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Martingales, Girsanov Type Theorems and Applications to The Markov branching diffusions martingales Girsanov type theorems and applications to the long term behaviour Janos Englander and Andreas E Kyprianou Octob er spinetex Abstract Consider a spatial branching particle pro cess where the underlying mo d tion is a conservative diusion on D R corresp onding to the elliptic op erator L on D and the branching is strictly binary dyadic with spatially varying rate x and which is assumed to b e b ounded from ab ove Weprove that under extremely mild circumstances the pro cess exhibits lo cal extinction if and only if where denotes the gen c c eralized principal eigenvalue for the op erator L on D This criterion is analogous to the one obtained by Pinsky for the lo cal extinction of sup erdiusions Furthermore we show that when the pro cess do es not exhibit lo cal extinction every nonempty op en subset is o ccupied innitely often with p ositive probability whic h can b e characterized by a solution b ounded in to the semilinear elliptic equation Lu u u on D Moreover in this case there is an exp onential rate of growth on suciently large compact domains and this rate can b e arbitrarily close to In order to reach these conclusions we rst develop some results c concerning innerpro duct and multiplicative martingales and their relation to the op erators L and L resp ectively where x x x In the case of the innerpro duct martingales weshow that for some cir cumstances they can be used as changes of measure for the law of the branching pro cess in a similar way that Girsanov densities act as changes of measure in the context of diusions More sp ecically the change of measure induces a drift consisten t with a certain Do obs htransform on the path of a randomized ancestral line of descent These concepts are essentially spatial versions of spine decomp ositions for GaltonWatson pro cesses given in Lyons et al AMS classication Primary J Corresp onding author Keywords spatial branching pro cesses branching diusions lo cal extinc tion spine decomp osition change of measure on trees generalized prin cipal eigenvalue Intro duction and main results Markov branching diusions d Let D R b e a domain and consider Y fY tt g the diusion pro cess with probabilities fP x D g corresp onding to the elliptic op erator L on D x satisfying d d X X a b on D L ij i x x x i j i ij i where the symmetric matrix axa x is p ositive denite for x D and ij a C D b C D We assumed these stronger than usual smo othness ij i assumptions for convenience they guarantee that there will be no problem dening the adjoint op erator We assume that Y is conservative that is that D D inf ft Y t D g satises P in other words x Y has an as innite lifetime F urthermore let C D be b ounded from ab ove on D and Here C D denotes the usual Holder space Then the binary L branching diusion under the measure P is dened informally as follows A single particle starts at p osition x D x performs a Y motion on D This particle is killed with spatially dep endent rate At the moment and spatial p osition of its death the particle pro duces precisely two ospring Each of these two individuals pro ceed indep endently to perform Y motions killed at rate at whichpoint they repro duce in the same t each time t the branching diusion way as their parent and so on A consists of a point pro cess Z dened on Borel sets with almost surely nite t total mass Z D t In the sequel the notation P E and Z will be used for the branching x x diusion and the notation P E and Y will be used for the diusion on D x x corresp onding to L Moreover we shall use the UlamHarris label ling notation That is an individual u is identied by its line of decent from the initial ancestor More precisely if u i i i then she is the i th child of the i th n n n n child of of the i th child of the initial ancestor Thus uv refers to the individual who from us p ersp ective has line of descent expressed as v Further the length juj is equal to the generation in which individual u lives We shall use N to t denote the set of individuals alive at time t and fY t u N g for their u t p ositions in D In this waywehave for example Z Acardfu N Y t t t u Ag where A is any Borel set A whole array of questions can b e asked ab out the large time b ehavior of the pro cess Z fZ g Basic questions which address the concepts of lo cal t t extinction and recurrence however undened as yet fo cus on whether this pro cess will visit nonempty op en sets innitely often and if so how can this b e quantied Surprisingly there are few results in this direction in existing liter ature In the late fties and sixties there were a small cluster of pap ers which considered simple prop erties of either general or sp ecic examples of branching diusions Sp ecically we sp eak of the former Soviet and Japanese contri butions of Sevastyanov Skoroho d Watanab e and Ikeda et al ab Some results can b e found amongst these references p ertaining to the kind of problems wehave alluded to ab ove in particular Watanab e In more recent times the numb er of articles concerning growth and spread of branching diusions are again largely restricted to sp ecial cases for example branching random walks and branching Brownian motions or simple variations thereof Notably Biggins has pro duced lo cal limit theorems analogous to the one in Watanab e which demonstrate that numb ers of particles in any Borel set can b e appropriately rescaled over time bytheiraverage to achievea Law of Large Numb ers typ e result Ogura also showed for branching diusions where the motion pro cess is a Brownian motion with a restricted class of space dep endent drift that in a given compact set the number of particles will b ecome zero and remain zero or a strong law of large numb ers can b e pro duced for the numb er of particles in that set By comparison the recent developments and p opularity of measurevalued diusions sup erdiusions havegiven a more thorough treatment of analogous issues concerning concentration and migration of mass It is almost imp ossible to give a full account of books and pap ers on measurevalued diusions We therefore restrict ourselves here by mentioning the two basic textb o oks Daw son and Dynkin and the recent monograph Etheridge on measurevalued pro cesses in general and the articles Delmas Englander and Pinsky Isco e Pinsky b Pinsky and T rib e regarding the long term b ehaviour of these pro cesses when then underlying mo tion pro cess is a diusion in particular In this article wehave three main goals The rst is to formalize the rela tionship b etween the op erators L and L where xx x and two classes of martingales related to the Markov branching diusion Secondly to intro duce the concept of spine decomp osition for Markov branching diusions Thirdly we aim to show the functionality of these twoby applying them to the fundamental question of local extinction versus recurrence Innerpro duct and multiplicative martingales In the mid nineties the article of Lyons et al app eared which formalized a new approach to analyzing some fundamental problems in the GaltonWatson pro cess It was shown that a classical martingale that has b een a p opular ob ject of study in earlier years numb ers aliveinthe nth generation divided by their exp ectation can serve as a RadonNikodym change of measure on the space of GaltonWatson branching trees The eect of this change of probabil ityistopick out a randomized line of decent called the spine and to sizebias ospring distributions asso ciated with each no de along the spine In the case of GaltonWatson pro cesses the spine dominates the b ehavior of the pro cess under the new measure That is to the extent that simple and intuitively app ealing probabilistic pro ofs of classical theorems can be achieved where the original pro ofs were more analytical and complicated In fact this phenomenon is not particular to GaltonWatson pro cesses Anumb er of authors haveshown that similar constructions can b e pro duced for a variety of nonspatial and discrete time spatial branching pro cesses see Lyons Kurtz et al Olof sson Athreya Kyprianou and Rahimsadeh Sani Biggins and Kyprianou In each of these cases a naturally o ccurring martingale is used as the appropriate change of measure The natural o ccurrence of these martingales is a consequence of the fact that they are constructed from pos itive harmonic functions Take for example the case of a discrete time typ ed branching pro cess Roughly sp eaking for this case the afore mentioned mar P n tingales are of the form hX where the sum is typically taken over i i individuals aliveinthe nth generation X is the a typ e of individual i which i could b e for example a birth p osition and h is an eigenfunction with resp ect to the exp ectation op erator with eigenvalue The last of these prop erties can P b e written using obvious notation E hX hx where is the eigen x i i value See Athreya and Ney Chapter VI Athreya and Biggins and Kyprianou In Section we show that the idea of martingales from harmonic functions transfers comfortably into the context of Markov branching diusions In this R case the analogous class of martingales takes the form expftg hy dy t where is an appropriately evolving p oint pro cess emb edded within the Markov branching pro cess and h is p ositive and harmonic with resp ect to L for some appropriate R These martingales we refer to as innerpro duct martingales for the case of branching Brownian
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