GAUSSIAN CHAOS AND SAMPLE PATH PROPERTIES OF ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES

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1 The Aals of Probabiliy 996, Vol, No 3, 3077 GAUSSIAN CAOS AND SAMPLE PAT PROPERTIES OF ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES BY MICAEL B MARCUS AND JAY ROSEN Ciy College of CUNY ad College of Sae Islad, CUNY Le X be a srogly symmeric u process wih -poeial desiy u Ž x, y Le ½ Ž 5 G u x, y d x d y ad le L deoe he coiuous addiive fucioal wih Revuz measure For a se of posiive measures M G, subjec o some addiioal regulariy codiios, we cosider families of coiuous Ž i ime addiive fucioals L L,, R M of X ad a secod-order Gaussia chaos Ž, M which is associaed wih L by a isomorphism heorem of Dyki A geeral heorem is obaied which shows ha, wih some addiioal regulariy codiios depedig o X ad M, if has a coiuous versio o M almos surely, he so does L ad, furhermore, ha moduli of coiuiy for are also moduli of coiuiy for L Special aeio is give o Levy processes i R ad T, he - dimesioal orus, wih M ake o be he se of raslaes of a fixed measure May cocree examples are give, especially whe X is Browia moio i R ad T for ad 3 For cerai measures o T ad processes, icludig Browia moio i T 3, ecessary ad sufficie codiios are give for he coiuiy of L,, R M, where M is he se of all raslaes of Iroducio I his paper we sudy he coiuiy of families of addiive fucioals of symmeric Markov processes Le us briefly cosider his quesio heurisically Le X, R be a symmeric Markov process wih locally compac sae space S Oe may hik of he local ime of X s, up o ime, a a poi x S, as x L lim x, Xs ds, 0 0 Received Jue 99; revised Jue 995 This research was suppored i par by gras from he Naioal Sciece Foudaio, he Guggeheim Foudaio, PSC-CUNY ad a Academic Iceive Award from The Ciy College of New York M B Marcus is graeful as well o Uiversie Louis Paseur ad CNRS, Srasbourg, ad he Saisical Laboraory ad Clare all, Cambridge Uiversiy, for he suppor ad hospialiy he received while much of his work was carried ou This research was suppored i par by gras from he Naioal Sciece Foudaio, PSC-CUNY ad he Lady Davis Fellowship Trus J Rose is graeful as well o he Isiue of Mahemaics of he ebrew Uiversiy, Jerusalem, for he suppor ad hospialiy he received while much of his work was carried ou AMS 99 subjec classificaios G0J55, G0G5 Key words ad phrases Coiuous addiive fucioals, Markov, Gaussia chaos 30

2 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 3 wheever he limi exiss i some sese, where x, is a family of approxi- mae dela fucios a x Cosiderable effor has bee spe over he las 30 years o fid reasoable codiios for he coiuiy of he sochasic process x T L,, x R S Some hisorical backgroud is give i, i which we obai ecessary ad sufficie codiios for he coiuiy of T for Markov processes wih symmeric rasiio probabiliy desiy fucios Local imes exis oly for a relaively arrow class of Markov processes A Levy process i R has a local ime oly whe Whe local imes do o exis, ad eve whe hey do, oe ca cosider coiuous addiive fucioals of a Markov process deermied by measures o he sae space of he process We may hik of hese as L lim x, Xs ds d x, 0 S 0 where is a posiive measure o S I his case, depedig o he measure, such limis exis for all Levy processes i R, for all For some family of measures M for which Ž exiss, edowed wih some opology, we cosider he quesio of he coiuiy of L L,, R M The papers of Bass ad Bass ad Khoshevisa 3 are he oly prior works we kow ha pursue his quesio I, i sudyig he coiuiy of he local ime process T, we used a isomorphism heorem of Dyki, which eabled us o show ha he coiuiy of T was equivale o he coiuiy of a associaed Gaussia process o S Sice he sample pah properies of Gaussia processes are very well udersood, we were able o use hem o obai may ew resuls abou local imes of Markov processes A differe versio of Dyki s isomorphism heorem associaes a secod-order Gaussia chaos o M wih L I his paper we firs prove a geeral heorem which shows ha he coiuiy of his Gaussia chaos implies he coiuiy of L, subjec o various addiioal codiios These codiios are removed whe we specialize o he case of Levy processes i R ad T, he -dimesioal orus, wih M ake o be he se of raslaes of a fixed measure Furhermore, usig kow resuls abou sample pah properies of Gaussia chaoses, cocree sufficie codiios for he coiuiy ad modulus of coiuiy of L are obaied Eve whe M is resriced o he se of raslaes of a fixed fiie measure, he diversiy of he processes L is vasly greaer ha is subse T I some cases descripios of L i erms of he associaed Gaussia chaoses lead o weak resuls I oher cases he esimaes obaied are quie sharp ad for cerai impora Levy processes, icludig Browia moio ad oher sable processes i T 3, ake ogeher wih cerai fiie measures, we show ha L is coiuous if ad oly if he associaed Gaussia chaos is coiuous Ž x Le X, F, X, P, R, deoe a srogly symmeric u process wih lifeime ad locally compac separable sae space S wih referece measure m The full defiiio of hese erms is give i For he purposes of his paper, i is eough o jus say ha X has a symmeric

3 3 M B MARCUS AND J ROSEN rasiio probabiliy desiy p Ž x, y Le 0 As usual, we defie he -poeial desiy 3 u x, y e p x, y d 0 ad, seig u x, y e p x, y d, assume ha, for all 0, Ž 5 u Ž x, y x, y S We also cosider u 0 Ž x, y for 0 ad defie u 0 Ž x, y as i Ž Whe dealig wih u 0 Ž x, y we assume ha Ž 5 holds As usual, we someimes drop he superscrip 0 whe dealig wih u 0 or u 0 We are primarily cocered wih Markov processes for which u Ž x, x for some, or all, x S This is he fudameal differece bewee he processes cosidered i his paper ad hose cosidered i We assume ha u Ž x, y fž y dmž y is a bouded coiuous fucio o S for some, equivalely all, 0, for all bouded measurable fucios f o S which vaish ouside of a compac subse of S Whe we cosider už x, y we assume ha his is also he case for už x, y fž y dmž y These are he smoohess hypoheses o he poeial give i Chaper 6, ad, of Theorem is expressed i erms of a auxiliary fucio h Ž By he smoohess hypoheses o he poeial, we ca always fid sricly posiive bouded fucios f i L Ž dm such ha Ž 6 h Ž x u Ž x, y f Ž y dmž y Ž is coiuous ad bouded We defie U h u Ž, y h Ž y dž y Ž Ž To ay coiuous addiive fucioal A of X we ca associae a posiive -fiie measure A called he Revuz measure of A The measure A is defied by he formula m A s s 0 0 Ž 7 Ž g lim E g Ž X da for all bouded coiuous fucios g o S, ad A is uiquely deermied by A We will use he oaio L for he coiuous addiive fucioal wih Revuz measure, ad ofe refer o L as he coiuous addiive fucioal deermied by No every -fiie measure is he Revuz measure of a coiuous addiive fucioal The se of all Revuz measures of coiuous addiive fucioals of X will be deoed by RevŽ X A complee characerizaio of RevŽ X is kow; see, for example, 6 ad 6 For our purposes i will be eough o oe ha a sufficie codiio for RevŽ X is ha U x is bouded, or, more geerally, ha U h Ž x Ž is bouded for some 0

4 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 33 For posiive measures o S, defie ½ 5 8 G u x, y d x d y ad ½ 5 9 G u x, y d x d y Le G, F, G, F deoe he se of fiie measures i G, G, respecively The CauchySchwarz iequaliy shows ha Ž 0 G, F G, F Throughou his paper is a fixed umber greaer ha or equal o 0 As usual, we deoe G0 ad G0 by G ad G We are cocered wih he sample pah properies of he sochasic process Ž L L, Ž, R G, R, where G G RevŽ X A isomorphism heorem of Dyki give i 5, R associaes L wih a secod-order Gaussia chaos Ž, G A Gaussia chaos is a family of secod-order erms i he ermie polyomial expasio of radom variables i L Ž, where is he caoical Gaussia produc measure o R N We describe i Secio i which we also give, i Theorems ad, proofs of versios of Dyki s heorem which we use We begi wih a geeral heorem which saes ha a family of coiuous addiive fucioals of a Markov process is joily coiuous if he associaed Gaussia chaos is coiuous, subjec o various addiioal codiios I several subseque heorems we impose regulariy codiios which eable us o elimiae or simplify hese addiioal codiios For ay se C we deoe by BŽ C he se of bouded fucios o C wih he opology iduced by he sup-orm Occasioally, we will simply say ha a sochasic process is coiuous o mea ha he process has a versio which is coiuous almos surely TEOREM Le X be a Markov process saisfyig all he codiios give above ad le M G Assume ha we are give a opology O for M uder which M is locally compac ad has a couable base Assume also ha: i U ad U h are coiuous maps from M o BŽ S Ž ; Ž ii he associaed secod-order Gaussia chaos is coiuous almos surely o M The here exiss a polar se Q S, such ha, if we resric X ad S Q, we ca fid a coiuous versio of L, Ž, 0, M M o REMARK Whe X is a Levy process o R or T, he -dimesioal orus, ad he se of measures M is he se of raslaes of a fixed fiie measure, he exclusio of a polar se is uecessary ad Ž i ca be elimiaed These resuls are give i Theorems 3 ad 3

5 3 M B MARCUS AND J ROSEN REMARK For may Markov processes, such as Levy processes, he killig of ime is ideically ifiie I his case he las erm i Theorem ca be replaced by L,, R M I geeral, whe is o ifiie, i order o fid a versio of L ha is also coiuous a, we mus resric M o be a se of measures wih commo compac suppor This is he coe of Theorem 3 REMARK 3 Sice measures wih bouded poeials do o charge polar ses, resricig M o S Q does o require us o aler he measures i M Theorem requires he coiuiy of a associaed Gaussia chaos Ž We ow describe a well-kow sufficie codiio for he coiuiy of a Gaussia chaos Defie a meric o G : dž, d Ž, Ž ž Ž Ž Ž Ž Ž / u x, y d x x d y y ž Ž / E Ž, where is he Gaussia chaos associaed wih he coiuous addiive fucioal wih poeial U The las equaliy is explaied i Ž 0 for 0 TEOREM Le Ž, M be a secod-order Gaussia chaos Ž ad le M G be a se of measures ha is compac wih respec o G, d, where d is give i Ž Assume ha here exiss a probabiliy measure o M such ha Ž 3 lim sup log d 0, 0 B Ž, M 0 d where B Ž, d deoes he ball i he meric d, wih ceer a ad radius 0 The has a versio which is coiuous almos surely This is Theorem of 9 See Remark for furher explaaio REMARK Noe ha a sufficie codiio for Ž 3 is ha Ž IŽ d, M log N Ž M, d, def 0 where N Ž M, d is he miimum umber of balls of radius ha covers M log N Ž M, is called he meric eropy of M wih respec o d d ow- ever, eiher Ž 3 or Ž is a geeral ecessary codiio for he coiuiy of he ype of secod-order Gaussia chaoses cosidered i Theorems ad I fac, we do o kow ecessary ad sufficie codiios for he coiuiy of hese chaoses owever, of paricular imporace i his paper, d

6 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 35 is ha by rece resuls 5 here are may examples of classes of hese chaoses for which a ecessary ad sufficie codiio for coiuiy is Ž 5 JŽ d, M log N Ž M, d def 0 REMARK 5 The saeme i Ž 3, bu wih he log erm replaced by is square roo, is ecessary ad sufficie for he coiuiy of Gaussia chaoses associaed wih ordiary local imes I his case he se of measures are he ui poi masses This implies ha Ž 5 is ecessary ad sufficie for he coiuiy of Gaussia chaoses associaed wih local imes of Levy processes This is discussed i Secio We ow specialize o he case of Levy processes i R, or T, ad for he class of measures we cosider he se of raslaes Ž 6 x, x R Ž or T of a sigle fiie measure ; ha is, Ž A Ž x A x for all measurable ses A R, or T This class of measures icludes he poi masses which, obviously, are he raslaios of he poi mass a he origi Thus he family of coiuous addiive fucioals ha are deermied by he raslaes of a sigle fiie measure iclude he local imes I is easy o check ha RevŽ X implies ha RevŽ X x for each raslae of For a se of measures such as Ž 6, we also hik of d Ž x, y d Ž, def x y as a meric o R, or T Usig he added srucure provided by Levy processes i R, or T, he basic coiuiy resul, Theorem, ca be simplified as follows TEOREM 3 Le X X, R be a symmeric Levy process i R Le G be a fiie measure If he associaed secod-order chaos, x R is coiuous almos surely, he Rev X ad L x x, x, R R is coiuous almos surely This also holds wih R replaced by T We ca do more wih coiuous addiive fucioals of Levy processes i T For a cerai class of hese processes ad for cerai smooh measures G, we ca show ha L x, x T is coiuous almos surely if ad oly if he associaed secod-order chaos Ž, x T x is coiuous almos surely Before preseig his we eed o develop some oaio ad o meio some resuls abou coiuiy of Gaussia chaoses Le X X, R be a symmeric Levy process i R wih 7 Ee i X e Ž, R Similarly, le Y Y, R be a symmeric Levy process i T wih 8 Ee i k Y e Ž k, k Z I each case we refer o as he characerisic expoe of he process Oe reaso for deoig each characerisic expoe by is ha for each Levy d

7 36 M B MARCUS AND J ROSEN process X i R as defied i 7 we ca defie a Levy process Y i T wih he same fucio by projecig X oo 0, This is explaied i Secio 6 The isomorphism heorem ca also be used o obai ieresig resuls abou Gaussia chaoses The ex heorem, a cosequece of he isomorphism heorem, gives ecessary codiios for he coiuiy ad boudedess of a class of secod-order Gaussia chaoses closely relaed o he associaed chaoses of cerai families of coiuous addiive fucioals I is proved i Secio 6 I wha follows le g k k Z be idepede ideically disribued ormal radom variables wih mea 0 ad variace TEOREM Le be he characerisic expoe of a Levy process i T ad b k k Z he Fourier coefficies of a fiie measure o T The bž k i k x sup Ý e xt kz Ž k Ž 9 Ž g j gk j, k bž k j iž kj x CE sup Ý e, Ž j Ž k ' xt ' j, kz where C is a cosa idepede of ad bž k Furhermore, a similar resul is obaied whe he uiform orm is replaced by he Lipshiz orm, i k x ha is, sup xy, x, y T, wih e replaced by Že i k x e i k y for all k Z I preparaio for he ex heorem, we say ha a posiive fucio hž k, k Z, is almos regularly varyig wih idex p if here is a regularly varyig fucio h x, x R, of idex p, such ha Ž 0 C hž k hž k ChŽ k for some cosa 0 C I he ex heorem we will assume ha he sequeces Ž k k Z ad bž k k Z i Theorem are symmeric ad almos regularly varyig wih idex p ad q 0, respecively I his case i follows from Theorem 3 of 5 ha he wo sides of Ž 9 are eiher boh fiie or boh ifiie Noe ha by he posiiviy of he Fourier coefficies bž k k Z we have ha U Ž x is bouded if U Ž 0 This allows us o obai he followig equivalece relaioships TEOREM 5 Le X, R be a Levy process i T wih characerisic sequece Ž k ad le Ž k k Z ˆ k Z be he Fourier coefficies of a fiie measure o T Assume ha Ž k ad Ž k k Z ˆ k Z are symmeric ad almos regularly varyig wih idex p ad q 0, respecively, ad ha here exiss a cosa C such ha, for all j, k ˆ Ž k j ˆ Ž j Ž sup C Ž k Ž j k j

8 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 37 The he followig are equivale: Ž i U Ž 0 ; Ži Ý Ž k Ž Ž k k Z ˆ ; ii G ad JŽ d, M, where M, x T x ; Ž iii he Gaussia chaos Ž, x T x is coiuous almos surely; x iv Rev X ad L, x, T R is coiuous almos surely The ieresig cases of his heorem are whe p q The ex resul is a corollary of his heorem ad is proof for Browia moio i T 3 3 COROLLARY Le X, R be Browia moio i T ad le fž u, u 0,, be regularly varyig a 0 such ha ufž u is decreasig o Ž0, ad fž u 0 for u Ž, Defie 3 L x, f X s x ds, x, T R Ž 0 The he followig are equivale: Ž i L Ž 0, is fiie almos surely; 3 ii L x,, x, T R is coiuous almos surely; Ž iii ufž u L 0, We should meio ha G implies ha 3 Also Theorem 5 does o apply o Browia moio o T Noe ha for local imes he exisece of he -poeial does o imply he coiuiy of L x, x T bu for Brow- ia moio i T 3 i does for he measures cosidered i Theorem 5 We do o kow wheher or o his is rue i T The equivalece of he wo apparely simple saemes Ž i ad Ž iv i Theorem 5 suggess ha perhaps i ca be obaied by a simple direc argume Our proof is complex ad circuious Also Theorem 5 suggess ha he meric d of Ž, which we have o see before i poeial heory, has a sigifica role i describig coiuiy properies of addiive fucioals of Markov processes A aalog of Theorem 5 ad Corollary also holds for Levy processes i R This is give i Theorem 5 depeds very srogly o he Fourier coefficies of he measures ad o he characerisic sequeces of he Levy processes beig smooh I 3, employig differe mehods from hose used i his paper, we show ha here exiss a large class of measures ad siged measures, a such ha, for arbirary Levy processes o T, L, a, T R is coiuous if ad oly if JŽ, M, where Ž a, b Ž 3 u Ž x, y d Ž x Ž x d Ž y Ž y ž a b a b / for some 0 ad his is valid for ay The meric i Ž 3 is associaed wih he eergy iegral of X ŽI 3 we cosider he more

9 38 M B MARCUS AND J ROSEN geeral class of coiuous addiive fucioals deermied by geeralized fucios, as defied i 6 Whe he codiios of Theorem 5 are o saisfied, we ca use Theorem x 3 which geerally allows us o ifer coiuiy of L, x, R R or x, T R whe Ž holds ad, i fac, Ž is o much weaker ha Ž 5 ere is a cocree applicaio of Theorem 3 Le X X, R be a symmeric Levy process i R wih characerisic expoe ad le be a fiie measure o R wih characerisic fucio ˆ Assume ha d Ž Ž Ž Ž Ž Ž Noe ha is he Fourier rasform of u x so ha ˆ Ž 5 Ž Ž d u Ž x, y dž x dž y TEOREM 6 Le X X, R be a symmeric Levy process i R wih characerisic expoe If ž x Ž ˆ Ž d / Ž 6 dx, x x he Rev X ad L, x, R R is coiuous almos surely I paricular, for Browia moio i R, his is he case whe Ž 7 ˆ Ž O as Ž log 3 By Theorem 5 we have for Browia moio i T ha, if RevŽ X, x 3 L, x, R R is coiuous almos surely if ad oly if ˆŽ k Ž 8 Ý k kz 3 as log as ˆ Ž k is almos regularly varyig i he sese of Ž 0 ad saisfies Ž The oly oher papers ha we kow of ha deal wih he joi coiuiy of coiuous addiive fucioals of Markov processes idexed by measures are ad 3, which cosider his quesio for Browia moio i R The maerial i 3 shows ha Ž, for all compac subses of measures M wih bouded poeial, is sufficie for he coiuiy of he associaed coiuous addiive fucioals L,, R M of Browia moio bu wih he meric d replaced by 9, sup u x y dž y y xr

10 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 39 Furhermore, his resul is valid i R for all owever, he meric is difficul o esimae I is o comparable o he meric d give i Ž Resriced o R for or 3, he mehods of 3 seem o give somewha weaker resuls abou he coiuiy of coiuous addiive fucioals of Browia moio ha he oes obaied i his paper For example, he resul i Ž is implied by he exisece of he -poeial, whereas he meric i Ž 9 is a fucio of he -poeial, ad for he coiuiy of L x oe mus also have IŽ, M Ž Pursuig his lie, oe is off by a facor of he square of a logarihm Also i some cases, such as hose ha occur i Secio 7, i he sudy of moduli of coiuiy of measure-idexed coiuous addiive fucioals, he meric is comparable oly o d A slighly differe versio of he isomorphism heorem, Theorem, ca be used o obai he modulus of coiuiy of measure-idexed coiuous addiive fucioals ere are wo examples of such resuls for Browia moio i R where, as above,, a R a deoes he se of raslaes of a fixed fiie measure o R Give a se A R, le meas A deoe he ausdorff measure of A i dimesio Le dim A deoe he ausdorff dimesio of A Tha is, dim A supmeas A We defie he idex of a measure o be he supremum of he umbers such ha Ž 30 sup BŽ x, r Cr r, xr where BŽ x, r is a Euclidea ball a x of radius r We oe ha by Frosma s lemma dim A if ad oly if A carries a fiie measure wih idex ŽSee, eg, Chaper 0 of 8 TEOREM 7 Le B B, R be Browia moio i R Le A R have ausdorff dimesio, 0 The here exiss a fiie measure RevŽ X, suppored o A, for which L a L b Ž 3 lim sup 0 Ž a b ab0 a, b 0, for almos all R almos surely for all 0 Equivalely, le be a measure wih idex The is suppored o a se A wih dim A ad 3 holds For he ex heorem le us oe ha i follows from older s iequaliy p ha, for Browia moio i R, U f x is bouded if f L for some p TEOREM 8 Le B B, R be Browia moio i R Le be a p fiie measure o R such ha f x dx, where f L for some p The L a L b Ž 3 lim sup C f Ž p 3 p a b loga b ab0 a, b0,

11 0 for almos all R deped o p M B MARCUS AND J ROSEN almos surely, where C is a fiie cosa which may The Gaussia chaos ha we have bee referrig o is carefully defied i Secio, i which we also give, i Theorems ad, wo versios of he isomorphism heorem These are more complicaed ha he versios give i, i which local imes are associaed wih Gaussia processes Viewig ha case i he ligh of his paper, we see ha he measures are poi masses ad, clearly, iegraio wih respec o poi masses is rivial ere we mus carry ou he releva iegraios Thus Secio i his paper does o follow easily from he maerial o he isomorphism heorem i Theorem ad he commes followig i are proved i Secio 3 I a brief Secio we show ha whe a Markov process has a local ime, coiuiy of he local ime ad he associaed Gaussia chaos are equivale Theorem 3, for processes i R, ad Theorem 6 are proved i Secio 5 I Secio 6 we prove Theorem 3 for processes i T We also prove Theorems ad 5, Corollary ad obai a aalog of Theorem 6 for Levy processes i T A cocree descripio of he Gaussia chaoses associaed wih coiuous addiive fucioals of Levy processes o T is also give I Secio 7 we briefly cosider he moduli of coiuiy of coiuous addiive fucioals of Markov processes ad give he proofs of Theorems 7 ad 8 Throughou his paper C will deoe a cosa greaer ha 0 which is o ecessarily he same a each occurrece Also we use he oaio fž x gž x as x o mea ha here exis cosas 0 C, C such ha C gž x fž x C gž x for all x x0 for some x0 sufficiely large, ad similarly a 0 The isomorphism heorem I we preseed a proof of a versio of Dyki s isomorphism heorem ha relaed he local ime of a symmeric Markov process o a mea-zero Gaussia process, which had as is covariace he -poeial desiy of he Markov process I his paper we are ieresed i Markov processes which may o have local imes bu for which we ca defie coiuous addiive fucioals deermied by posiive measures o he sae space I his secio we prove a versio of Dyki s isomorphism heorem which relaes hese fucioals o a Gaussia chaos o he space of measures G G, defied i Ž 9 0 The argume ha we give ca be used for ay 0 owever, o keep he oaio from becomig oo cumbersome, we carry ou he argume i deail oly i he case 0 To defie he secod-order Gaussia chaos referred o i Secio, we firs cosider he Gaussia process G, G, which has mea 0 ad covariace Ž E G G už x, y d Ž x d Ž y G

12 PAT PROPERTIES OF ADDITIVE FUNCTIONALS Le Ž dy p Ž x, y dmž y I is easy o see, by assumpio Ž 5 x, for 0, ha dy G for all 0 ad x S ad ha x, Ž Ž E G G u Ž x, y x, y, Le G G The, by Ž ad Lemma 5 of x, x,, we have ha Ž 3 E G G u Ž x, y u Ž x, x u Ž y, y Therefore x, y, ž Ž / Ž E Gx, E Gx, Gy, E Gy, u x, y I order o defie he Gaussia chaos which occurs i he isomorphism heorem, we firs cosider a simpler class of Gaussia chaoses x, Ž Ž x, Ž 5 Ž, G E G dž x for 0 ad G I follows from ha Ž Ž 6 E,, u x, y d x d y Therefore, for G, we have ha Ž 7 lim Ž, 0 exiss as a limi i L ad saisfies Ž 8 E 0 ad Ž 9 EŽ Ž Ž už x, y dž x d Ž y, G Thus we see ha Ž 0 Ž EŽ Ž ž Ž už x, y Ž dž Ž x Ž x Ž dž Ž y Ž y /, G We explai why we call Ž, ad Gaussia chaoses i Remark We coiue o defie he erms which appear i he isomorphism heorem Le f ad h be as give i Ž 6 Sice u is a excessive fucio i each variable, i follows ha h is a excessive fucio ad hece lower semicoiuous Moreover, h 0 ad h is locally bouded For g b S we defie Ž h Ž P g Ž x P Ž gh Ž x hž x

13 M B MARCUS AND J ROSEN I is easy o see ha P Ž h is a semigroup I follows from Theorem 69 of Ž x 7 ha here exiss a uique Markov process, F, X, P h, called he h-rasform of X, wih rasiio operaors P Ž h, for which x h x Ž P Ž FŽ Ž P Ž FŽ hž XŽ hž x for all F bf Le G be a compacly suppored probabiliy measure As usual, we se Ž 3 E h P x h d Ž x For a measure ad f a fucio o S, we deoe by f he measure o S give by fž x Ž dx, x S Also, if g is a kerel o S S ad is a measure o S, he g gž, y Ž dy ad gf gž, y fž y Ž dy Le h ad f m I is easy o verify ha, G The ex heorem is coaied i Theorem 6 of 5 We give a more deailed proof for he coveiece of he reader I his heorem, for a give Markov process wih 0-poeial desiy u x, y ad measures G wih bouded poeial, we cosider L lim L ad he associaed Gaussia chaos Ž TEOREM Le i i be a sequece of measures i G ad assume Ž ha U is a bouded o S for all i Se L L, L, i ad Ž,, The, for ay compacly suppored G ad C-measurable oegaive fucio F o R, Ž Ž h EG E F L EG F G G, where C deoes he -algebra geeraed by he cylider ses of R PROOF This heorem is a geeralizaio of he isomorphism heorem for Example give i Secio of We explai how he proof of ha heorem ca be modified o prove his oe Our argume is mea o be read i cojucio wih he maerial i Accordig o Ž 38 of, we have Gu, G i v i, EG Ł G G ž / i Ž 5 Ý ž Ý cov Ž B cov Ž B B / P BC,,, Ž Ý Ž Ž Ž ŽC u Ž y u Ž z, y u Ž z, where he secod sum is ake over he se of all possible pairigs P of u v i i B i i B The specific pairs i P are deoed by B,, B B If, for example, B Ž u, v, he cov Ž B EŽ G G i j k i u, v, The las sum is ake j k over all permuaios Ž Ž,, ŽC of C ad over all ways of assigig u, v o y, z The explaaio of wha y, z Ži Ži Ži Ži Ži Ži are ad wha we mea by assigig is give i he ex immediaely precedig Ž 38 of

14 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 3 For ay pairig P of u v ad subse A,,, i i B i i B, le A, P or 0 depedig o wheher or o ui is paired wih vi for all i A Applyig Ž 5 o he simple ideiy Gu, Gv, EŽ Gu, Gv, i i i i EG Ł G G ž i / ž ž / ia / ia Gu, Gv, Gu, G A i i i v i, Ý Ł Ž EG EG Ł G G, c A,, we see ha Gu, Gv, EŽ Gu, Gv, i i i i EG Ł G G ž i / Ž 6 A Ý Ý Ý A, P Ž Ž B BC,,, AB P Ý u y Ž u z Ž, y Ž u z ŽC cov B cov B Ž Ý ž Ý Ž Ž B / cov B cov B P BC,,, Ž Ý Ž Ž Ž ŽC u Ž y u Ž z, y u Ž z, where Ý is ake over he se of all possible pairigs P of u v such ha ui is o paired wih vi for ay i I he las sep we used A A Ý Ž Ý A, PwŽ P Ý wž P Ý Ž A, P AB P P AB P i i B i i B Ý Ł Ý wž P i, P wž P, P ib P where wž P deoes ay fucio of P A similar aalysis shows ha ž Ł Ž i i i i / ž Ý Ž Ž B / E G G E G G G u, v, u, v, ib Ž 7 cov B cov B P Therefore if we se u v x we ge E Ž 8 Gx, EŽ Gx, i i Ý EG ž Ł / BC,,, ib i i i Gx, EŽ Gx, i i G Ł G G i ž Ý Ž Ž Ž ŽC / Ž C u Ž x u Ž x, x u Ž x,

15 M B MARCUS AND J ROSEN where ow he las sum is ake over all permuaios of C Iegraig wih respec o he measures ad recallig Ž 5, we ge E Ž, i G Ł G G ž i / Ž 9 BC,,, i i Ž, i ž Ł / ib / Ý EG ž ž Ý Ž Ž Ž ŽC Ł i i ic / u Ž x u Ž x, x u Ž x d Ž x C We ow show ha we ca ake he limi as 0 i 9 o ge E Ž 0 Ž i G Ł G G ž i / BC,,, Ž i ž Ł / ib / Ý EG ž Ý Ž Ž Ž ŽC Ł i i ž ic / u Ž x už x, x u Ž x d Ž x C To see his, we oe ha, sice i, i i L, here exiss a sequece 0 such ha Ž, Ž almos surely ad E Ž, j i j i i j CE Ž i for some cosa C, for all i Also, for all iegers m, by he hypercoraciviy of he Gaussia chaos Žsee eg,, Ž m m Ž EG Ž i, j CmŽ EG Ž i, j m G i C E Ž I follows from Ž ad muliple uses of he Schwarz iequaliy ha he erms ivolvig he Gaussia chaoses i Ž 9 are uiformly iegrable ad we ca ake he limis as j 0 This also shows ha he limi as j 0 of he las iegral i Ž 9 exiss ad sice i is moooically icreasig i is equal o he las iegral i Ž 0 Also, sice he erms ivolvig he Gaussia chaoses are bouded, he iegrals i Ž 0 ca be see o be bouded by iducio o owever, i is easy o see his direcly sice h ad U i, i, are bouded ad has compac suppor We will show below ha E x h L i ž Ł / ic Ž Ý už x, x Ž už x Ž, x Ž u Ž x ŽC Ł di Ž x i hž x ic Ž C

16 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 5 Iegraig wih respec o gives Ž 3 E h L i ž Ł / ic Ý Ž Ž Ž ŽC Ł i i u Ž x už x, x u Ž x d Ž x Ž C Usig 3 i 0, we see ha Ž E Ž i G Ł G G ž i / ž ib / Ž i ž Ł ž i / i / ic Ž Ý EG Ł E Ł L BC,,, h EG E L i h i ic This is equivale o Ž 3 of i he case of Example, Secio of Theorem 5 ow follows from he las paragraph of he proof of Theorem of y i To obai, cosider 8 of wih L replaced by L i Thus i becomes Ž,, r r r 0 r r dl dl dl To obai he aalog of Ž 30 ad Ž 3 of, follow he argume of Ž 30 of ad use Theorem 3, Chaper 6 of o ge Ž 5 E Ž L E h X dl hž x 0 x h x Ž r r UhŽ x hž x už x, x u Ž x dž x, hž x sice u h Coiuig a proof by iducio, we ow assume ha x h Ž 6 E Ž už x, x už x, x u Ž x Ł di Ž x i hž x i ad le Ž,, Noe ha i he lie above Ž 3, of should be Followig Ž 33 of, we ge x h x h X r h Ž 7 E Ž E E Ž, dl r, hž x 0

17 6 M B MARCUS AND J ROSEN ad proceedig as i 5, we ge x h x h Ž 8 E Ž už x, x hž x E Ž, dž x hž x Usig 6 i 8, we ge This complees he proof of Theorem Le M be a family of measures coaied i G RevŽ X We prese a slighly differe versio of he isomorphism heorem o use i sudyig he moduli of coiuiy of L L,, R M Le be a expoeial radom variable wih mea, which is idepede of he Markov process X We cosider L, which is simply L, wih replaced by he idepede soppig ime We associae wih L a secod-order Gaussia chaos Ž, M which is defied exacly he same way as was i he begiig of his secio, excep ha i place of už x, y i Ž we use u Ž x, y TEOREM Le f be a posiive fucio o S such ha f m G Assume ha U is bouded for each M ad ha L ad are boh i C M, he se of coiuous fucios o some compac subse M G The, for ay compacly suppored G ad ay oegaive Borel-measurable fucio F o CŽ M, we have Ž Ž 9 EG E E F L f X EG F G G fm PROOF The proof is similar o he proof of he isomorphism heorem for Example give i Secio of REMARK We explai why we call Ž, ad Gaussia chaoses Le T be some idex se ad le g be idepede, ideically dis- ribued ormal radom variables wih mea 0 ad variace We say ha a sochasic process Ž, T is a secod-order Gaussia chaos if i ca be wrie i he form Ý Ý Ž 30 Ž g g Ž g Ž, T, jk j k j, k j j, j j where we assume ha he series coverges i L for each T Sice we will oly be cocered wih secod-order chaoses, we will o boher o repea he words secod order whe discussig hem Ž,, defied i Ž 5, is a Gaussia chaos o G To see his, we oe ha he Gaussia process G, G ca be wrie i erms of is KarhueLoeve ` expasio Therefore, i paricular, we ca cosider he Gaussia process Ž 3 G g Ž, x S Ý x, j j x, j

18 Referrig o 5, we see ha PAT PROPERTIES OF ADDITIVE FUNCTIONALS 7 Ž 3 Ý Ž, g g Ž Ž dž x jk j k j x, k x, Ý g Ž dž x j j j x, Sice he L limi of a Gaussia chaos is a Gaussia chaos, by Theorem 3 of, we see ha is a Gaussia chaos I fac, Ž, G has a expasio as i Ž 30 for G, alhough we do o kow wha i is explicily Noe ha i a Gaussia chaos is defied as he closure i L of expressios of he form Ž 30 ere we are usig he defiiio of Gaussia chaos give i 9 I follows from Theorem 3 of ha he wo defiiios are equivale REMARK The coiuiy codiio for Gaussia chaoses give i Theorem is coaied i Theorem of 9 To see his, i is oly ecessary o oe ha he meric d is smaller ha he meric d Ž i he oaio of 9 3 Coiuiy heorems I his secio we give he proofs of Theorems ad is refiemes, Theorems 3 ad 3 PROOF OF TEOREM We firs cosider he case 0 Recall ha we are deoig h 0, u 0, U 0, G0, G0 ad so o by h, u, U, G, G ad so forh Le us also recall ha, by Ž 3 ad Ž 5, UhŽ x x h 3 E L, hž x ad, for G a probabiliy measure wih compac suppor, P h P x h d Ž x By workig locally i suffices o cosider M compac Le D M be a couable dese se Usig he proof of Theorem 6 of ogeher wih a versio of Dyki s isomorphism heorem, Theorem of his paper, we ca see ha L,, R D is uiformly coiuous almos surely wih respec o P h, where R 0, is he compacificaio of R obaied by Ž x addig he poi a Thus, e : R 0, is a isomorphism of compac ses Le 3 L is uiformly coiuous o R D

19 8 M B MARCUS AND J ROSEN We have ha x h h Ž 33 P d Ž x P Ž for all fiie measures G wih compac suppor x h Le Q xp Ž By Ž 33 ad sadard argumes Žsee he discussio o page 85 of, we see ha Q is a polar se We heceforh resric our Markov process ad measures o S Q, oig ha uder our assumpios he measures i M do o charge polar ses Thus, i effec, we are cosiderig a Markov process defied o S S Q for which x h Ž 3 P Ž x S We he exed L,, R D o L,, R M by coiuiy Sice we may assume ha he L are perfec coiuous addiive fucioals for all D, we immediaely see ha he same is rue for L for each M We ow show ha L is a versio of L for he h-rasformed process For his, by Theorem 363 of 7, i suffices o show ha L has he same poeial as L, ha is, ha UhŽ x x h Ž 35 E Ž L x S hž x By he defiiio of L, Ž 35 holds for all D, ad, for ay M, if we choose a sequece i i i of measures i D such ha i, he L L almos surely Sice, by assumpio Ž i of his heorem, Uh Ž x UhŽ x i, i we ca complee he proof of 35 by showig ha L i are uiformly i iegrable For his i suffices o show ha L i are uiformly bouded i x h L wih respec o P This is easily see sice by Ž we have h x x h Ž 36 E Ž L i UŽ Ž Uhi i Ž x, which is uiformly bouded i i by assumpio Ž i Redefie L by seig i equal o s s Ž s raioal Ž 37 lim if L Ž for all Ž As i he proof of Theorem 6 of, we see ha L ad x L agree o 0,, P almos surely We ow see ha he above limi iferior is a rue limi ad ha L L for all R Therefore L is a versio of L for he Markov process X Fially, as i he proof of Theorem x h 6 of, we see ha he P almos sure coiuiy of L,, R x M implies he P almos sure coiuiy of L, Ž, 0, M This complees he proof i he case 0 We ow give he proof for The proof for ay oher 0 is similar Le X be a Markov process which saisfies he hypoheses of his heorem i he case Le Y be he Markov process obaied by killig X a a

20 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 9 idepede expoeial ime wih mea Le Ž, F, P x be he probabil- Ž x iy space of X As usual, we ake R, F B, P, where Ž d e d, o be he probabiliy space for Y, where Y Ž, X for ad Y Ž,, he cemeary sae, for I is easy o check ha he 0-poeial desiy of Y ad he -poeial desiy of X are equal ad ha Y saisfies he hypoheses of his heorem wih 0 Furhermore, we ca check ha if L is a coiuous addiive fucioal of X wih -poeial U, he L is a coiuous addiive fucioal of Y wih 0-poeial U We have already proved his heorem i he case 0 Therefore, for V, R L is uiformly coiuous o R D, we have Ž 38 Ž P x h Ž V x S Q for some polar se Q S We resric X, ad cosequely Y, ad he measures M o S Q ad hus ca cosider ha boh X ad Y have sae space S S Q Le ˆ L is locally uiformly coiuous o R D ˆ ad oe ha R V Fubii s heorem ad a moooiciy argume ow show ha x h Ž 39 P Ž ˆ x S ˆ ˆ From he defiiio of, we see ha, for each, L Ž, Ž, R D ca be exeded o a locally uiformly coiuous sochasic process Lˆ ˆ ˆ,, R M Se L 0 for c By akig he limi over sequeces of measures i D, we see ha Lˆ is a coiuous addiive fucioal for each M We ow show ha Lˆ has -poeial ŽU hž x hž x wih respec o he h-rasform of X ad cosequely is a coiuous versio of L Ž, Ž, R M Accordig o he proof i he case 0 applied o he h-ras- form of Y, L,, R D has a coiuous exesio o R M ˆ for all, R We deoe his exesio by L,,, R Ž M ad recall ha i has 0-poeial U hž x hž x Iegraig by pars ad usig Fubii s heorem, we have Ž 30 ž / ž / Ž E Ž L Ž, x h s x h s 0 0 x h E e dl Ž, s E e L Ž, s ds U hž x hž x ˆ ˆ Clearly, L, s Ls for all ad s R, sice boh sides are coiuous exesios of L ˆ s Ls for D Thus we see ha Ls Ž has -poeial U hž x hž x for he h-rasform of X The rasiio

21 50 M B MARCUS AND J ROSEN from he h-rasformed process o he origial process is he same as i he case 0 This complees he proof of Theorem The ex heorem relaes o Remark TEOREM 3 Le X be a Markov process as i Theorem ad le M G be a se of measures wih commo compac suppor Assume ha we are give a opology O for M uder which M is locally compac ad has a couable base Assume also ha: i U h is a coiuous map from M o BŽ S Ž ; Ž ii he associaed secod-order Gaussia chaos is coiuous almos surely o M The here exiss a polar se Q S such ha, if we resric X ad M o S Q, we ca fid a coiuous versio of L,, R M PROOF The codiio ha U is a coiuous map from M o BŽ S of Theorem was used oly o eable us o saisfy he requireme of he isomorphism heorem ha U is bouded for each M ad o obai upper bouds i Ž 36 Acually, we oly used he weaker codiio: Ž i U hž is a coiuous map from M o B S ad U is a bouded map from C o BŽ S for all compac ses C M; ha is, sup sup U Ž x C x S Ž The firs codiio of i implies ha sup sup U h Ž x C x S Ž The he secod codiio i Ži follows from his, sice if h Ž x K Ž 0, where K S deoes a compac eighborhood coaiig he suppors of all he measures M The res of he proof follows from he proof of Theorem 63 of ad he fac ha oe of he coiuous addiive fucioals L are icreasig uless X K ŽSee Chaper 6, Theorem 3, of We ow develop he maerial o explai Remark Le be a separable locally compac group ad le X be a Levy process i, wih -poeial Ž desiy u x, y u xy We will use he caoical represeaio for X i which is he se of cadlag pahs : R, X Ž ad E x Ž f Ž E 0 Ž f Ž x For hese processes L deoes he coiuous addiive fucioal of X Ž wih -poeial U x u xy dž y For each measure o ad x, we defie he measure o be he uique measure o for which x g Ž z d Ž z g Ž zx dž z x for all bouded coiuous fucios g o Noe ha Ž x y y x ad Ž Ax Ž A x for all Borel ses A Le Tx deoe he bijecio o he space of measures defied by T We say ha a se M of measures x x

22 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 5 o is raslaio ivaria if i is ivaria uder Tx for each x ad ha a opology O o such a se M is homogeeous if Tx is a isomorphism for each x The ex heorem relaes o Remark TEOREM 3 Le X be a symmeric Levy process i ad le M G be a raslaio-ivaria se of measures o Assume ha here is a homogeeous opology O for M uder which M is locally compac ad has a couable base ad ha: i U ad U h are coiuous maps from M o BŽ Ž ; Ž ii he associaed secod-order Gaussia chaos is coiuous almos surely o M The here exiss a coiuous versio of L,, R M PROOF We give he proof i he case The same proof is valid for all 0 ad also for 0 for rasie processes For ay x ad M, se A Ž L Ž x Clearly, A Ž 3 is a coiuous addiive fucioal Compuig is -poeial ž / ž / y x E ž e dl Ž / 0 y y 0 0 E e da Ž E e dl Ž x u yxz dž z u yž zx dž z u yz d Ž z, we see ha Ž 3 L x L x Ž as for each x ad M Le D M be he couable dese se of measures ha eers io he proof of Theorem The proof of Theorem shows ha L,, R a c D is locally uiformly coiuous, P almos surely, for all a Q, where Q is a polar se As i Theorem, we exed L by coiuiy o c L,, R M for all pahs sarig i Q owever, his proce- dure provides o way o exed L, Ž, R M o pahs sarig i Q Tha is why, i Theorem, we foud i ecessary o resric he Markov process o Q c I his heorem we use he raslaio ivariace of M o exed L, Ž, R M o pahs sarig i Q x

23 5 M B MARCUS AND J ROSEN We ca assume ha Ž 3 holds for all D, almos surely Therefore, by coiuiy, we have ha x y Ž 33 L Ž x L Ž M, P as for each x ad y Q c such ha yx Q c This suggess how we ca c exed L,, R M o pahs sarig i Q Fix a Q ad for each pah sarig a a ad each y Q se 3 L a y L a y Ž def We oe ha a y is a bijecio from he se of cadlag pahs i sarig a a o hose pahs sarig a y We mus verify ha wih his defiiio L, Ž, R M saisfies he requiremes of his heorem, ha is, i is coiuous almos surely, ad ha, for each M, L is a coiuous addiive fucioal of X wih -poeial U We firs show ha L, Ž, R M are coiuous addiive fucio- als for he Levy process X The oly par requirig proof is he addiiviy: Ž 35 L Ž L Ž L Ž, P y as s s for all y ad M c If y Q, Ž 35 follows by coiuiy sice i holds for all D Noe c y ha, sice Q is a polar se, 0 Q, P almos surely a Cosider ow ha y Q By defiiio 3 we have ha, P almos surely, L a y L a y Ž Ž 36 s s a y a y s L Ž L Ž L a y L a y s Ž Assume ha Ž 33 holds wihou resricio o x, ha is, ha x y Ž 37 L Ž x L Ž M P as for each x ad y Q c The, usig he Markov propery, we see ha Ž 38 Ž s Ž s a X P P L a y L a y Ž s s Ž a P L a y L a y Ž The fial equaliy follows from Ž 37, sice, by he polariy of Q, we have c a X Q, P almos surely Usig Ž 36 ad Ž 38, we see ha a 39 L a y L a y L a y, P as, s s Ž which is equivale o Ž 35 To show ha L has -poeial U, we oe ha, by Theorem, ž / y 0 Ž 30 E e dl Ž U Ž y

24 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 53 for all y Q, ad, sice boh sides of Ž 30 are excessive fucios, hey will be equal for all y by Theorem 3, Chaper of Aleraely, we ca use he calculaio i Ž 3 Agai, by Theorem, sice for Levy y processes, L,, R M is joily coiuous, P almos surely, for all y Q ad hece for all y by defiiio Ž 3 ad he fac ha T x is a isomorphism for each x We ow reur o he proof of c 37 If yx Q, his is precisely he coe of Ž 33 If yx Q, he by Ž 3 we have Ž 3 L a yx L a y x Ž, a P as c Ž c Now, sice boh a Q ad a a y y Q, i follows from Ž 33 ha Ž 3 x Ž x a L a y L y L a y x Ž, a P as Combiig Ž 3 ad Ž 3, we obai L a yx L x a y, a P as, which is equivale o 37 Local imes I his secio we explai how local imes fi io our framework Suppose ha he Markov process X has a local ime Cosider he se of measures M, a S a, where a deoes he ui poi mass a a We se L a L a ad oe ha i is he ordiary local ime of X a a Le Ž, M a a be he associaed Gaussia chaos Whe a Markov process has a local ime, coiuiy of he local ime ad he associaed Gaussia chaos are equivale, ad we obai he resuls discussed i Remark 5 Le / Ž T Ž d, M def lim sup log d, 0 ž B Ž, M 0 d where d,, M ad B are defied i Theorem TEOREM Le X be a Markov process as i Theorem wih sae space S ad assume ha a local ime exiss for X a all pois a S The he followig are equivale: a i L, a S has a coiuous versio almos surely; Ž ii Ž, M a a has a coiuous versio almos surely; Ž iii for all compac subses M of M, here exiss a probabiliy measure o M such T Ž d, M 0 Furhermore, if X is a Levy process i R, or T, Ž iii ca be replaced by Ž iii JŽ d, M for all compac subses M of M PROOF I is eough o prove his heorem for S compac I his case we ca ake M o be compac ad we deoe i by M If X has a local ime for all x S, he u Ž x, x for all x S ŽSee, eg, Theorem 3 of I his case he Gaussia process GŽ, M, defied i Secio, is he same a a

25 5 M B MARCUS AND J ROSEN as he Gaussia process GŽ a, a S wih covariace u Ž x, y ŽAs we remarked i Secio he cosrucio ca be carried ou for all I follows a from Theorem of ha L, a S has a coiuous versio almos surely if ad oly if GŽ a, a S is coiuous We see from he cosrucio a a of he chaos associaed wih L L ha Ž G Ž a EG Ž a a a Thus, obviously, Ž, M is coiuous if ad oly if GŽ a, a S a a is coiuous Thus we see ha Ž i ad Ž ii are equivale A ecessary ad sufficie codiio for he coiuiy of GŽ a, a S is ha here exiss a probabiliy measure o S such ha T Ž, S 0, where Ž For he measures i M, he meric defied i Ž is x, y u x, x u y, y u x, y Ž 3 Similarly, Ž x y ' ŽŽ u Ž x, x u Ž x, y u Ž x, x d, u Ž x, x u Ž y, y u Ž x, y ' Ž u y, y u x, y u y, y Ž sup u x, x x, y xs C Ž x, y Ž d x, y if u x, x x, y xs C Ž x, y We kow ha C 0 ŽSee, eg, Lemma 36 of Thus, if here exiss a probabiliy measure o S such ha T Ž, S 0, he T Ž d, M 0 O he oher had, if here exiss a probabiliy measure o S such ha T Ž, S 0, he L a, Ž a, S R is coiuous ad hece, by Theorem 37 of, u Ž x, x is coiuous o S ece C ad cosequely T Ž d, M 0 Sice GŽ a, a S is coiuous if ad oly if here exiss a probabiliy measure o S such ha T Ž, S 0, i follows ha Ž ii ad Ž iii are equivale I is clear from he firs lie of 3 ha d is also a meric o R, or T Thus we ca wrie Ž as Ž T Ž d, 0, lim sup log d, 0 0 ž Ž B Ž, / x0, d where is a probabiliy measure o 0, Furhermore, if X is a Levy process d is raslaio ivaria I is well kow ha for raslaio ivaria merics o R ad T we ca ake he measure i Ž o be ormalized Lebesgue measure ad ha T Žd, 0, 0 if ad oly if JŽd, 0, owever, ow cosiderig d as a meric o M, we see ha Ž J d, 0, J d, M Thus, whe X is a Levy process i R, or T, Ž iii ca be replaced by Žiii

26 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 55 5 Levy processes i R We ow specialize o Levy processes i R ad families of measures M, x R x which cosis of raslaes of a fixed measure o R We firs develop maerial leadig o he proof of Theorem 3 We he prove Theorem 6 ad lasly cosider he relaioship bewee he -poeial ad he meric eropy iegral Ž 5 Le us defie Ž 5 VhŽ x,, x k Ž z k Ýu Ž z, y u Ž y, y u Ž y k, yk hž yk Ł dx Ž y i, Ži i where he sum rus over all permuaios of,, k Noe ha V Ž x Ž z U h Ž z h ad also ha fucios such as V Ž x,, x Ž z h k arise i he proof of he isomorphism heorem, as i Ž 6 The ex heorem is used i he proof of Theorem 3 TEOREM 5 Le X X, R be a symmeric Levy process i R Le G be a fiie measure o R If Ž, x R x is coiuous almos surely, he for ay h U f, where f SŽ R is sricly posiive, x U h is a bouded ad uiformly coiuous map from R o BŽ R x, ad, more geerally, Ž 5 x, x,, x V Ž x,, x Ž z k h k Ž k Ž is a bouded ad uiformly coiuous map from R o B R PROOF We will firs show ha x U h is a bouded ad uiformly x Ž coiuous map from R o B R Tha is, Ž 53 sup U h Ž z for each x R, ad x, zr Ž 5 lim sup sup U hx Ž z U h yž z 0 0 xy zr Le us assume firs ha gž x dx, where gž x is bouded ad uiformly coiuous I paricular, his guaraees ha s x 55 Ls g Xr x dr, x R, 0 is coiuous almos surely ad has bouded -poeial U g Ž x, where g g x We ow obai bouds o U h i erms of Ž x def which will exed o all fiie G We begi by oig ha Ž 56 U h Ž z E z L f Ž X E y L z y f Ž X x x zy

27 56 M B MARCUS AND J ROSEN The firs equaliy is paricularly easy o see for L of he form Ž 55 The secod iequaliy follows by a chage of variables Le B x R x r As i Ž 56, for x, z, v B we have ha r Ž 57 U h Ž z E v L x zv f Ž X Therefore x zv Ž 58 sup U h Ž z E sup L f Ž X, v z x zb zxb where Ž 59 f Ž x sup f Ž x Ž d We oe ha, sice f S R zb z, f has he propery ha Ž 50 u Ž x, y f Ž x f Ž y dx dy Therefore, iegraig Ž 58 wih respec o dmž v resriced o B ad usig he isomorphism heorem, Theorem, ad he saioariy of, we see ha 5 sup U hx z C sup z, zb zb uiformly i x R Fially, usig he saioariy of ha agai ad he fac Ž 5 U h y xyž z U hx Ž z y ad ha Ž 50 is uchaged if we replace f by f, we see ha Ž 5 gives Ž 53 U hx Ž z C sup Ž z zb for all x, z R Similarly, for x, y, z, v B we have, as i Ž 56, ha 5 U h z U h z E v L x zv L y zv f Ž X ece y Ž x y zv v x Ž y x y xy xy x, y, zb x, yb3 sup U h Ž z U h Ž z E sup L L f Ž X, where f is defied i 59 As above, he isomorphism heorem, Theorem, ow shows ha Ž 55 sup U hx Ž z U h yž z C sup Ž x Ž y xy xy x, y, zb x, yb3

28 ad ha 55 implies ha PAT PROPERTIES OF ADDITIVE FUNCTIONALS 57 Ž 56 sup U hx Ž z U h yž z C sup Ž x Ž y xy xy x, yb 3 for all z R We ow prove he assumpio, ha has a bouded uiformly coiuous desiy, i Ž 53 ad Ž 56 Le bž x be a posiive coiuous ad symmeric fucio suppored o B wih bž x dx Le x b Ž x b d ž ad se b g Ž x dx Noe ha g Ž x b Ž x y Ž dy is bouded ad uiformly coiuous whe is a fiie measure We apply 53 wih replaced by o obai Ž 57 / sup U h Ž z C sup Ž x x xr xb C sup b x xb C sup Ž x xb 3 I he secod lie of Ž 57 we use he fac ha Ž 58 Ž x b Ž x, which follows easily from Ž 9 We ow ake he limi i Ž 57 as 0 We show below ha, for ay x R, Ž 59 U hx Ž z U hx Ž z Ž i L R, dz as 0 Therefore for some subsequece 0 we have 50 U h k Ž z U h Ž z x for almos all z wih respec o Lebesgue measure This implies ha Ž 5 U hx Ž z C sup Ž x xb 3 for almos all z owever, sice U h Ž z is -excessive, i follows ha Ž 5 x holds for all z Le us ow prove Ž 59 I suffices o cosider he case x 0 Sice Ž 5 U h Ž z U h Ž z b Ž x dx, i is eough o show ha x U h Ž z x is a bouded ad uiformly coiu- Ž ous map from R o L R, dz To see his, i suffices o oe ha Ž 53 U h hž y Ž dy h Ž R x x x x k

29 58 M B MARCUS AND J ROSEN ad Ž 5 U h U h x u z y h y x dy u z y h y dy dz ž / yr u z y x h y x dy u z y h y dy dz sup u Ž z y x hž y x u Ž z y hž y dz Ž R Ž x x u u h u h h Ž R Similarly, we ca ow remove he assumpio, ha has a bouded uiformly coiuous desiy i Ž 56, by arguig exacly as above, wih respec o he L -fucio z U h z U h Ž z This shows ha Ž 56 x y holds for all fiie G This complees he proof of Ž 53 ad Ž 56 i he geeral case, ad verifies Ž 53 ad Ž 5 We ow prove ha Ž 5 is bouded ad uiformly coiuous As above, le us assume firs ha gž x dx, where gž x is bouded ad uiformly coiuous Noe ha Ž 55 k z x i h k ž Ł i / k y x i zy ž Ł zy i / V x,, x z E L f X E L f X The firs equaliy is sraighforward for L of he form Ž 55 See also Ž The secod equaliy follows by a chage of variables Therefore k v Ł zi h k zb z ix ib i sup V Ž x,, x Ž z E sup L f Ž X, where, as before, Ž 56 f Ž x sup f Ž x zb As above, Theorem, older s iequaliy ad he saioariy of us ha z k 57 sup Vh x,, x k z C sup x, zb zb show uiformly i x Ž,, x k R k Agai, by saioariy, as above, we see ha k 58 sup Vh x,, x k z C sup x, zr zb uiformly i x Ž,, x k R k The assumpio ha has a bouded uiformly coiuous desiy ca be removed exacly as before This shows ha Ž 5 is bouded The proof of uiform coiuiy is similar

30 PAT PROPERTIES OF ADDITIVE FUNCTIONALS 59 PROOF OF TEOREM 3 By Theorem 5, U h Ž z Ž is bouded so ha RevŽ X Le x i i be a sequece of pois i R We will firs show ha he versio of he isomorphism heorem, Theorem, sill holds if we ake x i for he sequece of measures i i, eve wihou he assumpio i ha he U z are bouded o R We firs oe ha, for each i, x i Ž 59 U x i Ž x qe x, which follows from he fac, 0, ha G so ha U Ž x dž x x x i i for all G ŽSee, eg, Theorem 33 of 6 Furhermore, hž Ž 530 L dl s, h Ž X 0 Ž s sice boh sides are coiuous addiive fucioals wih he same Revuz measure Togeher wih Ž 59 ad Theorem 3, Chaper 6, of, his shows ha 53 E x L x i U Ž x qe x x i This, ogeher wih Theorem 5 which eables us o corol he iegrals i Ž 6, allows us o esablish Ž for qe x which is sufficie o esablish he isomorphism heorem, Theorem We he follow he proof of Theorem ere, he assumpios of ha heorem cocerig U Ž z are used oly i boudig Ž 36 x, ad, oce agai, Theorem 5 allows us o do his Fially, i he proof of Theorem 3 we ca h Ž use he relaioship 530 bewee L ad L o esablish Ž 3 ad o ideify L Puig all his ogeher complees he proof of Theorem 3 PROOF OF TEOREM 6 By Theorem, Remark ad Theorem 3, we eed oly show ha Ž holds wih d d By Ž ad Ž 5, R / h Ž 53 d Ž y h, y si Ž ˆ Ž d ž Recall ha d y h, y d, For x 0 defie y yh x Ž 533 FŽ x Ž ˆ Ž d ad oe ha h Ž / Ž 53 dž y h, y Cž h 0 u dfž u FŽ h Ž h A sligh modificaio of he argume o pages 5 of 7 ake Ž u u or of he proof of Lemma, Chaper 7, of 0 shows ha Ž 6 implies ha Ž u Ž 535 du u 0

31 60 M B MARCUS AND J ROSEN I is easy o see ha Ž 535 implies ha Ž holds ŽSee, eg, Lemma 53, Chaper, of 7 or he proof of Lemma 36, Chaper, of 0 The equivalece of Ž i ad Ž ii i Theorem 5 is a surprisig relaioship bewee he -poeial of a Levy process i T ad he square-roo meric eropy of he Gaussia chaos associaed wih cerai of is coiuous addiive fucioals I he ex heorem we see ha a similar resul holds for processes ad measures i R TEOREM 5 Le X be a Levy process i R wih characerisic sequece ad le be a fiie measure o R,, 3 Assume ha: Ž i ˆ Ž 0; Ž ii Ž ad ˆ Ž are radially symmeric; Ž iii here exis cosas 0 C, C such ha Ž Ž C C Ž for ; Ž iv here exis a decreasig sequece a j j of posiive umbers ad cosas 0 C, C such ha j j for The ˆŽ Ž j C aj C a U 0 log Nd R, d We use Boas s lemma, Lemma, Chaper, of 7 LEMMA 5 Le s j j be a sequece of posiive real umbers Suppose sj as j The here exis cosas 0 C, C such ha j j j j j j Ý Ý Ý Ý 537 C s s C s Furhermore, he lef-had side of Ž 537 remais valid wihou he codiio ha sj as j ha PROOF OF TEOREM 5 By Ž iv here exis cosas 0 C, C such ˆŽ Ž 538 Caj d C a j, j j j Ž