arXiv:1303.4181v1 [math.PR] 18 Mar 2013 hebc rcs,cne fms,lwo ag numbers. large of law mass, of center process, Uhlenbeck eteo aso h rcs si stesmls iuino t of diffusion simplest the consid is to it as natural process is the It of (repelle below. mass attractor of shown with centre is process mass (SOU) of Ornstein–Uhlenbeck attract centre either its is from that diffusion measure-valued interacting 03 o.4,N.2 989–1029 2, No. DOI: 41, Vol. Probability2013, of Annals The
c n yorpi detail. 989–1029 typographic 2, the and by No. published 41, Vol. article 2013, original Statistics the Mathematical of of reprint Institute electronic an is This nttt fMteaia Statistics Mathematical of Institute .Itouto n anresults. main and Introduction 1. e od n phrases. and words Key M 00sbetclassifications. subject 2011. 2000 December AMS revised 2011; August Received UE RSENULNEKPOESINTERACTING PROCESS ORNSTEIN–UHLENBECK SUPER A 10.1214/11-AOP741 rgn n rv otnu nlge frslso Engl¨an of results of analogues continuum attracti prove with and process origin, Ornstein–Uhlenbeck calculu super stochastic a rep historical modify the or we Using to, mass. attracted of center are the that particles representative with ovre oarno on a.s. point ex random the repe and a approaches attractive to it the converges both as in is, process normalized that the set; time, extinction the on proven n hoe fEg¨ne n itr[ Winter Stat. Engl¨ander and Probab. of Ornstein–Uh which theorem variable, a super random Gaussian ens normalized In a the to mass. a.s. of that converges center process proven its is of it value limiting setting, Ornstein–Uhle the the at of centered distribution cess, stationary the to surely conve process measure-valued interacting normalized mass motion. Probab. J. tron. otat ihtecne fms fa riaysprOrnste super diverge ordinary to an shown is of which mass repulsion, conjectu of with a process center give Uhlenbeck a. then the and converges strong with mass too of contrasts not center is the repulsion the that vided show we setting, pelling ecntutasprrtclitrcigmauevle d measure-valued interacting supercritical a construct We ti hw,o h uvvlst hti h trciecs t case attractive the in that set, survival the on shown, is It eso farsl fTie[ Tribe of result a of version A IHISCNE FMASS OF CENTER ITS WITH 42 nvriyo rts Columbia British of University 15 uepoes neatn esr-auddffso,Orns diffusion, measure-valued interacting Superprocess, 20)1115 nti atclrcs.I h re- the In case. particular this in 171–185] (2006) 21)13–90 o iaybacigBrownian branching binary for 1938–1970] (2010) 2013 , yHrepGill Hardeep By hsrpitdffr rmteoiia npagination in original the from differs reprint This . rmr 06;scnay60G57. secondary 60J68; Primary in h naso Probability of Annals The n.Probab. Ann. 1 h xsec n nqeeso self- a of uniqueness and existence The n.Is.HniPoincar´e Henri Inst. Ann. 20 19)2631 is 286–311] (1992) fPerkins, of s gsalmost rges le from, elled bc pro- nbeck ln cases lling dt rrepelled or to ed e [ der strength- h same the nt its to on )gvnb the by given r) tinction , e This re. lenbeck iffusion . pro- s., a.s. Elec- i sort. his in– he rasuper a er tein– 2 H. GILL
This type of model first appeared in a recent paper of Engl¨ander [1] where a d-dimensional binary Brownian motion, with each parent giving birth to exactly two offspring and branching occurring at integral times, is used to construct a binary branching Ornstein–Uhlenbeck process where each particle is attracted (repelled) by the center of mass (COM). This is done by solving the appropriate SDE along each branch of the particle system and then stitching these solutions together. This model can be generalized such that the underlying process is a branching Brownian motion (BBM), T (i.e., with a general offspring dis- tribution). We might then solve an SDE on each branch of T: t t n n−1 ¯ n n (1.1) Yi (t)= Yp(i) (n 1) + γ Yn(s) Yi (s) ds + dBi (s) − Zn−1 − Zn−1 n for n 1
K X ( )= 1(yt )Kt(dy), t · Z ∈ · which is a (supercritical) super Brownian motion. SOU INTERACTING WITH ITS COM 3
A key advantage to projecting down and constructing measure-valued pro- cesses is that it is possible to use the historical stochastic calculus to couple different projections together (and hence couple measure-valued diffusions). One can sensibly define the Ornstein–Uhlenbeck SDE driven by y accord- ing to Kt as the solution of a stochastic equation.
d d Definition 1.1. Let Z0 : R R be Borel measurable. We say that (X,Z) is a solution to the strong→ equation (SE)1 if the pair satisfies Z0,K t (a) Zt(y)= Z0(y0)+ yt y0 γ Zs(y) ds, K-a.e., − − Z0 ((SE)1 ) Z0,K d (b) Xt(A)= 1(Zt A)Kt(dy) A B(R ) t 0, Z ∈ ∀ ∈ ∀ ≥ where X and Z are appropriately adapted. We will henceforth call the projection X the ordinary super Ornstein–Uhlenbeck process.
If γ > 0, Zt is attracted to the origin, and if γ < 0 it is repelled. The approximate meaning of K-a.e. in (a) is that the statement holds for Kt- a.a. y, for all Kt, P-a.s. The exact definition is given in the next section. The projection Xt is a SOU process with attraction to (repulsion from) the origin at rate γ. Intuitively, K tracks the underlying branching structure and Zt is a function transforming a typical Brownian path into a typical Ornstein–Uhlenbeck path. Note that in the definitions of (SE)1 and (SE)2 given below, part Z0,K Y0,K (b) is unnecessary to solve the equations. It has been included to provide an easy comparison to the strong equation of Chapter V.1 of [9]. For all the results mentioned in the remainder of this work, the standing assumption (unless indicated otherwise) will be that
(1.2) 1 dK0 < . Z ∞
Theorem 1.2. There is a pathwise unique solution (X,Z) to (SE)1 . Z0,K That is, X is unique P-a.s. and Z K-a.e. unique. Furthermore, the map t Xt is continuous and X is a β-super-critical super Ornstein–Uhlenbeck process.→
Similar to the above, we establish a function of the path y that is a path of an OU process with attraction (repulsion) to the COM and project down.
d d ′ Definition 1.3. Let Y0 : R R be Borel measurable. Define (X ,Y ) as the solution of → t (a) Yt(y)= Y0(y0)+ yt y0 + γ Y¯s Ys(y) ds, K-a.e. − Z0 − 4 H. GILL
((SE)2 ) Y0,K ′ d (b) X (A)= 1(Yt A)Kt(dy) A B(R ) t 0, t Z ∈ ∀ ∈ ∀ ≥ where the COM is ′ ¯ xXs(dx) Ys = ′ . R 1Xs(dx) We will call the projection X′ the superR Ornstein–Uhlenbeck process with at- traction (repulsion) to its COM, or the interacting super Ornstein–Uhlenbeck process.
Note that by our definitions of X and X′ as solutions to (SE)i , i = 1, 2, Y0,K respectively, that (1.2) is the same as saying that 1 dX , 1 dX′ < , as 0 0 ∞ these quantities equal that in (1.2). R R
Theorem 1.4. There is a pathwise unique solution to (SE)2 . Y0,K One could prove this theorem using a combination of the proof of The- orem 1.2 and a localization argument. We find it more profitable however, to employ a correspondence with the ordinary SOU process X. This corre- spondence plays a central role in the analysis of X′, and indeed reveals a very interesting structure: We have that for any γ, t ′ ¯ (1.3) φ(x) dXt(x)= φ x + γ Zs ds dXt(x), Z Z Z0
R xXs(dx) where Z¯ is the COM of X, defined as Z¯s = . The correspondence R 1Xs(dx) essentially says that the SOU process with attraction (repulsion) to its COM is the same as the ordinary SOU process being dynamically pushed by its COM. From this equation, a relation between Y¯ and Z¯ can be established: t Y¯t = Z¯t + γ Z¯s ds. Z0 ′ ˜ Xt ˜ ′ Xt Define Xt = and Xt = ′ . As the goal of this work is to prove that Xt(1) Xt(1) ˜ ′ Xt has interesting limiting behavior as t approaches infinity, (1.3) yields a method of approach: show first that the time integral in (1.3) converges in some sense and establish limiting behavior for X˜ . One then hopes to combine these two facts with (1.3) to get the desired result. Let S be the event that Kt survives indefinitely. Note that this implies that on S, both X and X′ survive indefinitely by their definitions as solutions of the equations above. Let η be the time at which Kt goes extinct. The next two theorems settle the question of what happens on the ex- tinction set Sc. SOU INTERACTING WITH ITS COM 5
c Theorem 1.5. On S , Y¯t and Z¯t converge as t η< , P-a.s., for any γ R. ↑ ∞ ∈ Theorem 1.6. On the extinction set, Sc, ′ X˜ δ and X˜ δ ′ t → F t → F as t η< a.s., where F and F ′ are Rd-valued random variables such that ↑ ∞ η ′ F = F + γ Z¯s ds. Z0 This last theorem is an analogue of the result of Tribe [14] for ordinary critical superprocesses. Note that here it does not matter whether there is attraction or repulsion from the COM. The following three theorems for the attractive case (γ> 0) form the main results of this work.
Theorem 1.7. On S the following hold: (a) If γ> 0 ∞ a.s. a.s. Z¯t 0 and Y¯t γ Z¯s ds, −→ −→ Z0 and this integral is finite almost surely. (b) If γ = 0, then Z¯t = Y¯t and this quantity converges almost surely.
This says that the COMs of ordinary and interacting SOU process con- verge and is the result that allows us to fruitfully use the correspondence of (1.3) to show convergence of the interacting SOU process. The next theorem shows that the mass normalized SOU process converges almost surely, which is a new result among superprocesses. Engl¨ander and Winter in [3] have shown that this process converges in probability, and before them, Engl¨ander and Turaev in [2] shown convergence in distribution. One expects a result of this sort to hold since in the particle picture con- ditional on survival, at large time horizons, there are a very large number of particles that move as independent OU processes, each of which are located in the vicinity of the origin. Thus, we expect that in the limit the mass will be distributed according to the limiting distribution of an OU process.
Theorem 1.8. Suppose γ> 0. Then on S, d(X˜ , P ) a.s. 0, t ∞ −→ where Pt is the semigroup of an Ornstein–Uhlenbeck process with attraction to 0 at rate γ, and d is the Vasserstein metric on the space of finite measures on Rd (see Definition 2.6). 6 H. GILL
Remark 1.9. It is possible to show that Theorem 1.8 holds for a more general class of superprocesses. If the underlying process has an exponential rate of convergence to a stationary distribution, then the above theorem goes through. One can appeal to, for example, Theorem 4.2 of Tweedie and Roberts [12] for a class of such continuous time processes.
Using the correspondence of (1.3), one can then show that the mass nor- malized interacting SOU process with attraction converges to a Gaussian distribution, centered at the limiting value of the COM, Y¯∞.
Theorem 1.10. Suppose γ> 0. Then on S, ¯ (1.4) d(X˜ ′, P Y∞ ) a.s. 0, t ∞ −→ ¯ Y∞ ¯ where P∞ is the OU-semigroup at infinity, with the origin shifted to Y∞.
When there is repulsion, matters become more difficult on the survival set. It is no longer clear whether there exists a limiting random measure, or what the correct normalizing factor is. We can show however that in some cases the COM of the interacting SOU process still converges. That there should be a limiting measure comes from the fact that the ordinary repelling SOU process has been shown to converge in probability by Engl¨ander and Winter in [3] to a multiple of Lebesgue measure. One may expect something similar to hold for the interacting SOU process, given (1.3). Unfortunately, the correspondence is rendered ineffectual in this case by part (b) of the following theorem.
Theorem 1.11. The following hold on S if 0 >γ> β : − 2 (a) The process, Y¯t converges almost surely. (b) However, Z¯t diverges exponentially fast. That is, there is a random variable L, such that P(eγtZ¯ L,L = 0 S) = 1. t → 6 | This reveals an interesting byplay between X and X′ in the repelling case. That is, if one fixes a compact set A Rd, then for the ordinary SOU process, ⊂ Xt, A is exponentially distant from the COM of the process. However, the COM of X′ will possibly lie in the vicinity of A for all time. Therefore, one ′ might expect that A is charged by a different amount of mass by Xt than ′ Xt, and thus we might need to renormalize Xt differently to get a valid limit. It is also possible that the limit for each case is different (and not simply connected by a random translation). The proofs for these theorems and more are contained in the following sec- tions. In Section 2, we give some background information for the historical SOU INTERACTING WITH ITS COM 7 process K and some rigorous definitions. In Section 3, we prove Theorems 1.2 and 1.4 and state some important preliminary results regarding the nature of the support of a supercritical historical Brownian motion. These are con- sequently used to get moment bounds on the center of mass processes Y¯t and ′ Z¯t. We also derive a martingale problem for X˜ and X˜ . In Section 4, we give the proofs of the convergence theorems mentioned above and in the final section give the proofs of technical results that are crucial to prove these.
2. Definitions and background material.
Notation. We collect some terms below: E, E′ Metric spaces ′ ′ Cc(E, E ) Compact, cont. functions from E to E ′ ′ Cb(E, E ) Bounded, cont. functions from E to E (E, E) Arbitrary measure space over E bE Bounded, E-mble. real-valued functions MF (E) Space of finite measures on E µ(f)= f dµ where f : E R and µ a measure on E → µ(f) =R (µ(f1),...,µ(fn)) if f = (f1,...,fn), fi : E R → p p Rd, denotes Euclidean norm of p | | ∈ f = supE fi(x) if f = (f1,...,fn), fi : E R k k d | | → d C = C(R+, RP) Space of continuous paths in R C The Borel σ-field of C t y = y·∧t The path y stopped at t Ct = yt : y C Set of all paths stopped at t C = σ{(ys,s∈ t,y} C) Natural filtration associated with C t ≤ ∈
We take K to be a supercritical historical Brownian motion. Specifically, let K be a (∆/2, β, 1)-historical superprocess (here ∆ is the d-dimensional Laplacian), where β> 0 constant, on the probability space (Ω, F, (Ft)t≥0, P). Here β corresponds to the branching bias in the offspring distribution, and the 1 to the variance of the offspring distribution. A martingale problem characterizing K is given below. For a more thorough explanation of histor- ical Brownian motion than found here, see Section V.2 of [9]. t It turns out that Kt is supported on C C a.s. and typically, Kt puts mass on those paths that are “Brownian”⊂ (until time t and fixed there- after). As K takes values in MF (C), Kt( ) will denote integration over the y variable. · Let Bˆ = B( t) be the path-valued process associated with B, taking t ·∧ values in Ct. Then for φ bC, if s t let P φ(y)= Es,y(φ(Bˆ )), where the ∈ ≤ s,t t right-hand side denotes expectation at time t given that until time s, Bˆ follows the path y. 8 H. GILL
The weak generator, Aˆ, of Bˆ is as follows. If φ : R+ C R we say φ D(Aˆ) if and only if φ is bounded, continuous and (C )-predictable,× → and ∈ t for some Aˆsφ(y) with the same properties as φ, t φ(t, Bˆ) φ(s, Bˆ) Aˆrφ(Bˆ) dr, t s, − − Zs ≥ s is a (Ct)-martingale under Ps,y for all s 0,y C . If m M (Rd), we will say K satisfies≥ the∈ historical martingale prob- ∈ F lem, (HMP) , if and only if K = m a.s. and φ D(Aˆ), m 0 ∀ ∈ t t Mt(φ) Kt(φt) K0(φ0) Ks(Aˆsφ) ds β Ks(φs) ds ≡ − − Z0 − Z0
((HMP)m) is a continuous (Ft)-martingale t 2 with M(φ) t = Ks(φs) ds t 0, a.s. h i Z0 ∀ ≥
Using the martingale problem (HMP)m, one can construct an orthogonal martingale measure M ( ) with the method of Walsh [15]. Denote by P, the t · σ-field of (Ft)-predictable sets in R+ Ω. If ψ : R+ Ω C R is P C- measurable and × × × → × t 2 (2.1) Ks(ψs ) ds < t 0, Z0 ∞ ∀ ≥ then there exists a continuous local martingale Mt(ψ) with quadratic varia- t 2 tion M(ψ) t = 0 Ks(ψs ) ds. If the expectation of the term in (2.1) is finite, h i 2 then Mt(ψ) is anR L -martingale.
Definition 2.1. Let (Ωˆ, Fˆ, Fˆ )=(Ω C, F C, F C ). Let Fˆ ∗ denote t × × t × t t the universal completion of Fˆ t. If T is a bounded (Ft)-stopping time, then the normalized Campbell measure associated with T is the measure PˆT on (Ωˆ, Fˆ) given by
P(1AKT (B)) PˆT (A B)= for A F,B C, × mT (1) ∈ ∈ where mT (1) = P(KT (1)). We denote sample points in Ωˆ by (ω,y). There- ˆ −1 fore, under PT , ω has law KT (1) dP mT (1) and conditional on ω, y has law K ( )/K (1). · T · T ˆ ∗ 1 2 Definition 2.2. For two (Ft )-measurable processes Z and Z , we will say that Z1 = Z2, K-a.e. if Z1(s,ω,y)= Z2(s,ω,y) s t, K -a.a. y ∀ ≤ t for all fixed times t 0. ≥ SOU INTERACTING WITH ITS COM 9
Definition 2.3. We say that (X,Z) is a solution to the strong equation (SE)1 if both (a) and (b) of that equation are satisfied and where Z is Z0,K t ˆ ∗ an (Ft )-predictable process and Xt is an (Ft)-predictable process. We say (X′,Y ) is the solution to the stochastic equation (SE)2 if it Y0,K ˆ ∗ satisfies (a) and (b) of that equation and where Yt is an (Ft )-predictable ′ process, Xt is an (Ft)-predictable process. Unless stated otherwise, (X,Z) will refer to a solution of (SE)1 and Z0,K (X′,Y ) to the solution of (SE)2 . Y0,K
ˆ ∗ d Definition 2.4. For an arbitrary Ft -adapted, R -valued process zt, define the centre of mass (COM),z ¯t, with respect to Kt as follows:
Kt(zt) z¯t = . Kt(1) We also let ′ ˜ Xt( ) Kt(Zt ) ˜ ′ Xt( ) Kt(Yt ) Xt( ) · = ∈ · , Xt( ) ′ · = ∈ · and · ≡ Xt(1) Kt(1) · ≡ Xt(1) Kt(1)
Kt( ) K˜t( )= · . · Kt(1)
Note that as K is supercritical, Kt survives indefinitely on a set of positive probability S, and goes extinct on the set of positive probability Sc. Hence, we can only make sense ofz ¯t for t < η where η is the extinction time.
Definition 2.5. Let the process Mt be defined for t [0,ζ) where ζ possibly random. M is called a local martingale on its lifetime∈ if there exist≤∞ stopping times TN ζ such that MTN ∧· is a martingale for all N. The interval [0,ζ) is called the lifetime↑ of M.
The following definition introduces a metric on the space of finite mea- sures, which is equivalent to the metric of convergence in distribution on the space of probability measures.
Definition 2.6. Let d denote the Vasserstein metric on the space of finite measures on Rd. That is, for µ,ν finite measures,
d(µ,ν)= sup φ(x) d(µ ν)(x), φ∈Lip1 Z − where Lip = ψ C(Rd) : x,y, ψ(x) ψ(y) x y , ψ 1 . 1 { ∈ ∀ | − | ≤ | − | k k≤ } 10 H. GILL
3. Proofs of existence and preliminary results.
Proof of Theorem 1.2. Although closely related, this does not fol- low automatically from Theorem 4.10 of Perkins [8] where it is shown that equations like (SE)2 [but with more general, interactive, drift and diffusion terms in (a)] have solutions if K is a critical historical Brownian motion. ◦ −βt −βt Note that Kt = e Kt defines a (∆/2, 0, e )-Historical superprocess. ◦ ˆ1 Let PT be the Campbell measure associated with K (note that if T is taken to be a random stopping time, this measure differs from PˆT ). The proof of 1 Theorem V.4.1 of [9] with minor modifications shows that (SE) ◦ has a Z0,K pathwise unique solution. This is because (K3) of Theorem 2.6 of [8] shows ˆ1 that under PT , yt is a Brownian motion stopped at time T and Proposi- tion 2.7 of the same memoir can be used to replace Proposition 2.4 and Remark 2.5(c) of [9] for the setting where the branching variance depends on time. ◦ Once this is established, it is simple to deduce that if (X,Z) is the solution 1 of (SE) ◦ , and we let Z0,K
βt ◦ Xt( ) e Xt( )= 1(Zt )Kt(dy) · ≡ · Z ∈ · 1 then (X,Z) is the pathwise unique solution of (SE)Z0,K . The only thing to t check this is that Zt(ω,y)= Z0(ω,y0)+ yt y0 γ 0 Zs(ω,y) ds K-a.e., but ◦ − − this follows from the fact that K K , t. R t ≪ t ∀ It can be shown by using by Theorem 2.14 of [8] that X satisfies the following martingale problem: For φ C2(Rd), ∈ b t ∆ t Mt(φ) Xt(φ) X0(φ)+ γx φ(x) φ(x)Xs(dx) ds β Xs(φ) ds ≡ − Z0 Z ·∇ − 2 − Z0 is a martingale where M(φ) = t X (φ2) ds. Then by Theorem II.5.1 of [9] h it 0 s this implies that X is a version ofR a SOU process, with initial distribution given by K (Z−1( )). 0 0 · Remark 3.1. (a) Under the Lipschitz assumptions of Section V.1 of [9], one can in fact uniquely solve (SE)1 where K is a supercritical historical Z0,K Brownian motion. The proof above can be extended with minor modifica- tions. (b) The proof of Theorem 1.2 essentially shows that under PˆT , T fixed, + d the path process y : R Ωˆ R such that (t, (ω,y)) yt is a d-dimensional Brownian motion stopped× at→ T . 7→ SOU INTERACTING WITH ITS COM 11
(c) Under PˆT , T fixed, Zt can be written explicitly as a function of the driving path: For t T , ≤ t t γt γs γs e Zt = Z0 + e dZs + Zs(γe ) ds Z0 Z0 (3.1) t t t γs γs γs = Z0 + e dys + e ( γZs) ds + Zs(γe ) ds, Z0 Z0 − Z0 where we have used a differential form of (SE)1 (a) for the second equality. Z0,K Hence, t −γt −γ(t−s) (3.2) Zt(y)= e Z0 + e dys. Z0
Next, we show that there exists a unique solution to (SE)2 . Y0,K
Proof of Theorem 1.4. Suppose there exists a solution Y satisfy- ing (SE)2 . Then under Pˆ , Y can be written as a function of the driving Y0,K T t path y and Y¯ . Using integration by parts gives t t γt γs γs e Yt = Y0 + e dYs + γe Ys ds Z0 Z0 t t γs γs = Y0 + e dys + γe Y¯s ds Z0 Z0 and hence t t −γt γ(s−t) γ(s−t) (3.3) Yt = e Y0 + e dys + γe Y¯s ds. Z0 Z0 If (X,Z) is the solution to (SE)1 where Z = Y , then note that by Re- Z0,K 0 0 mark 3.1(c), t γ(s−t) Yt = Zt + γe Y¯s ds. Z0
By taking the normalized measure K˜t on both sides of the above equation, we get t −γ(t−s) (3.4) Y¯t = Z¯t + γ e Y¯s ds. Z0
Hence, Y¯t is seen to satisfy a Volterra Integral Equation of the second kind (see equation (2.2.1) of [11]) and therefore can be solved pathwise to give t Y¯t = Z¯t + γ Z¯s ds, Z0 12 H. GILL
¯ 1 which is easily verified using integration by parts. Also, if Yt is a second process which solves (3.4), then
t 1 −γ(t−s) 1 Y¯t Y¯ = γ e (Y¯s Y¯ ) ds | − t | Z − s 0 t −γ(t−s) ¯ ¯ 1 γ e Ys Ys ds. ≤ | | Z0 | − | ¯ ¯ 1 By Gronwall’s inequality, this implies Yt = Yt , for all t and ω. Pathwise uniqueness of X′ follows from the uniqueness of the solution to (SE)1 Z0,K and the uniqueness of the process Y¯t solving (3.4). We have shown that if there exists a solution to (SE)2 , then it is Y0,K necessarily pathwise unique. Turning now to existence to complete the proof, we work in the opposite order and define Y and X′ as functions of the pathwise unique solution to (SE)1 where Z = Y : Z0,K 0 0
t Yt = Zt + γ Z¯s ds, Z0 X′( )= K (Y ). t · t t ∈ ·
Then Y¯t satisfies the integral equation (3.4), and hence
t t −γ(t−s) Z¯s ds = e Y¯s ds. Z0 Z0 Therefore t −γ(t−s) Yt = Zt + γ e Y¯s ds Z0 t t −γt −γ(t−s) −γ(t−s) = e Y0 + e dys + γ e Y¯s ds, Z0 Z0 by equation (3.2), and so
t t γt γs γs e Yt = Y0 + e dys + γ e Y¯s ds. Z0 Z0 Multiplying by e−γt and using integration by parts shows
t Yt = Y0 + yt y0 + γ (Y¯s Ys) ds − Z0 − which holds for K-a.e. y, thereby showing (X′,Y ) satisfies (SE)2 . Y0,K SOU INTERACTING WITH ITS COM 13
Remark 3.2. Some useful equivalences in the above proof are collected below. If Y0 = Z0, then for t < η, t (a) Yt = Zt + γ Z¯s ds, Z0 t (b) Y¯t = Z¯t + γ Z¯s ds, Z0 (c) Y Y¯ = Z Z¯ , t − t t − t t t −γ(t−s) (d) e Y¯s ds = Z¯s ds. Z0 Z0
These equations intimately tie the behaviour of the interacting and ordi- nary SOU processes. Part (a) says that the interacting SOU process with attraction to the center of mass is the same as the ordinary SOU process pushed by the position of its center of mass. We now consider the martingale problem for X′. For φ : Rd R, recall → that φ¯t Kt(φ(Yt))/Kt(1) and that the lifetime of the process φ¯ is [0, η). Then the≡ following theorem holds:
Theorem 3.3. For φ C2(Rd, R), and t < η, ∈ b t φ¯t = φ¯0 + Nt + ¯bs ds, Z0 where b = γ φ(Y ) (Y¯ Y )+ 1 ∆φ(Y ) s ∇ s · s − s 2 s and Nt is a continuous local martingale on its lifetime such that t φ(Ys) φ¯s Nt = − dM(s,y) Z0 Z Ks(1) and hence has quadratic variation given by t 2 ¯ 2 φs (φs) [N]t = − ds. Z0 Ks(1)
Similarly, the following is true.
Remark 3.4. The method of Theorem 3.3 can be used to show that for the β-supercritical SOU process, X, for φ C2(Rd, R) and t < η, ∈ b t X˜t(φ)= K˜t(φ(Zt)) = K˜0(φ(Z0)) + Nt + K˜s(Lφ(Zs)) ds, Z0 14 H. GILL where Lφ(x)= γx φ(x)+ 1 ∆φ(x) − ·∇ 2 and Nt is a continuous local martingale on its lifetime such that t φ(Zs) K˜s(φ(Zs)) Nt = − dM(s,y) Z0 Z Ks(1) and hence has quadratic variation given by t 2 2 K˜s(φ (Zs)) K˜s(φ(Zs)) [N]t = − ds. Z0 Ks(1) Proof of Theorem 3.3. The proof is not very difficult; one need only use Itˆo’s lemma followed by some slight modifications of theorems in Chapter V of [9] to deal with the drift introduced in the historical martingale problem due to the supercritical branching. Let T be a fixed time and t T . Recall that under Pˆ , y is a stopped ≤ T Brownian motion by Remark 3.1(b), and hence Yt(y) is a stopped OU process (attracting to Y¯t). Therefore, under PˆT : t t 1 i j φ(Yt)= φ(Y0)+ φ(Ys) dYs + φij(Ys) d[Y ,Y ]s Z ∇ · 2 Z 0 i,jX≤d 0 t t 1 = φ(Y0)+ φ(Ys) dys + γ φ(Ys) (Y¯s Ys)+ ∆φ(Ys) ds Z0 ∇ · Z0 ∇ · − 2 by the classical Itˆo’s lemma. Then t t Kt(φ(Yt)) = Kt(φ(Y0)) + Kt φ(Ys) dys + Kt bs ds Z0 ∇ · Z0 t t = K0(φ(Y0)) + φ(Y0) dM(s,y)+ β Ks(φ(Y0)) ds Z0 Z Z0 t s + φ(Yr) dyr dM(s,y) Z0 Z Z0 ∇ · t s + β Ks φ(Yr) dyr ds Z0 Z0 ∇ · t t t + bs dM(s,y)+ β Ks(bs) ds + Ks(bs) ds Z0 Z Z0 Z0 t = K0(φ(Y0)) + φ(Ys) dM(s,y) Z0 Z t t + β Ks(φ(Ys)) ds + Ks(bs) ds. Z0 Z0 SOU INTERACTING WITH ITS COM 15
The equality in the third line to K ( t φ(Y ) dy ) follows from Proposi- t 0 ∇ s · s tion 2.13 of [8]. The equality of the fourthR line to the last term in the first line follows from a generalization of Proposition V.2.4 (b) of [9]. The last equality then follows by collecting like terms and using the definition of Y¯ . Note that Kt(1) is Feller’s β-supercritical branching diffusion and hence t 0 Kt(1) = K0(1) + Mt + β Ks(1) ds, Z0 0 0 t where Mt is a martingale such that [M ]t = 0 Ks(1) ds. Therefore for t < η, Itˆo’s formula and propertiesR of Kt(1) and Kt(φ(Yt)) imply that
Kt(φ(Yt)) φ¯t = Kt(1) t t ¯ φ(Ys) Ks(φ(Ys)) Ks(bs) = φ0 + 2 dM(s,y)+ ds. Z0 Z Ks(1) − Ks(1) Z0 Ks(1) Since φ is bounded, the stochastic integral term can be localized using the stopping times TN min t : Kt(1) N or Kt(1) 1/N N and hence it is a local martingale≡ on [0,{ η). It is easy≥ to check that≤ it}∧ has the appropriate quadratic variation.
The following lemmas will be used extensively in Section 4, but will be proven in Section 5.
Lemma 3.5. There is a nonnegative random variable W such that e−βtK (1) W a.s. t → and η< = W = 0 almost surely. { ∞} { } Note that as X and X′ are defined as projections of K, their mass pro- cesses are the same as that of K, and thus grow at the same rate.
Definition 3.6. Let h(δ) = (δ ln+(1/δ))1/2, where ln+(x)=ln x 1. Let ∨ S(δ, c)= y : y y < ch( t s ), t,s with t s δ . { | t − s| | − | ∀ | − |≤ } Lemma 3.7. Let K be a supercritical historical Brownian motion, with drift β, branching variance 1, and initial measure X0. For c0 > 6 fixed, c(t)= √t + c , there exists a.s. δ(ω) > 0 such that Supp(K (ω)) S(δ(ω), c(t)) for 0 t ⊂ all t. Further, given c0, P(δ < λ)
The following moment estimates are useful in establishing the convergence of Y¯t. Recall that η is the extinction time of K.
Lemma 3.8. Assume P(K˜ ( Y 2 + y 2)) < . Then, 0 | 0| | 0| ∞ P( Y 2; t < η) < A(γ,t), | t| where O(1 + t6e−2γt), if γ< 0, A(γ,t)= O(1 + t5), if γ 0. ≥ Remark 3.9. (a) The proof of Lemma 3.8, under the same hypotheses (if Z0 = Y0) yields P( Z 2; t < η) < A(γ,t). | t|
(b) Lemma 3.8 and its proof can be extended to show that for any positive k k integer k if P(K˜0( Z0 + y0 )) < (and Z0 = Y0), then there exists a function B(γ,t,k) polynomial| | | | in t if∞γ 0, exponential if γ< 0 such that ≥ P( Y k; t < η) ′ φ(y0)K0(dy)= φ(x)X0(dx)= φ(x)X (dx). Z Z Z 0 Also recall that our standing hypothesis is that K0 has finite initial mass ′ [and hence X0(1) = X0(1) < ]. In this section, we will first∞ settle what happens to X and X′ when there is extinction, and then the case when there is survival, under the attrac- tive regime and lastly address the interacting repelling SOU process on the survival set. 4.1. On the extinction set. Proof of Theorem 1.5. Assume for now that P(K˜ ( y 2)) < (the 0 | 0| ∞ case where K0 = 0 can be ignored without loss of generality). By Theo- rem 3.3, t Ys Y¯s Y¯t = Y¯0 + − dM(s,y) Z0 Z Ks(1) and therefore is a local martingale on its lifetime with reducing sequence T as defined in the proof of the same theorem. Using Doob’s weak in- { N } SOU INTERACTING WITH ITS COM 17 equality and Lemma 3.8, P sup Y¯s >n = lim P sup Y¯s >n N→∞ s 1 2 lim E( Y¯t∧T ) ≤ N→∞ n2 | N | A(γ,t) < . n2 By the first Borel–Cantelli lemma, P sup Y¯t >n i.o. = 0. s lim inf Y¯s > and lim sup Y¯s < s→t∧η −∞ s→t∧η ∞ which implies that on the set η P(K )= PK (K ) dν(K0), ∈ · Z 0 ∈ · where PK0 is the law of a historical Brownian motion with initial distribution ¯ ˜ 2 δK0 . Hence, the a.s. convergence of Yt in the case where K0( y0 ) is finite in mean imply | | ¯ c ¯ c P lim Yt exists; S = PK0 lim Yt exists; S dν(K0) t↑η Z t↑η c (4.1) = PK (S ) dν(K0) Z 0 = P(Sc) 2 if K0( y0 + 1) < ,ν-a.s. Finally,| | to get rid∞ of the assumption that K ( y 2) < note that Corol- 0 | 0| ∞ lary 3.4 of [8] ensures that if K0(1) < , then at any time t> 0, Kt (and ′ ∞ hence Xt, Xt) is compactly supported. Therefore, letting Sr = Kr = 0 we see that { 6 } ¯ c ¯ c P lim Yt does not exist,S = P lim Yt does not exist,S1/r,S t↑η t↑η r[∈Nn o c P PK lim Y¯t does not exist,S 1S ≤ 1/r t↑η 1/r Xr∈N = 0 18 H. GILL by (4.1) since K a.s. compact implies that K ( y 2) < holds. This 1/r 1/r | 1/r| ∞ completes the proof for the convergence of Y¯ on Sc in its full generality. The convergence of Z¯t now follows from the convergence of Y¯t and equa- tion (3.4). Proof of Theorem 1.6. As in the previous proof, note that we need 2 only consider the case that P(K˜0( y0 )) < . We will follow the proof of Theorem| | 1 of∞ Tribe [14] here. Define t 1 ζ(t)= ds, t<η. Z0 Ks(1) It is known by the work of Konno and Shiga [7] in the case where β = 0, that ζ : [0, η) [0, ) homeomorphically (recall that η< a.s. in that case). This latter→ result∞ also holds when β> 0 on the extinction∞ set Sc by a Girsanov argument. Define D : [0,ζ(η )) [0, η) as the unique inverse of ζ (on Sc, this defines the inverse on [0, −)) and→ for t ζ(η ), let D = . Let ∞ ≥ − t ∞ XD D ′ ˜ D t G F Xt = XDt , Xt = D and t = Dt Xt (1) and define L φ(x)= γ(Y¯ x) φ(x)+ 1 ∆φ(x). t t − ·∇ 2 Let ηN 1 TN = ds, Z0 Ks(1) where ηN = inf s : Ks(1) 1/N . Then note that TN ζ(η ) and each TN is a G -stopping{ time. On≤Sc for}φ C2, Theorem 3.3 ↑implies− t ∈ b Dt∧TN X˜ D (φ)= X˜ (φ)+ X˜ ′ (L φ) ds + M (φ) t∧TN 0 s s Dt∧TN Z0 t∧TN ˜ ˜ D D (4.2) = X0(φ)+ Xs (LDs φ)Xs (1) ds + Nt∧TN (φ) Z0 t∧TN ˜ D = X0(φ)+ Xs (LDs φ) ds + Nt∧TN (φ) Z0 D G since dDt = Xt (1) dt and where Nt = MDt . It follows that Nt∧TN is a t- local martingale. Then, by Theorem 3.3, Dt∧T ˜ ′ 2 ˜ ′ 2 ˜ D N Xs(φ ) Xs(φ) [X (φ)]t∧TN = − ds Z0 Xs(1) t∧TN ˜ D 2 ˜ D 2 = Xs (φ ) Xs (φ) ds, Z0 − SOU INTERACTING WITH ITS COM 19 which is uniformly bounded in N. Hence, sending N , one sees that →∞ Nt∧ζ(η−) is a Gt-martingale. Note that on Sc, ζ(η )= and hence on that event, − ∞ ∞ ∞ D ¯ 1 D Xs ( LDs φ ) ds γ(YDs x) φ(x) + ∆φ(x) Xs (dx) ds Z0 | | ≤ Z0 Z | − ·∇ | 2| | ∞ ¯ 1 D γ φ KDs ( YDs YDs )+ ∆φ Xs (1) ds ≤ Z0 | |k∇ k | − | 2k k ∞ D ˜ ¯ 1 = Xs (1) γ φ KDs ( YDs YDs )+ ∆φ ds Z0 | |k∇ k | − | 2k k ∞ D 2 1/2 1 Xs (1) γ φ ( YDs ) + ∆φ ds, ≤ Z0 | |k∇ k | | 2k k where in the second line we have used the definition of X′ and the Cauchy– Schwarz inequality in the fourth. Using the definition of Ds yields ∞ ∞ D 2 1/2 1 D Xs ( LDs φ ) ds γ φ sup( Ys ) + ∆φ Xs (1) ds Z0 | | ≤ | |k∇ k s<η | | 2k k Z0 2 1/2 1 (4.3) = γ φ sup( Ys ) + ∆φ η | |k∇ k s<η | | 2k k < ∞ 2 2 as φ Cb and Ys is continuous on [0, η) (which follows from Theorem 3.3). Hence,∈ this implies| | that for φ positive, on Sc ∞ ˜ D Nt(φ) > X0(φ) Xs ( LDs φ ) ds − − Z0 | | for all t and hence by Corollary IV.34.13 of [13], Nt converges as t . ˜ D →∞ Therefore by (4.2) and (4.3), Xt (φ) converges a.s. as well. ˜ D ˜ D Denote by X∞(φ) the limit of Xt (φ). It is immediately evident that ˜ D d ˜ D X∞( ) is a probability measure on R . To show that X∞( )= δF ′ where F ′ is· a random point in Rd, we now defer to the proof of· Theorem 1 in Tribe [14], as it is identical from this point forward. c Similar (but simpler) reasoning holds to show X˜t δF a.s. on S where d t ¯ → F is a random point in R . Let f(t)= γ 0 Zs ds. Note that f is independent of y and that f(t) f(η) a.s. when t ηRbecause Y¯t = Z¯t + f(t) and both Y¯t → ↑ and Z¯t converge a.s. by Theorem 1.5. Then for φ bounded and Lipschitz, φ(x f(t))X˜ ′(dx) φ(x f(η))X˜ ′(dx) C f(η) f(t) Z − t − Z − t ≤ | − | a.s. 0 −→ 20 H. GILL as t η. Therefore it is enough to note that since f(η) depends only on ω, ↑ ˜ ′ the convergence of Xt gives φ(x f(η))X˜ ′(dx) a.s. φ(F ′ f(η)) Z − t −→ − and hence a.s. φ(x f(t))X˜ ′(dx) φ(F ′ f(η)). Z − t −→ − By Remark 3.2(a), ′ φ(x f(t))X˜ (dx) = φ(Yt(y) f(t))K˜t(dy) Z − t Z − = φ(Zt(y))K˜t(dy) Z = φ(x)X˜t(dx) Z a.s. φ(F ) −→ as t η. Since there exists a countable separating set of bounded Lipschitz functions↑ φ , and the above holds for each φ , { n} n η ′ ¯ F = F + γ Zs ds a.s. Z0 Remark 4.1. (a) Theorem 1.6 holds in the critical branching case. That is, if β = 0, a.s. ′ a.s. X δ and X δ ′ , t −→ F t −→ F where η ′ F = F + Z¯s ds. Z0 The convergence of the critical ordinary SOU process to a random point follows directly from Tribe’s result. That this holds for the SOU process with attraction to the COM follows from the calculations above. (b) The distribution of the random point F has been identified in Tribe [14] by approximating with branching particle systems. In fact, the law of F can be identified as xη, where xt is an Ornstein–Uhlenbeck process with initial distribution given by X˜0 and η is the extinction time. Finding the distribu- tion of F ′ remains an open problem however. SOU INTERACTING WITH ITS COM 21 4.2. On the survival set, the attractive case. Proof of Theorem 1.7. Let γ 0 and as in the proof of Theorem 1.5, 2 ≥ assume P(K˜0( y0 )) < . Also, without loss of generality, assume that d = 1 for this proof.| | ∞ By Theorem 3.3, Y¯t is a continuous local martingale with decomposition given by Y¯t = Y¯0 + Mt(Y ) where t 2 ¯ 2 t Ys Ys V (Ys) [M(Y )]t = − ds = ds, Z0 Ks(1) Z0 Ks(1) with V (Y )= Y 2 Y¯ 2. Theorem IV.34.12 of [13] shows that on the set s s − s [M(Y )]∞ < S, Mt(Y ) a.s. converges. { Note that∞}∩ by Lemma 3.5, for a.e. ω S, W (ω) > 0, recalling that W = lim e−βtK (1). Hence, it follows that∈ [M(Y )] < on S if t→∞ t ∞ ∞ ∞ −βs e V (Ys) ds < . Z0 ∞ Then ∞ ∞ −βs −βs 2 P e V (Ys) ds; S P e Ys ds; S Z0 ≤ Z0 ∞ e−βsA(γ,s) ds ≤ Z0 < ∞ by Cauchy–Schwarz and Lemma 3.8 since γ 0. Therefore on S, Y¯ con- ≥ t verges a.s. to some limit Y¯∞. Note that if γ = 0, Remark 3.2(b) gives Y¯t = Z¯t and so (b) holds. That Z¯t converges on S for γ> 0 follows from the fact that Y¯t converges and equation (3.4) by setting t −γ(t−s) Z¯t = Y¯t γ e Y¯s ds − Z0 t −γ(t−s) −γt = Y¯t Y¯∞ + γ e (Y¯∞ Y¯s) ds + e Y¯∞ − Z0 − 0 as t . → →∞ a.s. By Remark 3.2(b), we see that for γ> 0 since Y¯ = Z¯ +γ t Z¯ ds, Z¯ 0 t t 0 s t −→ and Y¯ a.s. γ ∞ Z¯ ds. R t −→ 0 s Now argueR by conditioning as in the end of Theorem 1.5 to get the full result. 22 H. GILL The next few results are necessary to establish the almost sure convergence of X˜t on the survival set. This will in turn be used to show the almost sure ˜ ′ convergence of Xt using the correspondence of Remark 3.2(a). Let Pt be the standard Ornstein–Uhlenbeck semigroup (with attraction to the origin). Note that P P in norm where t → ∞ d/2 γ −γ|z|2 P∞φ(x)= φ(z) e dz, Z π −βt which is independent of x. Recall that W = limt→∞ e Xt(1) and S = W > 0 a.s. from Lemma 3.5. { } 4 4 Lemma 4.2. If γ> 0, P(X0( x )) < and P(X0(1) ) < , then on S, L2 | | ∞ ∞ for any φ Lip , e−βtX (φ) W P φ and ∈ 1 t −→ ∞ P( e−βtX (φ) W P φ 2) Ce−ζt, | t − ∞ | ≤ where C depends only on d and X0, and ζ is a positive constant dependent only on β and γ. Remark 4.3. As the L2 convergence in Lemma 4.2 is exponentially fast, it follows from the Borel–Cantelli lemma and Chebyshev inequality that for ∞ ζk/2 a strictly increasing sequence tn n=0 where tn [k, k + 1) = e , for φ Lip , { } |{ }∩ | ⌊ ⌋ ∈ 1 e−βtn X (φ) W P φ a.s. as n . tn → ∞ →∞ The idea is to use the above remark to bootstrap up to almost sure con- vergence in Lemma 4.2 with some estimates on the modulus of continuity −βt of the process e Xt(φ). 8 8 Lemma 4.4. Suppose γ > 0, P(X0( x )) < and P(X0(1) ) < . If φ Lip and h> 0, then | | ∞ ∞ ∈ 1 ∗ (4.4) P([e−β(t+h)X (φ) e−βtX (φ)]4) C(t)h2e−ζ t, t+h − t ≤ where ζ∗ is a positive constant depending only on β and γ and C is polyno- mial in t, and depends on γ, β and d. Let Ψ : Rd R, p : R R be positive, continuous functions. Further, → + → suppose Ψ is symmetric about 0 and convex with lim|x|→∞ Ψ(x)= and p(x) is increasing with p(0) = 0. The following is a very useful result∞ of Garsia, Rodemich and Rumsey [5]. SOU INTERACTING WITH ITS COM 23 Proposition 4.5. If f is a measurable function on [0, 1] such that f(t) f(s) (4.5) Ψ − ds dt = B< ZZ 2 p( t s ) ∞ [0,1] | − | then there is a set K of measure 0 such that if s,t [0, 1] K then ∈ \ |t−s| −1 B (4.6) f(t) f(s) 8 Ψ 2 dp(u). | − |≤ Z0 u If f is also continuous, then K is in fact the empty set. With this result in hand, we can now bring everything together to prove convergence of X˜t. Proof of Theorem 1.8. The strategy for this proof is simple: We use Remark 4.3 to see that we can lay down an increasingly (exponentially) dense −βtn sequence e Xtn (φ) which converges almost surely, and that we can use −βt Lemma 4.4 to get a modulus of continuity on the process e Xt(φ), which then implies that if the sequence is converging, then the entire process must be converging. 8 8 8 Assume that P(K0( y0 )) = P(X0( x )) < and P(X0(1) ) < and ar- | | | | ∞ ∞ −βt gue as in Theorem 1.5 in the general case. Let φ Lip1. Denote e Xt ◦ ∈ 4 by Xt for the remainder of the proof. Let T > 0, and let Ψ(x)= x and 3/4 λ 1/2 4 | | p(t)= t (log( t )) where λ = e . Let BT (ω) be the constant B that ap- pears in| | Proposition 4.5, with aforementioned functions Ψ and p, for the ◦ path XT t(ω),t [0, 1]. Then note that∈ ◦ ◦ XT t XTs P(BT ) P Ψ − ds dt ≡ ZZ 2 p( t s ) [0,1] | − | ◦ ◦ 4 P[ XT t XTs ] = | −2 | ds dt ZZ 2 t s 3 log (λ/ t s ) [0,1] | − | | − | ∗ C(T (s t))e−ζ (s∧t)T 2 t s 2 ∧ | − | ds dt 3 2 ≤ ZZ 2 t s log (λ/ t s ) [0,1] | − | | − | ∗ 1 t C(Ts)e−ζ (Ts) t s 2 = 2T 2 | − | ds dt Z Z t s 3 log2(λ/ t s ) 0 0 | − | | − | 1 t 1 2C(T )T 2 ds dt ≤ Z Z t s log2(λ/ t s ) 0 0 | − | | − | C(T )T 2 , ≤ 2e4 24 H. GILL ◦ where C is the polynomial term that appears in Lemma 4.4. Since Xt is continuous, by Garsia–Rodemech–Rumsey [5], for all s,t 1, ≤ |t−s| 1/4 ◦ ◦ BT XT t XTs 8 2 dp(u) | − |≤ Z0 u λ 1/2 AB1/4 t s 1/4 log , ≤ T | − | t s | − | where A is a constant independent of T (see Corollary 1.2 of Walsh [15] for this calculation). Rewriting the above, 1/2 ◦ ◦ λT (4.7) X X D t s 1/4 log s ◦ ◦ ◦ ◦ ◦ ◦ X X X X ′ + X ′ X | t − ∞| ≤ | t − tn | | tn − ∞| ′ 1/8 ◦ ◦ t t + X ′ X ≤ | − n| | tn − ∞| −ζ⌊t⌋/16 ◦ ◦ ce + X ′ X . ≤ | tn − ∞| SOU INTERACTING WITH ITS COM 25 ◦ Sending t gives almost sure convergence of e−βtX (φ) to X = W P (φ) →∞ t ∞ ∞ by Theorem 4.2, since t′ with t. Note that this implies for φ Lip n →∞ ∈ 1 (4.8) X˜ (φ) a.s. P (φ) t −→ ∞ βt since on S, e W −1 a.s. Kt(1) → d By Exercise 2.2 of [4] on M1(R ), the space of probability measures on Rd, the Prohorov metric of weak convergence is equivalent to the Vasser- stein metric. It is easy to construct a class Θ that is a countable algebra of Lipschitz functions and that therefore is strongly separating (see page 113 of [4]). Hence by Theorem 3.4.5(b) of [4], Θ is convergence determining. Since there exists a set S0 S with P(S S0)=0 such that on S0, equation (4.8) holds simultaneously⊂ for all φ Θ,\ ∈ X˜ ( ) P ( ) t · → ∞ · in the Vasserstein metric, for ω S0 because Θ is convergence determining. To drop the dependence on the∈ eighth moment, we argue as in the proof of Theorem 1.5, where we make use of the Markov Property and the Compact support property for Historical Brownian Motion. Proof of Theorem 1.10. This follows almost immediately from The- orem 1.8 and the representation given in Remark 3.2(a). Let φ Lip , then ∈ 1 t ˜ ′ ˜ ˜ ¯ Xt(φ)= Kt(φ(Yt)) = Kt φ Zt + γ Zs ds Z0 = φ(x + f(t,ω)) dX˜t(dx), Z a.s. where f(t)= γ t Z¯ ds. Remark 3.2(b) gives f(t)= Y¯ Z¯ , and hence f(t) 0 s t − t −→ Y¯∞ follows fromR Theorem 1.7. Note that ¯ X˜ ′(φ) P Y∞ (φ) | t − ∞ | φ(x + f(t)) dX˜t(dx) φ(x + f( )) dX˜t(dx) ≤ Z − Z ∞ + φ(x + f( )) dX˜t(dx) φ(x + f( )) dX˜∞(dx) Z ∞ − Z ∞ f( t) f( ) + d(X˜ , X˜ ) ≤ | − ∞ | t ∞ since φ Lip . Taking the supremum over φ and the previous theorem give ∈ 1 a.s. d(X˜ ′, X˜ ′ ) f(t,ω) f( ,ω) + d(X˜ , X˜ ) 0. t ∞ ≤ | − ∞ | t ∞ −→ 26 H. GILL 4.3. The repelling case, on the survival set. Much less can be said for the SOU process repelling from its center of mass than in the attractive case. We can, however, show that the center of mass converges, provided the rate of repulsion is not too strong, which we recall was the first step toward showing the a.s. convergence of the normalized interacting SOU process in the attractive case. The situation here is more complicated since we prove that the COM of the ordinary SOU process with repulsion diverges almost surely, implying that results for convergence of X˜ ′ will not simply be estab- lished through the correspondence. We finish with some conjectures on the limiting measure for the repelling case. As in the previous section, assume Z0 = Y0 = y0, unless stated otherwise. Proof of Theorem 1.11. Assume that P(K˜ ( y 2)) < , like in the | 0| ∞ proof of Theorem 1.5. As in that theorem, this condition can be weakened to just the finite initial mass condition using similar reasoning. For part (a), note that as in Theorem 1.7, Y¯t will converge if P([Y¯ ]t) < and which holds if the following quantity is bounded: ∞ ∞ Y 2 Y¯ 2 ∞ Y 2 P s s P s | | βs− ds; S βs ds; S Z0 e ≤ Z0 e ∞ 1+ s6e−2γs c βs ds ≤ Z0 e < , ∞ by Lemma 3.8 and by the conditions on γ. For (b), we require the following lemma. Lemma 4.6. Let β/2 <γ< 0 and X = 0. For a measure m on Rd, − 0 6 let τ (m) be m translated by a Rd. That is, τ (m)(φ)= φ(x + a)m(dx). a ∈ a Then: R (i) For all but at most countably many a, (4.9) P (eγtZ¯ L = 0 S) = 1. τa(X0) t → 6 | (ii) For all but at most one value of a P (eγtZ¯ L = 0 S) > 0. τa(X0) t → 6 | Proof. We first note that, by the correspondence (3.4), we have that t −γt γs Z¯t = Y¯t γe e Y¯s ds. − Z0 SOU INTERACTING WITH ITS COM 27 Under our hypotheses Y¯t converges by Theorem 1.11(a), and hence t γt γs (4.10) lim e Z¯t + γe Y¯s ds = 0 t→∞ Z0 a.s. on S. Therefore, on S, γt (4.11) lim e Z¯t exists a.s. t→∞ Note that one can build a solution of (SE)2 with initial conditions given by τ (X ) by seeing that if Y gives the solution of (SE)2 , then Y + a a 0 t Y0,K t gives the solution of (SE)2 , and that the projection Y0+a,K ′ X ( )= 1(Yt + a )Kt(dy) t · Z ∈ · gives the appropriate interacting SOU process. By (4.10) and (4.11), t γt ¯ c γs ¯ PX0 ( e Zt L = 0 S)= Pτa(X0) lim e Ys ds = 0 S { → 6 } | t→∞ Z0 t γs Ks(a + Ys) = PX0 lim e ds = 0 S t→∞ Z0 Ks(1) t a γs ¯ = PX0 + lim e Ys ds = 0 S . −γ t→∞ Z0 ∞ γs ¯ The random variable 0 e Ys ds is finite a.s. and so only a countable number of values a existR with the latter expression positive, implying the first result. The second result also follows as well since the last expression in the above display can be 1 for at most 1 value of a. To complete the proof of Theorem 1.11(b), choose a value a Rd such ∈ that (4.9) holds. By Theorem III.2.2 of [9] and the fact that X0Ps τa(X0)Pt, for all 0 β Note that for 0 >γ> 2 , this implies that even if mass is repelled at rate γ, the COM of the interacting− SOU process still settles down in the long run. That is, driving Yt away from Y¯t seems to have the effect of stabilizing it. One can think of this as a situation where the mass is growing quickly enough that the law of large numbers overcomes the repelling force. More surprising is that the COM of the ordinary SOU process diverges exponentially fast, even while the COM of the interacting one settles down. This follows from the correspondence t Y¯t = Z¯t + γ Z¯s ds, Z0 and the cancellation that occurs in it due to the exponential rate of Z¯t. The next lemma shows that Theorem 1 of Engl¨ander and Winter [3] can be reformulated to yield a result for the SOU process with repulsion at rate γ (where γ is taken to be a negative parameter in our setting). Lemma 4.7. On S, for the SOU process, X, with repulsion rate β < − d γ< 0 and compactly supported initial measure µ, and any ψ C+(Rd) ∈ c d|γ|t P e X˜t(ψ) ξ ψ(x) dx, −→ ZRd where ξ is a positive random variable on the set S. Proof. Note that by Example 2 of Pinsky [10] it is shown that the hypotheses of Theorem 1 of [3] hold for the SOU process with repulsion from the origin at rate 0 < γ< β . The theorem says that there is a function − d φ C∞(Rd) such that c ∈ b Xt(ψ) P Wξ (4.13) µ , E (Xt(ψ)) −→ µ(φc) where W is as in Lemma 3.5. Example 2 also shows that for ψ C+(Rd), ∈ c −(β+γd)t µ lim e E (Xt(ψ)) = µ(φc)m(ψ), t→∞ where m is Lebesgue measure on Rd. Hence, manipulating the expression in (4.13) by using the previous equation and Lemma 3.5 gives |γ| dt µ |γ| dt P ξW e E (Xt(ψ)) e X˜t(ψ) lim −→ µ(φc) t→∞ Xt(1) −(β+γd)t µ ξW e E (Xt(ψ)) = lim −βt µ(φc) t→∞ e Xt(1) ξW µ(φ )m(ψ) = c . µ(φc) W SOU INTERACTING WITH ITS COM 29 This lemma indicates that on the survival set, when γ< 0, one cannot naively normalize Xt by its mass since the probability measures X˜t are not tight. That is, a proportion of mass is escaping to infinity and{ is} not seen by compact sets. Note that the lemma above implies that for Xt, the right normalizing factor is e(β+γd)t. ◦ Definition 4.8. We say a measure-valued process X goes locally ex- ◦ d tinct if for any finite initial measure X0 and any bounded A B(R ), there ◦ ∈ is a P ◦ -a.s. finite stopping time τA so that Xt(A) = 0 for all t τA a.s. X0 ≥ Remark 4.9. Example 2 of Pinsky [10] also shows that for γ β/d the SOU undergoes local extinction (all the mass escapes). Hence≤− for ψ d ∈ Cc(R ), there is no normalization where Xt(ψ) can be expected to converge to something nontrivial. From Remark 3.4, one can show that t Z¯t = Z¯0 + Nt γ Z¯s ds, − Z0 where N is a martingale. Therefore, you can think of the COM of X, the SOU process repelling from origin, as being given by an exponential drift term plus fluctuations. The correspondence of Remark 3.2(b) implies then that Y¯t = Y¯0 + Nt, or in other words, the center of mass of the SOU process repelling from its COM is given by simply the fluctuations. We finish with some conjectures. ′ Conjecture 4.10. On the survival set, if X0 is fixed and compactly supported, then the following is conjectured to hold: (a) If 0 < γ< β , then there exists constant β + γd α < β so that for − d ≤ φ C (Rd), ∈ c e−αtX′(φ) P ν(ψ), t −→ where ν is a random measure depending on Y¯∞. (b) If β/d γ, then X′ undergoes local extinction. ≤− t We expect that α < β simply because of the repulsion from the COM built in to the model results in a proportion of mass being lost to infinity. One would expect that the limiting measure ν is a random multiple of Lebesgue measure as in the ordinary SOU process case, due to the correspondence, but it is conceivable that it is some other measure which has, for example, a dearth of mass near the limiting COM. As stated earlier, it is difficult to use Lemma 4.7 to prove this conjecture as the correspondence becomes much less useful in the repulsive case. The 30 H. GILL problem is that while the equation t ′ ¯ φ(x) dXt(x)= φ x + γ Zs ds dXt(x) Z Z Z0 still holds for t finite, the time integral of Z¯s now diverges. 5. Proofs of technical lemmas. We now prove the lemmas first stated in Section 3. ◦ −βt Proof of Lemma 3.5. We first note that Kt = e Kt is a (∆/2, 0, ◦ −βt e )-historical process. The martingale problem then shows that Kt(1) is a nonnegative (Ft)-martingale and therefore converges almost surely by the Martingale Convergence theorem to a random variable W . It follows that η< W = 0 , since 0 is an absorbing state for Kt(1). Exercise II.5.3 in{ [9]∞}⊂{ shows that } P(η< )= e−2βK0(1). ∞ The same exercise also shows ◦ ◦ 2λβK (1) P (exp ( λK (1))) = exp 0 . K0 − t −2β + λ(1 e−βt) − Now sending t gives →∞ ◦ 2βλK (1) P (e−λW )=exp 0 , K0 − 2β + λ and sending λ gives →∞ ◦ P(W =0)= e−2βK0(1) = P(η< ) ∞ ◦ since K = K . Therefore, η< = W = 0 almost surely. 0 0 { ∞} { } Proof of Lemma 3.7. We follow the proof of Theorem III.1.3(a) in Perkins [9]. First, note that if H is another supercritical historical Brownian motion starting at time τ with initial measure m under Qτ,m and A a Borel subset of C, then the process defined by H′( )= H ( y : y A ) t · t · ∩ { τ ∈ } is also a supercritical historical Brownian motion starting at time τ with ′ ′ initial measure m given by m ( )= m( A) under Qτ,m. Then using the extinction probabilities for H′ (refer,· e.g.,·∩ to Exercise II.5.3 of [9]) we have 2βm(A) (5.1) Qτ,m(Ht( y : yτ A ) = 0 t s)=exp . { ∈ } ∀ ≥ −1 e−β(s−τ) − SOU INTERACTING WITH ITS COM 31 j Using the Markov property for K at time 2n and (5.1) gives j + 1 j j 1 j P t> s.t. K y : y y − > c h(2−n) > 0 ∃ 2n t 2n − 2n 2n n n = P[1 exp (2βK n ( y : y(j/2 ) y((j 1)/2 ) − {− j/ 2 { | − − | > c(j/2n)h(2−n) )) }n /(1 e−β/2 ) ] (5.2) − } n n n −n 2βK n ( y : y(j/2 ) y((j 1)/2 ) > c(j/2 )h(2 ) ) P j/2 { | − − | } ≤ 1 e−β/2n n − n n −n 2βK n ( y : y(j/2 ) y((j 1)/2 ) > c(j/2 )h(2 ) ) P j/2 { | − − | } ≤ (β/2n) (β2/22n+1) − 2n+1 ≤ 1 (β/2n+1) − j j 1 j −n P(K n (1))Pˆ n y y − > c h(2 ) , × j/2 j/2 2n − 2n 2n where in the last step we have simply rearranged the constants and multi- plied and divided by the mean mass at time j2−n and used the definition of the Campbell measure (Definition 2.1). Since under the normalized mean measure, y is a stopped Brownian mo- tion by Remark 3.1(b), we use tail estimates to see that the last quantity in (5.2) is n+1 2 d/2−1 −nc(j/2n)2/2 P(K n (1))c n 2 ≤ 1 (β/2n+1) j/2 d − n n 2n+22βj ln 2/2 P(X (1))c nd/2−12−n((j/2 )+c0)/2. ≤ 0 d Hence, summing over j from 1 to n2n gives j + 1 P There exists 1 j n2n s.t. t> s.t. ≤ ≤ ∃ 2n j j 1 j K y : y y − > c h(2−n) > 0 t 2n − 2n 2n n2n n n P(X (1))c nd/2−12n+2−nc0/2 2βj ln 2/2 −n(j/2 ) ≤ 0 d Xj=1 n2n n P(X (1))c nd/2−12−2n+2 2−j/2 ≤ 0 d Xj=1 P(X (1))c nd/2−12−n+2, ≤ 0 d 32 H. GILL where we have used the fact that c0 > 6. Hence, the sum over n of the above shows by Borel–Cantelli that there exists for almost sure ω, N(ω) such that for all n > N, for all 1 j n2n, for all t j+1 , for K -a.a. y, ≤ ≤ ≥ 2n t j 1 j j y − y < c h(2−n). 2n − 2n 2n Letting δ(ω) = 2−N (ω), note that on the dyadics, by above, we have that ln λ P(δ < λ)= P N > − ln 2 ln λ j + 1 = P n> , j n2n, t> , s.t. ∃ − ln 2 ∃ ≤ ∃ 2n j j 1 j K y : y y − > c h(2−n) > 0 t 2n − 2n 2n C′(d, c )nd/2−122n−nc 0/2 ≤ 0 n=⌊−Xln λ/ln 2⌋ C(d, c , ε)λ(c0/2)−ε, ≤ 0 where ε can be chosen to be arbitrarily small (though the constant C will in- crease as it decreases). The rest of the proof follows as in Theorem III.1.3(a) of [9], via an argument similar to Levy’s proof for the modulus of continuity for Brownian motion. Proof of Lemma 3.8. Assume that Z0 = Y0 and t < η. Recall t −γt γ(s−t) Zt(ω,y) e Z0 + e dys. ≡ Z0 Note that below “.” denotes less than up to multiplicative constants inde- pendent of t and y. Suppose that y S(δ, c(t)), where S(δ, c(t)) is the same ∈ t ¯ as in the previous lemma. Then, as Yt = Zt + γ 0 Zs ds, t R t 2 2 2 2 2 2 2 Yt . Zt + γ t Z¯s ds . Zt + γ t Zs ds | | | | Z0 | | | | Z0 | | by Cauchy–Schwarz and Jensen’s inequality. Therefore, integrating with re- spect to the normalized measure gives t 2 2 2 2 (5.3) Yt Zt + γ t Zs ds | | ≤ | | Z0 | | and therefore we need only find the appropriate bounds for expectation of Z 2 to get the result. | t| SOU INTERACTING WITH ITS COM 33 After another few applications of Cauchy–Schwarz and integrating by parts, t 2 2 −2γt 2 −γt γs Zt . e Z0 + e e dys | | | | Z 0 t 2 −2γt 2 −2γt γt γs . e Z0 + e e yt y0 γ yse ds | | − − Z 0 t −2γt 2 2 2 2 2 −2γ (t−s) . e ( Z0 + y0 )+ yt + γ t ys e ds. | | | | | | Z0 | | As y S(δ, c(t)), ∈ tc(t)h(δ) 2 Z 2 . e−2γt( Z 2 + y 2)+ y 2 + | t| | 0| | 0| | 0| δ t 2 2 2 sc(s)h(δ) −2γ(t−s) + γ t y0 + e ds Z0 | | δ . e−2γt( Z 2 + y 2)+ y 2 + y 2γt(1 e−2γt)/2 | 0| | 0| | 0| | 0| − h(δ) 2 + c(t)2 (t2 + γt3(1 e−2γt)) δ − . (1 + γ t)(1 + e−2γt)( Z 2 + y 2) | | | 0| | 0| h(δ) 2 + c(t)2 (t2 + γt3(1 e−2γt)). δ − Integrating by the normalized measure K˜t, Z 2 . (1 + γ t)(1 + e−2γt)K˜ ( Z 2 + y 2) | t| | | t | 0| | 0| h(δ) 2 + c(t)2 (t2 + γt3(1 e−2γt)). δ − Then using (5.3) and using the above bound on Z 2 gives | t| t 2 2 2 2 Yt Zt + γ t Zs ds | | ≤ | | Z0 | | t 2 −2γt 2 2 2 2 (5.4) . (1 + t )(1 + e ) K˜t( Z0 + y0 )+ K˜s( Z0 + y0 ) ds | | | | Z0 | | | | h(δ) 2 +(1+ t)c(t)2 (t2 + γt3(1 e−2γt)). δ − 2 2 Note that φ(y) Z0(y0) + y0 and φn(y) φ(y)1( y n) are Fˆ0 measur- ≡ | | | | ≡−1 | |≤ able. By applying Itˆo’s formula to Kt(φn)Kt (1) and using the decompo- 34 H. GILL t t sition Kt(φn)= K0(φn)+ 0 φn(y) dM(s,y)+ β 0 Ks(φn) ds (which follows from Proposition 2.7 of [8]),R we get R K˜t(φn)= K˜0(φn)+ Nt(φn), where Nt(φn) is a local martingale until time η, for each n. In fact, the sequence of stopping times T appearing in Theorem 3.3 can be used to { N } localize each Nt(φn). Applying first the monotone convergence theorem and then localizing gives P(K˜t(φ); t < η) = lim P(K˜t(φn); t < η) n→∞ = lim lim P(K˜t(φn)1(t < TN )) n→∞ N→∞ = lim lim P(K˜t∧T (φn) K˜T (φn)1(t TN )) n→∞ N→∞ N − N ≥ lim lim P(K˜t∧T (φn)) ≤ n→∞ N→∞ N = lim lim P(K˜0(φn)+ Nt∧T (φn)) n→∞ N→∞ N = lim P(K˜0(φn)) n→∞ = P(K˜0(φ)), where we have used the positivity of φn to get the fourth line and the monotone convergence theorem in the last line. Further, note that t t P K˜s(φ) ds; t < η P K˜s(φ)1(s < η) ds Z0 ≤ Z0 t = P(K˜s(φ); s < η) ds Z0 tP(K˜ (φ)), ≤ 0 by the calculation immediately above. Thus, taking expectations in (5.4) and plugging in c(t)= √c0 + t gives h(δ)2 P( Y 2; t < η) . (1 + t3)(1 + e−2γt)P(K˜ (φ)) + t5(1 + te−2γt)P . | t| 0 δ2 α Now let c0 be chosen so that Supp(Kt) S(δ, c(t)) and pc0 (λ)= Cλ for λ [0, 1]. Note that ⊂ ∈ h(δ)2 ln+(1/δ) ∞ ln+(1/λ) P P P 2 = = d (δ < λ) δ δ Z0 λ 1 ln+(1/λ) dP(δ < λ)+ P(δ> 1) ≤ Z0 λ SOU INTERACTING WITH ITS COM 35 ln+(1/λ) = lim P(δ < λ) P(δ< 1) λ↓0 λ − 1 ln+(1/λ) P(δ < λ) d + P(δ> 1) − Z0 λ < , ∞ by choosing the constant c so that α is large (α 2 is enough). 0 ≥ Remark 3.9 follows from the above proof, after noting that the exponent α in Lemma 3.7 can be made arbitrarily large by choosing a sufficiently large constant c0. Hence by choosing α appropriately, we can show that h(δ) k P < , δ ∞ which can then be used to adapt the proof above. d Recall that Lip1 = ψ C(R ) : x,y, ψ(x) ψ(y) x y , ψ 1 . We will, with a slight abuse{ ∈ of notation,∀ | allow −M to denote| ≤ | − the| orthogonalk k≤ } martingale measure generated by the martingale problem for X. Let A be the infinitesimal generator for an OU process, and hence recall that for φ C2(Rd), ∈ ∆ Aφ(x)= γx φ(x)+ φ(x). − ·∇ 2 The next two proofs are for lemmas stated in Section 4.2. Proof of Lemma 4.2. Let φ Lip1. By the extension of the martingale problem for X given in Proposition∈ II.5.7 of [9], for functions ψ : [0, T ] Rd R such that ψ satisfies the definition before that proposition, × → t t Xt(ψt)= X0(ψ0)+ ψs(x) dM(x,s)+ Xs(Aψs + βψs + ψ˙s) ds, Z0 Z Z0 where M is the orthogonal martingale measure derived from the martingale problem for the SOU process. It is not difficult to show that ψs = Pt−sφ where φ as above satisfies requirements for Proposition II.5.7 of [9]. Plugging this in gives t t Xt(φ)= X0(Ptφ)+ Pt−sφ(x) dM(s,x)+ βXs(Pt−sφ) ds Z0 Z Z0 ∂ −βt since ∂s Psφ = APsφ. Multiplying by e and integrating by parts gives −βt −βt e Xt(φ)= e Xt(ψt) t t −βs −βs (5.5) = X0(ψ0) βe Xs(ψs) ds + e dXs(ψs) − Z0 Z0 36 H. GILL t −βs = X0(Ptφ)+ e Pt−sφ(x) dM(s,x). Z0 Z Note that as the OU-process has a stationary distribution P where P ∞ t → P∞ in norm. When s is large in (5.5), Pt−sφ(x) does not contribute much −βt to the stochastic integral and hence we expect the limit of e Xt(φ) to be ∞ −βs (5.6) X0(P∞φ)+ e P∞φ(x) dM(s,x), Z0 Z which is a well defined, finite random variable as · ∞ −βs 2 −2βs e P∞φ(x) dM(s,x) < φ e Xs(1) ds, Z0 Z ∞ k k Z0 which is finite in expectation. As P∞φ(x) does not depend on x, it follows that ∞ −βs (5.6) = (P∞φ)X0(1) + (P∞φ) e dM(s,x)= W P∞φ. Z0 Z Given this decomposition for W P∞φ, we write P((e−βtX (φ) W P φ)2) t − ∞ ∞ 2 −βs 3P e P∞φ(x) dM(s,x) ≤ Zt t 2 −βs + 3P e (Pt−sφ(x) P∞φ(x)) dM(s,x) Z0 Z − + X (P φ P φ)2 . 0 ∞ − t If z is a d-dimensional OU process satisfying dz = γz dt + dB , where t t − t t Bt is a d-dimensional Brownian motion, then t −γt −(t−s)γ zt = e z0 + e dBt Z0 −γt 1 and hence zt is Gaussian, with mean e z0 and covariance matrix 2γ (1 −2γt 1 − e )I. Evidently, z∞ is also Gaussian, mean 0 and variance 2γ I. We use a simple coupling: suppose that wt is a random variable independent of −γt zt such that z∞ = zt + wt (i.e., wt is Gaussian with mean e z0 and covariance 1 e−2γtI). Then using the fact that φ Lip and− the Cauchy– 2γ ∈ 1 Schwarz inequality, followed by our coupling with z0 = x gives 2 2 x X0(P∞φ Ptφ) = E (φ(z∞) φ(zt))X0(dx) − Z − SOU INTERACTING WITH ITS COM 37 x 2 E ( z∞ zt ) X0(dx)X0(1) ≤ Z | − | x 2 = E ( wt )X0(dx)X0(1) Z | | −2γt 2 d = e x + X0(dx)X0(1) Z | | 2γ −2γt 2 2 ce x X0(dx)X0(1) + X0(1) . ≤ Z | | Taking expectations and using Cauchy–Schwarz and the assumptions on X0 gives exponential rate of convergence for the above term. r −βs Since we can think of 0 e Pt−sφ(x) dM(s,x) as a martingale in r up until time t, various martingaleR R inequalities can be applied to get bounds for t −βs the terminal element, 0 e Pt−sφ(x) dM(s,x). Note that this process is not in general a martingaleR R in t. Therefore, we have t t 2 −βs −βs P e P∞φ(x) dM(s,x) e Pt−sφ(x) dM(s,x) Z0 Z − Z0 Z t 2 −βs (5.7) = P e (P∞φ(x) Pt−sφ(x)) dM(s,x) Z0 Z − t −2βs 2 P e (P∞φ(x) Pt−sφ(x)) Xs(dx) ds . ≤ Z0 Z − Then as φ Lipschitz, by the coupling above, t −2βs −2γ(t−s) 2 d (5.7) P e e x + Xs(dx) ds ≤ Z0 Z | | 2γ t −2βs−2γ(t−s) 2 d = e P x + Xs(dx) ds Z0 Z | | 2γ t −2βs−2γ(t−s) 2 d = e P Ks( Zs )+ Xs(1) ds. Z0 | | 2γ Applying the Cauchy–Schwarz inequality followed by Remark 3.9(b) gives P(K ( Z 2)) s | s| = P( Z 2K (1); s < η) | s| s 2 P( Z 2 ; s < η)1/2P(X (1)2)1/2 ≤ | s| s B(s,γ, 4)1/2P(X (1)2)1/2 ≤ s 1 1/2 cB(s,γ, 4)1/2eβsP X (1)2 + X (1) , ≤ 0 β 0 38 H. GILL where the last line follows by first noting that t −βt −βs e Xt(1) = X0(1) + e dM(s,x) Z0 Z is a martingale. That is, s 2 −2βs 2 2 −βr e P(Xs(1) ) 2P X0(1) + e dM(r, x) ≤ Z0 · 2 −βr = 2P X0(1) + e dM(r, x) Z0 s s 2 −2βr = 2P X0(1) + e Xr(1) dr Z0 (5.8) s 2 −βr −βr = 2P(X0(1) ) + 2 e P(e Xr(1)) dr Z0 s 2 −βr = 2P(X0(1) ) + 2 e P(X0(1)) dr Z0 1 2P X (1)2 + X (1) . ≤ 0 β 0 Therefore, t 1/2 −2βs−2γ(t−s) βs 1/2 2 1 (5.7) e e B(s,γ, 4) P X0(1) + X0(1) ds ≤ Z0 β t −2βs−2γ(t−s) βs d + e e P(X0(1)) ds Z0 2γ t 1/2 −βs−2γ(t−s) 1/2 2 1 e B(s,γ, 4) P X0(1) + X0(1) ≤ Z0 β d + P(X (1)) ds 2γ 0 < Ce−ζ1t, where ζ1 = min(β, 2γ) ε where ε is arbitrary small and comes from the polynomial term in the− integral. Finally, ∞ 2 ∞ −βs 2 −2βs P e P∞φ(x) dM(s,x) = (P∞φ) P e Xs(1) ds Zt Zt ∞ 2 −βs −βs (P∞φ) e P(e Xs(1)) ds ≤ Zt (P φ)2 ∞ e−βtP(X (1)), ≤ β 0 SOU INTERACTING WITH ITS COM 39 −βs since e Xs(1) is a martingale. Therefore, since ζ1 < β, we see that ζ = ζ1 gives the correct exponent. Proof of Lemma 4.4. The proof will follow in a manner very similar to the proof of the previous lemma. From the calculations above, we see that e−β(t+h)X (φ) e−βtX (φ) t+h − t = X (P φ P φ) 0 t+h − t t+h t −βs −βs + e Pt+h−sφ(x) dM(s,x) e Pt−sφ(x) dM(s,x) Z0 Z − Z0 Z t −βs = X0(Pt+hφ Ptφ)+ e ((Pt+h−s Pt−s)φ(x)) dM(s,x) − Z0 Z − t+h −βs + e Pt+h−sφ(x) dM(s,x) Zt Z I + I + I . ≡ 1 2 3 4 Using the Cauchy–Schwarz inequality, we can find bounds for P( Ik ), k = 1, 2, 3, separately: | | I 4 = X (P φ P φ)4 | 1| 0 t+h − t [X ((P φ P φ)2)X (1)]2. ≤ 0 t+h − t 0 Recalling the simple coupling in the previous lemma to see that (P φ(x) P φ(x))2 Ex( φ(z ) φ(z ) 2) t+h − t ≤ | t+h − t | Ex( z z 2) ≤ | t+h − t| Ex( w 2), ≤ | t,t+h| where z is as above, an OU process started at x, and ws,t is independent of zs −γt −γs but such that zt = zs +ws,t. Hence, ws,t is Gaussian with mean x(e e ) and covariance matrix I (e−2sγ e−2tγ ). Therefore, − 2γ − d (P φ(x) P φ(x))2 x 2(e−γ(t+h) e−γt)2 + (e−2tγ e−2(t+h)γ ) t+h − t ≤ | | − 2γ − d = e−2γt x 2(1 e−γh)2 + (1 e−2hγ ) . | | − 2γ − Hence, 4 −2γt −γh 2 2 P( I1 )= P e (1 e ) x X0(dx)X0(1) | | − Z | | 40 H. GILL d 2 + e−2γt(1 e2γh)X (1)2 2γ − 0 C (d, γ)h2e−4γt, ≤ 1 where C1 is a constant that is finite by assumptions on the initial measure. To get bounds on the expectation of I2, we use martingale inequalities. Note that · e−βs((P P )φ(x)) dM(s,x)= N( ) is a martingale 0 t+h−s − t−s · until time tR. Therefore,R using the Burkholder–Davis–Gundy inequality and the coupling above gives P( I 4) cP([N]2) | 2| ≤ t t 2 −2βs 2 = cP e ((Pt+h−s Pt−s)φ(x)) Xs(dx) ds Z0 Z − t 2 −2βs −γ(t+h−s) −γ(t−s) 2 2 cP e (e e ) x Xs(dx) ds ≤ Z0 − Z | | t −2βs 2 de −2(t−s)γ −2(t+h−s)γ + (e e )Xs(1) ds Z0 2γ − t 2 −4βs −γ(t+h−s) −γ(t−s) 4 2 cP t e (e e ) x Xs(dx) ds ≤ Z0 − Z | | 2 t d −4βs −2(t−s)γ −2(t+h−s)γ 2 2 + t e (e e ) Xs(1) ds 2γ Z0 − t 2 −2βs −γ(t+h−s) −γ(t−s) 4 2 −βs = ct e (e e ) P x e Xs(dx) ds Z0 − Z | | 2 t d −2βs −2(t−s)γ −2(t+h−s)γ 2 −βs 2 + t e (e e ) P[(e Xs(1)) ] ds. 2γ Z0 − Since X ( x 2)= K ( Z 2), by Remark 3.9(b), s | | s | s| 2 P(e−2βsX ( x 2)2) P[Z2 (e−βsX (1))2; s < η] s | | ≤ s s 4 P(Z2 ; s < η)1/2P(e−4βsX (1)4)1/2 ≤ s s P(Z8; s < η)1/2P(e−4βsX (1)4)1/2 ≤ s s cB(γ,s, 8)1/2(P(X (1)4 + sX (1)2 + sX (1)))1/2. ≤ 0 0 0 −4βs 4 The bound on the expectation of e Xs(1) follows by an application of −βs s −βr the BDG Inequality to e Xs(1) = X0(1) + 0 e dM(r, x) and similar calculations used to determine the bound on (5.8R ). Therefore, t 4 −2βs 1/2 −γ(t+h−s) −γ(t−s) 4 P( I2 ) ct e B(γ,s, 8) (e e ) | | ≤ Z0 − SOU INTERACTING WITH ITS COM 41 (P(X (1)4 + sX (1)2 + sX (1)))1/2 ds × 0 0 0 ct t 1 −2βs −2(t−s)γ −2(t+h−s)γ 2P 2 + 2 e (e e ) X0(1) + X0(1) ds 4γ Z0 − β ctB(γ,t, 8)1/2(P(X (1)4 + tX (1)2 + tX (1)))1/2 ≤ 0 0 0 t (e−γh 1)4 e−2βse−4γ(t−s) ds × − Z0 ct 1 t −2γh 2P 2 −2βs −4(t−s)γ + 2 (e 1) X0(1) + X0(1) e e ds 4γ − β Z0 C (t,γ,β)h2e−ζ1t, ≤ 2 where C2 is polynomial in t. By another application of the BDG inequality, and noting that φ = 1, k k t+h 4 4 −βs P( I3 )= P e Pt+h−sφ(x) dM(s,x) | | Zt Z t+h 2 −βs 2 cP (e Pt+h−sφ(x)) Xs(dx) ds ≤ Zt Z t+h −4βs 2 chP e Xs(1) ds ≤ Zt t+h −2βt −2βs 2 = che P[e Xs(1) ] ds Zt t+h −2βt 2 che P[X0(1) + X0(1)/β] ds ≤ Zt C (β)h2e−2βt, ≤ 3 where the second last line follows from the same calculations performed in estimating moments of I2. Note that the constant C3 does not depend on t here. Putting the pieces together shows that there exists a function C polyno- mial in t and a positive constant ζ∗ such that (4.4) holds. Acknowledgement. I am greatly indebted to my advisor, Ed Perkins, for his numerous (very helpful) suggestions and generous support during the course of this work. I am particularly grateful for his help with Part (b) of Theorem 1.11. 42 H. GILL REFERENCES [1] Englander,¨ J. (2010). The center of mass for spatial branching processes and an application for self-interaction. Electron. J. Probab. 15 1938–1970. MR2738344 [2] Englander,¨ J. and Turaev, D. (2002). A scaling limit theorem for a class of su- perdiffusions. Ann. Probab. 30 683–722. MR1905855 [3] Englander,¨ J. and Winter, A. (2006). Law of large numbers for a class of su- perdiffusions. Ann. Inst. Henri Poincar´eProbab. Stat. 42 171–185. MR2199796 [4] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. MR0838085 [5] Garsia, A. M., Rodemich, E. and Rumsey, H. Jr. (1970/1971). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565–578. MR0267632 [6] Gill, H. S. (2009). Superprocesses with spatial interactions in a random medium. Stochastic Process. Appl. 119 3981–4003. MR2565556 [7] Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201–225. MR0958288 [8] Perkins, E. (1995). On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 vi+89. MR1249422 [9] Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffu- sions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lec- ture Notes in Math. 1781 125–324. Springer, Berlin. MR1915445 [10] Pinsky, R. G. (1996). Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24 237– 267. MR1387634 [11] Polyanin, A. D. and Manzhirov, A. V. (2008). Handbook of Integral Equations, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL. MR2404728 [12] Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Probab. 37 359–373. MR1780996 [13] Rogers, L. and Williams, D. (1985). Diffusions, Markov Processes and Martin- gales. Cambridge Univ. Press, Cambridge. [14] Tribe, R. (1992). The behavior of superprocesses near extinction. Ann. Probab. 20 286–311. MR1143421 [15] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In Ecole´ D’´et´ede Probabilit´es de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin. MR0876085 Department of Mathematics University of British Columbia 1984, Mathematics Road Vancouver, British Columbia V6T1Z2, Canada E-mail: [email protected]