ] 1/2, 1.15 P{X = X } = O α, 1.16 where X denoe an independen copy of X. If, in addiion, u l 1 Z d and he underlying probabiliy pace i complee, hen lim up u x = lim u x 2 = almo urely x Z d x Z d 5

6 Remark 1.8 Hyerei. Conider he parabolic Anderon model [σx x], where he underlying ymmerized walk X X i ranien, he noie level i mall, and u i a conan. I i well known ha, under hee condiion, u x converge weakly a o a non-void random variable u x for every x Z d. See, for example, Greven and den Hollander [28, Theorem 1.4], Cox and Greven [18], and Shiga [35]. Thee reul provide a parial affirmaive anwer o a queion of Carmona and Molchanov [13, p. 122] abou he exience of long-erm invarian law in he low-noie regime of he ranien parabolic Anderon model, in paricular. By conra, Theorem 1.7 how ha if u i far from aionary [here, i decay a infiniy], hen he yem i very rongly diipaive in he low-noie regime. Among oher range hing, hi reul ha he conequence ha he parabolic Anderon model remember i iniial ae forever. Remark 1.9. Coninuou-ime walk ha have he propery 1.16 include all ranien finie-variance cenered random walk on Z d [d > 2, necearily]. For hoe walk, α := d/2, hank o he local cenral limi heorem. There are more inereing example a well. For inance, uppoe 1/p X converge in diribuion o a able random variable S a. [See Gnedenko and Kolmogorov [27, 35] for neceary and ufficien condiion.] Then, S i necearily able wih index p, p, 2], and 1/p X X converge in law o a ymmeric able random variable S wih abiliy index p. If, in addiion, he group of all poible value of X X generae all of Z d, hen a heorem of Gnedenko [27, p. 236] enure ha 1/p P{X = X } converge o f <, where f denoe he probabiliy deniy funcion of S, a long a p, 1. We cloe hi inroducion wih ome background on Burkholder conan. According o he Burkholder Davi Gundy inequaliy [8 1], z p := up x up > E x p E x p/2 1/p <, 1.18 where he upremum up x i aken over all non-zero marinagle x := {x } ha have coninuou rajecorie and are in L 2 P a all ime, x denoe he quadraic variaion of x a ime, and / := / :=. Davi [22] ha compued he numerical value of z p in erm of zeroe of pecial funcion. In he pecial cae ha p = k where k 2 an ineger, Davi heorem implie ha z k i equal o he large poiive roo of he modified Hermie polynomial He k. Thu, for example, we obain he following from direc evaluaion of he zero: z 2 = 1, z 3 = 3, z 4 = , z 5 = , z ,.... I i known ha z p 2 p a p, and up p 2 z p / p = 2; ee Carlen and Kree [11, Appendix]. 6

7 2 Preliminarie 2.1 The mild oluion A i cuomary, by a oluion o SHE we mean a oluion in inegraed or mild form. Tha i, a predicable proce u, wih value in R Zd, ha olve he following infinie yem of Iô SDE: u x = p u x + where p x := P{X = x}, f gx := p y xσu y db y; 2.1 fx ygy x Z d 2.2 denoe he convoluion on Z d ; and for every funcion h : Z d R we define a new funcion h, hx := h x x Z d, 2.3 a he reflecion of h. I migh be helpful o noe alo ha P φx := p φ x define he emigroup of he random walk X via he ideniy P φx = Eφx + X. Thu we can wrie 2.1 in he following, perhap more familar, form: u x = P u x A BDG inequaliy p y xσu y db y, 2.4 Suppoe Z := {Z x},x Z d i a predicable random field, wih repec o he infinie-dimenional Brownian moion {B }, ha aifie he momen bound E Z 2 l 2 Z d d <. Then, he Iô inegral proce defined by Z db := Z y db y 2.5 exi and define a coninuou L 2 P maringale. Thi i par of he andard folklore of infinie-dimenional ochaic analyi; ee for example Prévô and Röckner [33]. The following variaion of he Burkholder Davi Gundy inequaliy yield momen bound for ha maringale ha alo pay pecial aenion o he conan in uch inequaliie. Lemma 2.1 BDG Lemma. For all finie real number k 2 and, k E Z db 4k k/2 { E Z y k} 2/k d

8 Proof. We follow a mehod of Foondun and Khohnevian [24]. A andard approximaion argumen ell u ha i uffice o conider he cae ha y Z y ha finie uppor. To be concree, le F Z d be a finie e of cardinaliy m 1, and uppoe Z y = for all y F. Conider he [andard, finie-dimenional] Iô inegral proce Z db := y F Z y db y. According o Davi [22] form of he Burkholder Davi Gundy inequaliy m-dimenional Brownian moion [8 1], k E Z db zke k k/2 [Z y] 2 d. 2.7 Finally, we ue he Carlen Kree bound z k 2 k [11] ogeher wih he Minkowki inequaliy o finih he proof in he cae ha F i finie. A andard finiedimenional approximaion complee he proof. y F 3 Proof of Theorem 1.1: Par 1 Exience and uniquene, and alo coninuiy, of he oluion are deal wih exenively in he lieraure and are well known; ee for example Shiga and Shimizu [36], and he general heory of Prévo and Röckner [33] for ome of he lae developmen. However, in order o derive our eimae of he Lyapunov exponen we will need a priori eimae which will alo yield exience and uniquene. Therefore, in hi ecion, we hah ou ome hough no all of he deail. Le u proceed by applying Picard ieraion. Le u x := u x, and hen define ieraively for all n, u n+1 x := p u x + p y xσ u n y db y. 3.1 I follow from he properie of he Iô inegral ha M n+1 := up E u n+1 x k 2 k 1 up I x + J x, 3.2 x Z d x Z d where I x := p u x k, J x := E The fir erm i eay o bound: p y xσ u n y k 3.3 db y. up I x u k l Z d, 3.4 x Z d 8

9 ince x p x = 1. Nex we bound J x. Becaue σ i Lipchiz coninuou and σ =, we can ee ha σz Lip σ z for all z R. Thu, we may ue he BDG lemma [Lemma 2.1] in order o ee ha J 2/k x 4kLip 2 σ [p y x] 2 {E u n y k} 2/k d. 3.5 Therefore, we may recall he inducive definiion 3.2 of M o ee ha ince J 2/k x 4kLip 2 σ 4kLip 2 σ M n [p y x] 2 M n 2/k d 2/k d, 3.6 z Z d [p r z] 2 = P{X r = X r} 1, 3.7 where X denoe an independen copy of X. [Thi la bound migh appear o be quie crude, and i i when r i large. However, i urn ou ha he behavior of r near zero maer more o u. Therefore, he inequaliy i igh in he regime r of inere o u.] We may combine 3.2, 3.4, and 3.6 in order o ee ha for all β, >, e β M n /k k/2 2 k 1 u k l Z d + 16kLip2 σ k/2 e 2β /k e β M n d. Conequenly, he equence defined by N m β := up e β M m aifie he recurive inequaliy N n+1 β 2 k 1 u k l Z d + 16kLip2 σ k/2 e 2β/k d 8k 2 k 1 u k 2 l Z d + Lip 2 k/2 σ N n β β. In paricular, if we denoe [emporarily for hi proof] where δ > i fixed bu arbirary, hen N n+1 αk 2 m 3.9 k/2 N n β 3.1 α := 81 + δlip 2 σ, k 1 u k l Z d δ k/2 N n αk

10 We may apply inducion on n now in order o ee ha up n N n αk < ; 2 equivalenly, for all k 2 here exi c k, uch ha u n up E x k c k e 81+δLip2 σ k2 for all x Z d Similarly, u n+1 E x u n x k 3.14 { } k = E p y x σ u n y σ u n 1 y db y 4kLip 2 σ k/2 E { 2 [p y x] 2 u n y u y} d k/2 n 1. Define L n+1 u n+1 := up E x u n x Z d x k 3.15 o deduce from he preceding, 3.7, and Minkowki inequaliy ha 4kLip 2 σ k/2 L n+1 4kLip 2 σ k/2 L n [p y x] 2 L n 2/k d k/2. 2/k d k/ Therefore, aifie K m αk 2 = up e αk2 L m 3.17 K n+1 αk 2 4kLip 2 σ k/2 k/2 e 2αk d K n αk 2 4Lip 2 k/2 σ K n 2α αk 2 2 k K n αk From hi we can conclude ha n= Kn αk <. Therefore, here exi a 2 random field u x uch ha lim n u n x = u x, where he limi ake place in L k P. I follow readily ha u olve SHE, and u aifie 1.6 hank o 3.13 and Faou lemma. Uniquene i proved by imilar mean, and we kip he deail. 1

11 4 A local approximaion heorem In hi ecion we develop a decripion of he local dynamic of he random field u in he form of everal approximaion reul. Our fir approximaion lemma i a andard ample-funcion coninuiy reul; i ae baically ha ouide a ingle null e, u +τ x = u x + O τ 1+o1/2 a τ, for all and x Z d. 4.1 The reul i well known, bu we need o be cauiou wih variou conan ha crop up in he proof. Therefore, we include he deail o accoun for he dependencie of he implied conan. Lemma 4.1. There exi a verion of u ha i a.. coninuou in wih criical Hölder exponen 1 /2. In fac, for every T 1, ε, 1 and k 2, [ ] k u x u x up up E up x Z d I, I 1 ε/2 <, 4.2 where up I denoe he upremum over all cloed ubinerval I of [, T ] ha have lengh 1. Proof. Owing o Minkowki inequaliy, where { E u +τ x u x k} 1/k Q1 + Q 2 + Q 3, 4.3 Q 1 := p +τ u x p u x, Q 2 := E k [p +τ y x p y x] σu y db y Q 3 := E +τ k p +τ y xσu y db y 1/k 1/k. 4.4 We eimae each iem in urn. Le J,+τ denoe he even ha he random walk X jump ome ime during he ime inerval, + τ. Becaue x Z d p +τ x p x = x Z d E 1{X+τ =x} 1 {X=x}; J,+τ 2PJ,+τ = 2 1 e τ 2τ,,

12 we obain he following eimae for Q 1 : By he BDG Lemma 2.1, Q 2 2 4k 4k Q 1 2 u l Z d τ. 4.6 [p +τ y x p y x] 2 { E Q up σu y k} 2/k d { E σu y k} 2/k d, 4.7 where Q := p +τ z p z 2 < <. 4.8 z Z d I i poible o find a real-variable eimae for Q uing 4.5; namely, Q z Z p +τ z p d z 2τ. Unforunaely, hi i no good enough for our preen need; we need o do a lile beer by howing ha Q 2τ 2 : Recall ha we can repreen X := N j= Y j, where Y :=, {Y j } j=1 i a equence of i.i.d. random variable and {N} i an independen rae-one Poion proce. Le ϕξ := E expiξ Y 1 denoe he characeriic funcion of he incremen of he coninuou-ime random walk X. I i an exercie in Poionizaion ha Ee iξ X = e 1 ϕξ for all ξ R d and. 4.9 Therefore, we appeal o he Pareval ideniy and find ha Q = 2π d e +τ 1 ϕξ e 1 ϕξ 2 dξ 2τ 2, 4.1 [ π,π] d uniformly for all,. Thi how ha { Q 2 2 8kτ 2 up E σu y k} 2/k d. Becaue σz Lip σ z for all z R, he already-proved bound 1.6 ell u ha here exi conan c, c k, [k 2] uch ha up E σu y k c k ke ck2 for all ineger k 2 and Therefore, Q 2 2 4c 1 c 2 ke 2ck τ Finally, we apply he BDG Lemma 2.1 o ee ha Q 2 3 4k +τ { [p +τ y x] 2 E σu y k} 2/k d 4kc 2 k +τ [p +τ y x] 2 e 2ck d,

13 owing o Becaue [p hy x] 2 1 for all h, we find ha Q 2 3 2c 2 ke 2ck+τ τ We combine 4.6, 4.12, and 4.14 and find ha for all ineger k 2, here exi a finie and poiive conan ã := ãt, k uch ha for every τ, 1, up up E u +τ x u x k ãeãt τ k/ x Z d,t The lemma follow from hi bound, and an applicaion of a quaniaive form of he Kolmogorov coninuiy heorem [34, Theorem 2.1, p. 25]. We omi he remaining deail, a hey are nowaday andard. Our econd approximaion lemma yield a runcaion error eimae for he nonlineariy σ. Lemma 4.2. Le U N x denoe he a..-unique oluion o SHE where σ i replaced by σ N, where σ N = σ on N, N, σ N = on [ N 1, N + 1] c, and defined by linear inerpolaion on [ N 1, N] [N, N + 1]. Then, lim N U N x = u x almo urely and in L k P for all k 2,, and x Z d. Proof. Since σ N i Lipchiz coninuou, Theorem 1.1 enure he exience and uniquene of U N for every N 1. Nex we noe, uing 2.1, ha u x U N x = T 1 + T 2, 4.16 where: T 1 := T 2 := p y x{σu y σ N u y}db y; { p y x σ N u y σ N U N } y db y Becaue σz Lip σ z, Lemma 2.1 implie ha { E T 1 k } 2/k i a mo 4kLip 2 σ [p y x] 2 { E u y k ; u y N} 2/k d We have E Y k ; Y N N k EY 2k, valid for all 2k-ime inegrable random variable Y. Therefore, { E T1 k } 2/k 4kLip 2 σ N 2 [p y x] 2 { E u y 2k} 2/k d

14 Becaue [p y x] 2 1, he already-proved bound 1.6 ell u ha { E T1 k } 2/k a k N 2 e 128Lip2 σ k d Aa ke Ak N 2, 4.2 where a k and A are uninereing finie and poiive conan; moreover, a k depend only on k. Thi eimae he norm of T 1. A for T 2, we ue he imple inequaliy σ N r σ N ρ C r ρ, ogeher wih he BDG Lemma 2.1 in order o find ha { E T2 k} 2/k bk up { u E y U N y k} 2/k d, 4.21 where b k i a conan dependen on σ and k. Conequenly, we combine hee bound o deduce ha { u D N := up E x U N x k} 2/k 4.22 x Z d aifie he recurion D N ãkeãk 2 N 2 + b k D N d, 4.23 where ã k, b k, and à are poiive and finie conan, and he fir wo depend only on k [wherea he laer i univeral]. An applicaion of he Gronwall inequaliy how ha up [,T ] D N = ON 2 a N, for every fixed value T,. Thi i more han enough o yield he lemma. Our nex approximaion reul i he highligh of hi ecion, and refine 4.1 by inpecing more cloely he main conribuion o he Oτ 1+o1/2 error erm in 4.1. In order o decribe he nex approximaion reul, we fir define for every fixed an infinie-dimenional Brownian moion B a follow: B τ x := B τ+ x B x x Z d, τ If we coninue o hold fixed, hen i i eay o ee ha {B x} x Z d i a collecion of independen d-dimenional Brownian moion. Furhermore, he enire proce B i independen of he infinie-dimenional random variable u, ince i i eay o ee from he proof of he fir par of Theorem 1.1 ha u i a meaurable funcion of {B y} [,],, which i herefore independen of B by he Markov propery of B. Now for every fixed and x Z d, conider he oluion u x o he following [auonomou/non-ineracing] Iô ochaic differenial equaion: du τ x = d p τ u x + σ dτ dτ ubjec o u x = u x. 14 u τ db τ x x, dτ 4.25

15 Noe, once again, ha B i independen of u. Moreover, up E p τ u x 2 up E u y 2 <, 4.26 τ> hank o he already-proved bound 1.6 and Cauchy Schwarz inequaliy. Therefore, 4.25 i a andard Iô-ype SDE, and hence ha a unique rong oluion. Theorem 4.3 The local-diffuion propery. For every, he following hold a.. for all x Z d : u +τ x = u τ x + O τ 3 /2+o1 a τ The proof of Theorem 4.3 hinge on hree echnical lemma ha we ae nex. Lemma 4.4. Chooe and fix, τ [, 1], and x Z d, and define A := B := +τ +τ p +τ y xσu y db y, σu x db x Then, for all real number k 2 here exi a finie conan C k > depending on k bu no on, τ, x and a finie conan C > no depending on, τ, x, k uch ha E A B k C k e Ck2 +1 τ 3k/ Lemma 4.5. For every k 2 and T 1 here exi a finie conan Ck, T uch ha for every τ, 1], up up u+τ x u τ x k Ck, T τ 3k/ E [,T ] x Z d Lemma 4.6. There exi a verion of u ha i a.. coninuou in, τ. Moreover, for every T 1, ε, 1 and k 2, [ ] u ν x u k µ x up up up E up [,T ] x Z d I ν,µ I ν µ 1 ε/2 <, 4.31 ν µ where up I denoe he upremum over all cloed ubinerval I of [, T ] ha have lengh 1. In order o mainain he flow of he dicuion we prove Theorem 4.3 fir. Then we conclude hi ecion by eablihing he hree upporing lemma menioned above. 15

16 Proof of Theorem 4.3. Throughou he proof we chooe and fix ome [, T ] and x Z d. Our plan i o prove ha for all δ, 1 /2, u +τ x u τ x = O τ 3 /2 δ a τ, a Henceforh, we chooe and fix ome δ, 1 /2, and denoe by A k, A k, A k, ec. finie conan ha depend only on a parameer k 2 ha will be eleced laer on during he coure of he proof. Thank o Lemma 4.5, for all k 2 and τ [, 1], { P u+τ x u τ x 1 } 3 τ 3 /2 δ Ck, T τ δk We can chooe k large enough and hen apply he Borel Canelli lemma in order o deduce ha wih probabiliy one, u +τn x u τ n x < τ 3 /2 δ n for all bu a finie number of n, 4.34 where τ n := n δ 1 /2. Becaue τ n τ n+1 con n 1 τ n a n, Hölder coninuiy enure he following [Lemma 4.1 and 4.6]: Uniformly for all τ [τ n+1, τ n ], u +τ x u +τn x + u τ n x u τ x = O [τ n /n] 1 /2 δ a.. = O τ 3 /2 δ n, 4.35 hank o he paricular choice of he equence {τ n } n=1. diplay can now be combined o imply The preceding wo Proof of Lemma 4.4. We may rewrie B a follow: B = +τ 1 {} y xσu y db y Therefore, he BDG Lemma 2.1 can be ued o how ha { E A B k} 2/k 4k +τ 4kc 2 ke 2ck+1 4kc 2 ke 2ck+1 2 [ p+τ y x 1 {} y x ] 2 { E σu y k} 2/k d \{} τ τ [1 p ] 2 d, τ [p y] 2 d + [1 p ] 2 d

17 where c, c k appear in Nex we migh oberve ha p = P{X = } P{N = } = e, where {N } denoe he underlying Poion clock. Therefore, we obain τ [1 p ] 2 d 1/3τ 3, and hence E A B k 8/3 k/2 k k/2 c k ke ck2 +1 τ 3k/ Thi implie he lemma. Proof of Lemma 4.5. In accord wih 2.1, we may wrie u +τ x a p +τ u x + p +τ y xσu y db y + A, 4.39 where A wa defined in Lemma 4.4. By he Chapman Kolmogorov propery of he raniion funcion {p }, p τ u x = p +τ u x + p +τ y xσ u y db y. 4.4 The exchange of ummaion wih ochaic inegraion can be juified, uing he already-proved momen bound 1.6 of Theorem 1.1; we omi he deail. Inead, le u apply hi in 4.39 o ee ha +τ u +τ x = p τ u x + = p τ u x + τ σu x db x + A B σu + x d B x + A B. Lemma 4.4 implie ha for all k 2,, τ, and x Z d, τ E u +τ x p τ u x σu + x d B x k a k e ak2 +1 τ 3k/2, where a, i univeral and a k, depend only on k. On he oher hand, τ u τ x p τ u x σ u x db x = a.., 4.43 by he very definiion of u, and hank o he fac ha u y = u y. The preceding wo diplay and Minkowki inequaliy ha ψτ := { E u+τ x u τ x k} 1/k a 1/k k e ak+1 τ 3/2 + Q,

18 where Q := { E τ Q 2 4kLip 2 σ [ σu + x σ τ u ] x d B x k } 1/k According o he BDG Lemma 2.1 [acually we need a one-dimenional verion of ha lemma only], and ince σr σρ Lip σ r ρ, τ { E u+x u x k} 2/k d Thu, we find ha = 4kLip 2 σ [ψ] 2 d. [ψτ] 2 2a 2/k k e 2ak+1 τ 3 + 8kLip 2 σ τ 4.46 [ψ] 2 d for all τ The lemma follow from hi and an applicaion of Gronwall lemma. Proof of Lemma 4.6. One can model cloely a proof afer ha of Lemma 4.1. However we omi he deail, ince hi i a reul abou finie-dimenional diffuion and a uch impler han Lemma 4.1. We conclude hi ecion wih a final approximaion lemma. The nex aerion how ha he oluion o SHE depend coninuouly on i iniial funcion [in a uiable opology]. Lemma 4.7. Le u and v denoe he unique oluion o SHE, correponding repecively o iniial funcion u and v. Then, up E u x v x 2 u v 2 l Z d elip2 σ for all x Z d Proof. Chooe and fix. The fac ha p y = 1 alone enure ha up p u x p v x u v l Z d x Z d Therefore, 2.1 and Iô iomery ogeher imply ha E u x v x u v 2 l Z d + Lip2 σ p 2 l 2 Z d up E u y v y 2 d. Since p 2 l 2 Z d = P{X = X } 1, where X i an independen copy of X, we may conclude ha f := up x Z d E u x v x 2 aifie f u v 2 l Z d + Lip2 σ Therefore, he lemma follow from Gronwall inequaliy. f d

19 5 Proof of Theorem 1.1: Par 2 We now reurn o he proof of Theorem 1.1, and complee i by verifying he wo remaining aerion of ha heorem: i The oluion i nonnegaive becaue u x and σ = ; and ii The lower bound 1.7 for he lower Lyapunov exponen hold. I i be o keep he wo par eparae, a hey ue differen idea. Theorem 5.1 Comparion principle. Suppoe u and v are he oluion o SHE wih repecive iniial funcion u and v. If u x v x for all x Z d, hen u x v x for all and x Z d a.. The nonnegaiviy aerion of Theorem 1.1 i well known [35], bu alo follow from he preceding comparion principle. Thi i becaue he condiion 1.1 implie ha v x i he unique oluion o SHE wih iniial condiion v x. Therefore, he comparion principle yield u x v x = a.. Proof of Theorem 5.1. Conider he following infinie dimenional SDE: w x = w x + L w x d + σw xdb x x Z d. 5.1 I i a well-known fac ha he mild oluion o SHE i alo a oluion in he weak ene. See, for example, Theorem 3.1 of Iwaa [3] and i proof. Therefore, u x and v x repecively olve 5.1 wih iniial condiion u x and v x. Le {S n } n=1 denoe a growing equence of finie ube of Z d ha exhau all of Z d. Conider, for every n 1, he ochaic inegral equaion, wn x = w x + L w n x d + σw n xdb x if x S n ; w n x = w x if x / S n. 5.2 Similarly, we le v n olve he ame equaion, bu ar i a v x. Each of hee equaion i in fac a finie-dimenional SDE, and ha a unique rong oluion, by Iô heory. Moreover, Shiga and Shimizu proof of heir Theorem 2.1 [36] how ha, for every x Z d and >, here exi a ubequence {n k } k=1 of increaing ineger uch ha w n k x P w x and v n k x P v x, 5.3 a k. Therefore, we may appeal o a comparion principle for finiedimenional SDE, uch a ha of Geiß and Manhey [26, Theorem 1.2], in order o conclude he reul; he quai-monooniciy condiion of [26] i me imply becaue L i he generaor of a Markov chain. The verificaion of ha mall deail i lef o he inereed reader. We are now in poiion o eablih he lower bound 1.7 on he boom Lyapunov exponen of he oluion o SHE. 19

20 Proof of Theorem 1.1: Verificaion of 1.7. Le v olve he ochaic hea equaion dv x = L v x d + l σ v x db x, 5.4 ubjec o v x := u x. Alo define V N o be he oluion o x = L v x d + ζ N V N x db x, 5.5 dv N where ζ N x = l σ x on N, N, ζ N x = when x N + 1, and ζ N i defined by linear inerpolaion everywhere ele. Define σ N and U N a in Lemma 4.2. Becaue σ N ζ N everywhere on R +, and ince boh U N and V N are a.. and poinwie, he comparion heorem of Cox, Fleichmann, and Greven [17, Theorem 1] how u ha V N E x k U N E x k, 5.6 for all, x Z d, k 2, and N 1. Le N and apply Lemma 4.2 o find ha V N x v x and U N x u x in L k P for all k 2. Therefore, he preceding diplay how u ha E v x k E u x k. 5.7 Therefore, i uffice o bound γ k v from below. Le {X i } k i=1 denoe k independen copie of he random walk X. Then i i poible o prove ha E v x k k = E u X j + x e M k, 5.8 j=1 where M k denoe he muliple colliion local ime, M k := 2l 2 σ 1 i<j k 1 {} X i X j d. 5.9 In he cae ha X i he coninuou-ime imple random walk on Z d, hi i a well-known conequence of a Feynman Kac formula; ee, for inance, Carmona and Molchanov [13, p. 19]. When X i replaced by a Lévy proce, Conu [15] ha found an elegan derivaion of hi formula. The cla of all Lévy procee include ha of coninuou-ime random walk, whence follow 5.8. Finally, we noe ha if every walk X 1,..., X k doe no jump in he ime inerval [, ], hen cerainly k j=1 u X j + x e Mk [u x] k e kk 1l2 σ

21 Since he probabiliy i exp ha X j doe no jump in [, ], i follow from he independence of X 1,..., X k ha E v x k [u x] k exp {[ kk 1l 2 σ k ] }, 5.11 whence γ k u γ k v kk 1l 2 σ k, 5.12 ince u i no idenically zero. If k i a lea ε 1 + εl 2 σ 1, hen we cerainly have kk 1l 2 σ k 1 εk 2 l 2 σ, and he heorem follow. 6 Proof of Theorem 1.2 Theorem 1.2 i a conequence of he following reul. Propoiion 6.1. For every, he following hold a.. for all x Z d : u +τ x u x = σ u x {B +τ x B x} + o τ 1+o1 a τ. 6.1 Indeed, we obain 1.8 from hi propoiion, imply becaue well-known properie of Brownian moion imply ha for all ε, 1 /2 and, lim τ τ 1 ε = in probabiliy. 6.2 B +τ x B x Moreover, 1.9 follow from he local law of he ieraed logarihm for Brownian moion. I remain o prove Propoiion 6.1. Proof. According o 4.42, for every ineger k 2, and all, τ and x Z d, τ E u +τ x u x σu + x d B 2 k 1 [ a k e ak2 +1 τ 3k/2 + E x u x p τ u x k]. k 6.3 We may wrie E u x p τ u x k = E u x k p τ y xu y y Z d 6.4 = E u xp{x τ } k p τ y xu y. \{x} 21

22 Becaue P{X τ } = 1 exp τ τ, Minkowki inequaliy how ha { E u x p τ u x k} 1/k τ { E u x k} 1/k + \{x} p τ y x { E u y k} 1/k 2τ up { E u y k} 1/k. 6.5 We can conclude from hi developmen, and from Theorem 1.1, ha here exi A k <, depending only on k, and a univeral A < uch ha τ E u +τ x u x σu + x d B x k A k e Ak2 +1 [ τ 3k/2 + τ k] A k e Ak2 +1 τ k, 6.6 for all τ [, 1]. Now, we may apply he BDG Lemma 2.1 in order o ee ha { E τ σu + x d B = { E 4kLip 2 σ τ τ x σu xb τ {σu + x σu x} d B x + σu xbx x k } 2/k k } 2/k { E u + x u x k} 2/k d. 6.7 Thank o 4.15, up x Z d { E τ σu + x d B x σu xb τ x k } 2/k τ ã k eã d con τ Therefore, we can deduce from 6.6 ha E Dτ k c k, τ k τ 1, 6.9 where Dτ := u +τ x u x σu x {B +τ x B x}, 6.1 and c k, i a finie conan ha depend only on k and ; in paricular, c k, doe no depend on τ. Now we chooe and fix ome η > ξ > uch ha 22

23 η+ξ < 1 /2, and hen apply he Chebyhev inequaliy, and he preceding wih any choice of ineger k > ξ 1, in order o ee ha n=1 P{ Dn η > n η ξ } c k, n=1 n ξk <. Thu, D n η = O n η ξ a.., 6.11 hank o he Borel Canelli lemma. Becaue n η n + 1 η = On 1 η, he modulu of coninuiy of Brownian moion, ogeher wih Lemma 4.1, imply ha up D n η Dτ = O n+1 η τ n η n 1 /2 = o n η+ξ a Therefore a andard monooniciy argumen and 6.11 ogeher reveal ha D = O η ξ/η a, a.. Since η > ξ are arbirary poiive number, i follow ha lim up log D/ log 1 a.. Thi i anoher way o ae he reul. 7 Proof of Theorem 1.5. Fir we prove a preliminary lemma ha guaranee ric poiiviy of he oluion o he SHE. We follow he mehod decribed in Conu, Joeph, and Khohnevian [16, Theorem 5.1], which in urn borrowed heavily from idea of Mueller [31] and Mueller and Nualar [32]. Lemma 7.1. inf T u x > a.. for every T, and all x Z d ha aify u x >. Proof. We are going o prove ha if u x > for a fixed x Z d, hen here exi finie and poiive conan A and C uch ha { } P inf u x ε Aε C log log ε, 7.1 << for ha ame poin x, uniformly for all ε, 1. I urn ou o be convenien o prove he following equivalen formulaion of he preceding: { } P inf u x e n An Cn, 7.2 << imulaneouly for all n 1, afer a poible relabeling of he conan A, C,. If o, hen we can imply le n and deduce he lemma. Wihou lo of oo much generaliy we aume ha u >, and aim o prove 7.2 wih x =. In fac, we will implify he expoiion furher and eablih 7.2 in he cae ha u = 1; he general cae follow from hi one and caling. Finally, we appeal o he comparion heorem Theorem 5.1 in order o reduce our problem furher o he pecial cae ha u x = δ x for all x Z d

24 Thu, we conider hi cae only from now on. Le F := σ{b x : x Z d, < } decribe he filraion generaed by ime by all he Brownian moion, enlarged o ha u i a CR-valued [rong] Markov chain. Se T :=, and define ieraively for k he equence of {F } > -opping ime T k+1 := inf { > T k : u e k 1}, 7.4 uing he uual convenion ha inf :=. We may oberve ha he preceding definiion imply ha, almo urely on {T k < }, u Tk x e k δ x for all x Z d. 7.5 We plan o apply he rong Markov propery. In order o do ha, we fir define u k+1 o be he unique coninuou oluion o he SHE for ame Brownian moion, pahwie, wih iniial daa u k+1 x := e k δ x. Nex we noe ha, for every k, he random field w k+1 x := e k u k+1 x 7.6 olve he yem dw k+1 x = L w k+1 x + σ k d w k+1 x = δ x, w k+1 db x x d 7.7 where σ k y := e k σe k y. Becaue σ =, we have Lip σk = Lip σ, uniformly for all k 1. Thu, we can keep rack of he conan in he proof of Lemma 4.1, in order o deduce he exience of a finie conan K := Kε o ha for all, wih < 1, E up < <1 for all real number m 2. For each k le u define w k+1 w k+1 m Km 2 e Km2 m1 ε/2, 7.8 T k+1 1 = inf { > : w k+1 e 1}. 7.9 Equaion 7.5, he rong Markov propery, and he comparion principle [Theorem 5.1] ogeher imply ha ouide of a null e, he oluion o he revied SPDE 7.7 aifie e k w k+1 x u Tk +x. 7.1 Therefore, in paricular, T k+1 1 T k+1 T k,

25 and he opping ime T k+1 1 and T l+1 1 are independen if k l. For every < 1, { } P T k+1 1 < P { up w k+1 << } w k+1 1 e 1 Kεm 2 e Km2 1 e 1 m 1 εm/2, 7.12 where he la inequaliy follow by Chebyhev inequaliy and 7.8, and i valid for all < ε < 1. Le u emphaize ha he conan of he bound in 7.12 doe no depend on he parameer k which appear in he upercrip of he random variable T k+1 1. Now we compue { P inf u e n < } = P{T n } 7.13 = P{T n T n T 1 T } { } P T n 1 + T n T 1 1, owing o The erm T n 1,..., T 1 1, ha appear in he ulimae line of 7.13, are independen non-negaive random variable. Thank o he pigeon-hole principle, if he um of hoe erm i a mo, hen cerainly i mu be ha a lea n/2 of hoe erm are a mo /2n. If n i an even ineger, larger han > 2, hen a imple union bound on 7.13 and 7.12 yield { } P inf u e n < n Kε n/2 m n e Km2 n 1 e 1 mn 1 εmn/2 2n 1 εmn/2 n/2 Kε n 4 n m n e Km2 n 1 e 1 mn 1 εmn/4 2n 1 εmn/ Now we e m := log n/ log log n in 7.14 in order o deduce 7.2 for x = and every n 1 ufficienly large. Thi readily yield 7.2 in i enirey, and conclude hi demonraion. Nex we how ha if we ar wih an iniial profile u uch ha u x > for a lea one poin x Z d, hen u z > for all z Z d and > a.. Becaue we are inereed in eablihing a lower bound, we may apply caling and a comparion heorem Theorem 5.1 in order o reduce our problem o he pecial cae ha u = δ In hi way, we are led o he following repreenaion of he oluion: u x = p x + p y xσ u y db y

26 Propoiion 7.2. If u = δ, hen u x > for all x Z d and > a.. Propoiion 7.2 follow from a few preparaory lemma. Lemma 7.3. If u = δ, hen E u y 2 e Lip2 σ [p y] 2 for all > and y Z d Proof. We begin wih he repreenaion 7.16 of he oluion u, in inegral form, and appeal o Picard ieraion in order o prove he lemma. Le u x := 1 for all, x Z d, and hen le {u n+1 } n be defined ieraively by u n+1 x := p x + Le u define M k p y xσ := up x Z d E u n+1 u n y db y x 2, 7.19 p x and apply Iô iomery in order o deduce he recurive inequaliy for he M k : [ ] 2 M n Lip 2 p y xp y σ up M n d. 7.2 x Z p d x y Becaue [fy]2 [ fy]2 for all f : Z d R +, he emigroup propery of {p } > yield he bound [p y xp y] 2 [p x] 2, 7.21 whence M n Lip 2 σ M n d for all > and n. I follow readily from hi ha M n explip 2 σ, uniformly for all n and > ; equivalenly, E u n x 2 e Lip2 σ [p x] 2, 7.22 uniformly for all n, x Z d, and >. The lemma follow from hi and Faou lemma, ince u n x u x in L 2 P a n. Our nex lemma how ha he random erm on he righ-hand ide of 7.16 i mall, for mall ime, a compared wih he nonrandom erm in Lemma 7.4. Aume he condiion of Propoiion 7.2. Then here exi a finie conan C > uch ha for all, 1, up P x Z p y xσ u y db y d y Z > p x C d 26

27 Proof. By Lemma 7.3 and Iô iomery, 2 E p y xσ u y db y 7.24 y Z d Lip 2 2 σ σ d Lip 2 σ[p x] 2 e Lip2 σ d, [p y xp y] 2 e Lip where we have ued 7.21 in he la inequaliy. Becaue explip2 σ d c for all, 1 wih c := explip 2 σ, he lemma follow from Chebyhev inequaliy. Now we can eablih Propoiion 7.2. Proof of Propoiion 7.2. Le u chooe and fix an arbirary x Z d. By he rong Markov propery of he oluion, and hank o Lemma 7.1, we know ha once he oluion become poiive a a poin, i remain poiive a ha poin a all fuure ime, almo urely. Thu, i uffice o how ha u x > for all ime of he form = 2 k, when k i a large enough ineger. Bu hi i immediae from 7.16 and 7.23, hank o he Borel-Canelli lemma. The preceding lemma lay he groundwork for he proof of Theorem 1.5. We now proceed wih he main proof. Proof of Theorem 1.5. Le u fir conider he cae ha m = 1 and wihou lo of generaliy, x 1 =. In ha cae, we wrie limp { Su +τ Su q τ } 7.25 τ { } u+τ dy = lim P τ u σy q τ { u+τ 1 = lim P τ σy 1 dy + u +τ u q } τ. σu σu u Lemma 7.1 and he poiiviy condiion on σ enure ha σu > a.. Therefore, he heorem follow from Theorem 1.2 if we were o how ha 1 u+τ τ u 1 σy 1 dy almo urely, a τ σu Le I, + τ denoe he random cloed inerval wih endpoin u and u +τ. Our ric poiiviy reul [Lemma 7.1] implie ha I, + τ, for all, τ > a..,

28 and hu pave way for he a.. bound u+τ 1 σy 1 dy σu Lip u u +τ 2 σ inf y I,+τ σy 2 u = O τ log log τ τ ; ee 1.9 for he la par. Thi implie 7.26 and hu complee our proof for m = 1. The proof for general m i an eay adapion ince {Bx j } m j=1 are i.i.d. Brownian moion. 8 Preliminarie for he proof of Theorem 1.7 The following funcion will play a prominen role in he enuing analyi: P τ := p τ 2 l 2 Z d = x Z d [p τ x] 2 for all τ. 8.1 Becaue of he Chapman Kolmogorov propery, we can alo hink of P a P τ := P{X τ X τ = }, 8.2 where X i an independen copy of X. There i anoher ueful way o hink of P a well. Namely, we apply 4.9 and he Plancherel heorem o ee ha P τ = 2π d π,π d E expiξ X τ 2 dξ = 2π d π,π d e 2τ1 Re ϕξ dξ, 8.3 where ϕξ = E[expiξ Z 1 ], recall ha Z 1 i he diribuion of jump ize. Therefore, in paricular, he Laplace ranform of P i Υβ := e βτ P τ dτ β 8.4 = 2π d dξ β + 21 Re ϕξ. π,π d The inerchange of he inegral i juified by Tonelli heorem, ince 1 Re ϕξ. Noe ha Υ agree wih Alo, he claical heory of random walk ell u ha X X i ranien if and only if Υ = P τ dτ <, which i in urn equivalen o he condiion, dξ < ; Re ϕξ π,π d hi i he Chung Fuch heorem [14], ranlieraed o he eing of coninuouime ymmeric random walk hank o a andard Poionizaion argumen which we feel free o omi. 28

29 Lemma 8.1. If u l 2 Z d, hen u l 2 Z d a.. for all. Moreover, for every β uch ha Lip 2 συβ < 1, E u 2 l 2 Z d u 2 l 2 Z d eβ 1 Lip 2 συβ for all. 8.6 Proof. Le u x := u x for all and x Z d, and define u k o be he reuling kh-ep approximaion o u via Picard ieraion. I follow ha E u n+1 x 2 = p u x 2 + σ [p y x] 2 E u n 2 y d p u x 2 + Lip 2 σ [p y x] 2 E u n y 2 d. 8.7 We may add over all x Z d o deduce from hi and Young inequaliy ha u u n+1 E u 2 l 2 Z d +Lip2 n σ P E d l 2 Z d 2 l 2 Z d Since Υβ = β 1 exp P /β d β 1 <, we can find β > large enough o guaranee ha Lip 2 συβ < 1. We muliply boh ide of 8.8 by exp β for hi choice of β and noice from 8.8 ha A k := up [ e β E u k 2 l 2 Z d ] k 8.9 aifie A n+1 u 2 l 2 Z d + Lip2 συβa n for all n. 8.1 Since A = u 2 l 2 Z d, he preceding how ha up n A n i bounded above by 1 Lip 2 συβ 1 u 2 l 2 Z d. Propoiion 8.2. If u l 1 Z d, hen for every β uch ha Lip 2 συβ < 1, Moreover, e β E u 2 l 2 Z d d u 2 l 1 Z d Υβ 1 Lip 2 συβ for all β uch ha l 2 συβ 1. e β E u 2 l 2 Z d d =,

30 Proof. We proceed a we did for lemma 8.1. Bu inead of deducing 8.8 from 8.7, we ue a differen bound for p u l2 Z d u n+1 E 2 l 2 Z d p 2 l 2 Z d u 2 l 1 Z d + Lip2 σ = P u 2 l 1 Z d + Lip2 σ P E u n P E u n 2 l 2 Z d 2 l 2 Z d d, d 8.13 hank o a lighly differen applicaion of Young inequaliy. If we inegrae boh ide [exp βd], hen we find ha u I k := e β k E d k 8.14 aifie I n+1 u 2 l 1 Z d 2 l 2 Z d e β P d + In Lip 2 σ = u 2 l 1 Z d Υβ + I nlip 2 συβ; e β P d 8.15 ee 8.4. The fir porion of he lemma follow from hi, inducion, and Faou lemma ince Lip 2 συβ < 1. Nex, le u uppoe ha l 2 συβ 1. The following complimenary form of 8.13 hold [for he ame reaon ha 8.13 held]: E u 2 l 2 Z d p u 2 l 2 Z d + l2 σ I i no hard o verify direcly ha P E u 2 l 2 Z d d p u 2 l 2 Z d u2 x p 2 l 2 Z d, 8.17 whence, by u x > for ome x >, i follow ha F := E u 2 l 2 Z d 8.18 olve he renewal inequaliy, F u 2 x P + l 2 σ P F d Therefore, F β := exp βf d aifie F β u 2 x Υβ + l 2 συβ F β. 8.2 Since u x > and Υβ > for all β, i follow ha F β = whenever l 2 συβ 1. 3

31 Propoiion 8.3. If u l 1 Z d, hen up up u x <, x Z d σu y 2 d < a Moreover: i If, in addiion, q := Lip 2 συ < 1, hen E up up u x 2 E up u 2 l 1 Z d u 2 l 1 Z d + 4q x Z d 1 q ii If, in addiion, l 2 συ 1, hen E up u 2 l 1 Z d = E u 2 l 2 Z d d = Remark 8.4. Clearly, 8.21 implie ha if u l 1 Z d, hen lim inf up σu x 2 lim inf σu x 2 = a x Z d x Z d Suppoe, in addiion, ha l σ > [ay]. Then, we can deduce from he preceding fac ha lim inf up x Z d u x = a.. Recall ha X X i ranien if and only if Υ <. Therefore, in order for he condiion Lip 2 συ < 1 o hold, i i neceary hough no ufficien ha X X be ranien. Proof of Propoiion 8.3. Fir of all, Theorem 1.1 aure u ha u x a.., and hence u l 1 Z d = x Z d u x. Therefore, if we add boh ide of 2.1 hen we find ha u l 1 Z d = u l 1 Z d + σu y db y [I i eay o apply he momen bound of Theorem 1.1 o juify he inerchange of he um and he ochaic inegral.] In paricular, i follow ha M := u l1 Z d 8.26 define a non-negaive coninuou maringale wih mean u l1 Z d. I quadraic variaion aifie he following relaion: M = σu y 2 d Lip 2 σ u 2 l 2 Z d d The bound 1.6 of Theorem 1.1 i more han enough o how ha M := {M } i a coninuou L 2 P maringale. Since M a.. [Theorem 31

32 1.1] i follow from he maringale convergence heorem ha lim M exi a.. and i finie a.., which prove he fir par of And herefore, M = σu y 2 d ha o be alo a.. finie., ince we can realize M a W M for ome Brownian moion W, hank o he Dubin, Dambi Schwarz repreenaion heorem [34, p. 17]. i If we know alo ha Lip 2 συ < 1, hen Propoiion 8.2 guaranee ha E M i bounded from above by 1 Lip 2 συ 1 Lip 2 συ <, whence i follow ha M := {M } i a coninuou L 2 P-bounded maringale wih E up M 2 u 2 l 1 Z d + 4Lip2 συ 1 Lip 2 συ, 8.28 hank o Doob maximal inequaliy. Thi prove par i becaue u l Z d i bounded above by u l 1 Z d. ii Finally conider he cae ha l σ Υ 1. Since E u 2 l 1 Z d = E M 2 = u 2 l 1 Z d + u 2 l 1 Z d + l2 σ E u 2 l 2 Z d E σu y 2 d d, 8.29 i uffice o how ha hi final inegral i unbounded [a a funcion of ]. Bu ha follow from he econd par of Propoiion 8.2. Corollary 8.5. If u l 1 Z d, hen he following i a P-null e: { } { ω : ω : lim up u xω = x Z d } lim u l2 Z d ω =. 8.3 Proof. Le E 1 denoe he even ha lim up x Z d u x = and E 2 he even ha lim u l2 Z d =. Becaue of he real-variable bound, u 2 l Z d u 2 l 2 Z d u l Z d u l 1 Z d, we have { } E 1 E 2 ω : lim up u l 1 Z ω = d Bu we have noed already ha M := u l 1 Z d define a non-negaive maringale, under he condiion of hi corollary. Therefore, he final even in 8.31 i P-null, hank o Doob maringale convergence heorem. Thu, we find ha E 1 E 2 i a meaurable ube of a P-null e, and i hence P-null. Propoiion 8.6. Suppoe u l 1 Z d and he random walk X i ranien; i.e., Υ <. Then, 1 lim up u log E 2 l 1 Z d inf { β > : Lip 2 συβ < 1 } <

33 If, in addiion, l 2 συ > 1, hen lim inf 1 u log E 2 l 2 Z d inf { β > : l 2 συβ < 1 } > Proof. We have already proved a lighly weaker verion of Indeed, ince l 1 Z d l 2 Z d, 8.6 implie ha lim up 1 u log E 2 l 2 Z d inf { β > : Lip 2 συβ < 1 } Then 8.25 and 8.34 ogeher ell u ha for every C > inf{β > : Lip 2 συβ < 1}, here exi K = KC, uch ha E u 2 l 1 Z d u 2 l 1 Z d + Lip2 σ E u 2 l 2 Z d d 8.35 u 2 l 1 Z d + K e C d = O e C+o1 a. Thu follow he fir bound of he propoiion. Becaue of 8.16 and 8.17, we find ha F := E u 2 l 2 Z d olve he renewal inequaliy, where F g h F d, 8.37 g := u 2 x P, h := l 2 σ P A comparion reul Lemma A.2 ell u ha F f for all, where f i he oluion o he renewal equaion f = g + h f d The condiion ha l 2 συ > 1 i equivalen o h d > 1. Becaue of ranience [Υ < ] and he fac ha Υβ i ricly decreaing and coninuou, we can find β > uch ha exp β h d = 1. Noe ha f β := exp β f olve he renewal equaion f β = g β + h β f β d, 8.4 where g β := exp β g and h β := exp β h. Since h β i a probabiliy deniy funcion and g β i non increaing [ee 8.3], Blackwell 33

34 key renewal heorem [23] implie ha lim inf e β F lim f β = = u 2 x l 2 σ 1 h β d 1 e β P d Υβ. g β d 8.41 Since P 1, he righ-mo quaniy i a lea u 2 x l 2 σ β 2 Υβ >. Thi complee he proof of Noe ha we have ued he fac ha Υβ i coninuou in β and ricly decreaing, o ha β = inf{β > : l 2 συβ < 1}. Propoiion 8.7. If Lip 2 συ < 1, hen lim E u 2 l 2 Z d =. Furhermore, a : E P = O E u 2 l 2 Z d u 2 l 2 Z d ; and = O α for all α uch ha P = O α Proof. The fir aerion of 8.42 i imple o prove; in fac, E u 2 l 2 Z d [u x ] 2 P for any x Z d and all > ; ee 8.16 and We concenrae our effor on he remaining aemen. Thank o 8.13, E u 2 l 2 Z d P u 2 l 1 Z d + Lip2 σ P E u 2 l 2 Z d d Tha i, F := E u 2 l 2 Z d i a ub oluion o a renewal equaion; viz., for F g + h F d, 8.44 g := P u 2 l 1 Z d, h := Lip2 σ P A comparion lemma Lemma A.2 how ha F f for all, where f = g + h f d Therefore, i remain o prove ha f a. I i eay, a well a claical, ha we can wrie f in erm of he renewal funcion of h; ha i, f = g + n= h n g d, 8.47 where h 1 := h h d denoe he convoluion of h wih ielf, and h k+1 := h k h d for all k. We migh noe ha 34

35 g g = u 2 l 2 Z d becaue P i non increaing [ee 8.3] and one a zero. Therefore, h n g d u 2 l 2 Z d h n d n+1 u 2 l 2 Z d h d [Young inequaliy] = u 2 l 2 Z d Lip 2 σ Υ n I i no hard o ee ha lim g = lim P = ; hi follow from 8.3 and he monoone convergence heorem. Becaue Lip 2 συ < 1, we can deduce from 8.48 and 8.47, in conjuncion wih he dominaed convergence heorem, ha f hence F = E u 2 l 2 Z d converge o zero a. I remain o prove he econd aerion in Wih hi in mind, le u uppoe P aifie he following: There exi c, and α [, uch ha P c1 + α For here i nohing o conider oherwie. We aim o prove ha E u 2 l 2 Z d con 1 + α, 8.5 for ome finie conan ha doe no depend on. Thi prove he propoiion. Define F k := E u k 2 l 2 Z d, where uk denoe he kh approximaion o u via Picard ieraion 3.1, aring a u x. We can wrie 8.13, in hor hand, a follow: F n+1 P u 2 l 1 Z d + Lip2 σ Now le u chooe and fix ε, 1 and wrie c P F n d = ε ε F n d + up 1 + α c ε α 1 + α w P F n d + P F n d ε [1 + w α F n w] 1 ε P F n d P 1 + α d F n d + up [1 + w α Υ F n w] w 1 ε α 1 + α. The proof of Propoiion 8.2 how ha Conequenly, up n 8.52 F n d u 2 l 1 Z d Υ 1 Lip 2 συ R k := up [1 + w α F k w] k 8.54 w 35

36 aifie where R n+1 A + R n Lip 2 συ 1 ε α for all n, 8.55 A = Aε := c u 2 l 1 Z d + c u 2 l 1 Z d Lip2 συ ε α 1 Lip 2 συ Since Lip 2 συ < 1, we can chooe ε ufficienly cloe o zero o enure ha Lip 2 συ < 1 ε 1+α. For hi paricular ε, we find ha R n+1 A+1 εr n for all n. Since R =, hi prove ha up n R n A/ε. Eq. 8.5 whence he propoiion follow from he laer inequaliy and Faou lemma. 9 Proof of Theorem 1.7 Le u begin wih an elemenary real-variable inequaliy. Lemma 9.1. For all real number k 2 and x, y, δ >, k δ x + y k 1 + δ k 1 x k + y k. 9.1 δ Thi i a conequence of Jenen inequaliy when δ = 1. We are inereed in he cae ha δ 1. Proof. The funcion fz := z + 1 k 1 + δ k 1 z k z > i maximized a z := δ 1, and max z fz = fz = {1 + δ/δ} k 1 ; i.e., fx {1 + δ/δ} k 1 for all x >. Thi i he deired reul when y = 1. We can facor he variable y from boh ide of 9.1 in order o reduce he problem o he already-proved cae ha y = 1. Lemma 9.2. p +τ p 2 l 2 Z d d 4Υτ 2 for all τ. Proof. We apply he Plancherel heorem and 4.9 in order o deduce ha p +τ p 2 l 2 Z d = 2π d e +τ1 ϕξ e 1 ϕξ 2 dξ = 2π d π,π d π,π d e 21 Reϕξ 4τ 2 2π d e 21 Reϕξ dξ. π,π d Inegrae [d] o finih; compare wih e τ1 ϕξ 2 dξ Recall ha z k denoe he opimal conan in he BDG inequaliy [2.7]

37 Lemma 9.3. If k 2, aifie z k Lip σ Υ < 1 + δ k 1/k for ome δ >, hen up E up u x k up E u k l k Z d <. 9.3 x Z d Proof. Le u x := u x and define u n o be he nh ep Picard approximaion o u, a in 3.1. Define M n := E u n k l k Z d for all and k Then we can apply Lemma 9.1 and wrie, in analogy wih 8.26, k 1 M n δ I x δ k 1 J x, 9.5 δ x Z d x Z d where I x and J x were defined earlier in 3.3. One eimae x Z I d x via Jenen inequaliy, uing p x a he bae meaure, in order o find ha I x u k l k Z d. 9.6 x Z d In order o eimae x Z d J x, we define for all, x R + Z d a Borel meaure ρ,x on R + Z d a follow: ρ,x d dy := [p y x] 2 1 [,] d χdy; 9.7 where χ denoe he couning meaure on Z d. X X, he meaure ρ,x i finie; in fac, Becaue of he ranience of ρ,x R + Z d = p 2 l 2 Z d d = P d Υ. 9.8 Therefore, we apply 3.5 and Jenen inequaliy, in conjuncion, in order o ee ha { u J x zk k Lip 2 n σ E y k} 2/k k/2 ρ,x d dy [,] Z d u z k Lip σ k [Υ] [,] Z k 2/2 n E y k 9.9 ρ,x d dy. d Thu, J x z k Lip σ k [Υ] k 2/2 x Z d k z k Lip σ Υ up E r P E u n u n r k l k Z d, k l k Z d d