Branching Markov processes on graphs and manifolds:from queueing theory to symmetric spaces

M. Kelbert, Mathematics DepartmentUniversity of Wales–Swansea

Singleton Park, Swansea SA2 8PP, UKE-mail addresses: M.Kelbertswansea.ac.uk

Abstract.

Branching Markov processes on graphs and manifolds naturally emerge ina number of applications including queueing networks with global synchro-nization control, reaction-diffusion equations, etc. Present talk extends ear-lier results by Lalley–Sellke and Karpelevich–Pechersky–Suhov for branchingdiffusions on a real hyperbolic space to the case of the complex hyperbolicspace. Consider a time- and space-homogeneous random branching Markovprocess Θ on a complex hyperbolic space E = G/K where G =SU(d, 1) is thegroup of (d+ 1)× (d+ 1) complex matrices leaving invariant the Hermitian

form (z′, z′′) 7→d∑

j=1z′jz

′′j − z′d+1z

′′d+1 and K is a maximal compact subgroup of

G. An example of such a process is the branching diffusion corresponding tothe Laplace–Beltrami operator. We determine when process Θ is transient,i.e., leaves any given compact K ⊂ E , and calculate the Hausdorff dimensionof its limiting set Λ ⊂ ∂E .

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Example 1. The Lindley equation arises in the theory of thesingle-server FCFS queue:

wn+1 = max[wn + sn − τn, 0]. (1.1)

Here wn is the waiting time of the n-th task, wn+1 is that of the(n+ 1)-st one, sn is the service time of the n-th task and τn is thetime between the arrival epochs of the n-th and (n+ 1)-st task. Ina stationary regime one deals with the stochastic equation

w ∼= max[w + s− τ, 0]. (1.2)

The symbol ∼= means the equality in distributions. If sn and τn are independent sequences of IID variables, then w, s andτ are independent as well. In particular, for M/M/1/∞ queue withPoisson input flow of rate λ and exponential distribution with meanµ−1 for s under condition

λ < µ (1.3)

there is a unique stationary distribution. If

λ ≥ µ (1.4)

there is no stationary distribution.The higher-order Lindley equation of order m ≥ 2 has the form

w ∼= max[w(1) + s(1) − τ (1), ..., w(m) + s(m) − τ (m), 0]. (1.5)

Here the r.v.’s w, w(1), ..., w(m) have the same (unknown) distribu-tion, the random variables s(1), ..., s(m) are IID and so are the r.v.’sτ (1), ..., τ (m). All variables that appear in the r.h.s. of (1.5) are as-sumed to be independent. A common marginal distribution of thes(j)’s and that of the τ (j)’s may be considered as ‘coefficients’ inequation (1.5).

Let again s(j) and τ (j) be exponential r.v.’s with means µ−1 andλ−1, respectively. Then under condition

λ

µ≤ 2m− 1−

√(2m− 1)2 − 1 (1.6)

2

there exist an infinite number of stationary solutions for (1.5). Form = 2 the bound is 3−

√8 ≈ 0.17157. If

λ

µ> 2m− 1−

√(2m− 1)2 − 1 (1.7)

there is no solutions.Motivation. After receiving numerous complaints that the local

hospitals are overloaded and their transport resources are stretched,the medical authorities in a major city of New Kork decided to re-organize the ambulance service. They allocated to each of N citydistricts ( N is supposed to be large) an out-patient surgery thatregisters the patients from the district, establishes a preliminarydiagnosis indicating to which of N specialized hospitals a patientis to be directed and communicates this information to the city’sMedical Dispatching Centre (MDC). Each surgery has its own am-bulance that transports the patients one by one to their assignedhospitals.

Each hospital admits and treats patients one by one. The MDCphones the surgeries from time to time to tell them that a givenpatient may be transported to the corresponding hospital. Thefollowing ‘rule of justice’ is strictly observed: a patient cannotbe transported until all patients in the following two categoriesfinish their treatment: (i) those who registered at the same surgeryearlier, no matter what their hospitals are, and (ii) all patients whoare to be transported to the same hospital and were registeredearlier, no matter at what surgery.

However, as soon as these conditions are fulfilled, the MDCphones the surgery, and the patient should be transported to hisor her hospital immediately. The treatment time comprises thetransportation time to the hospital and back and the time spentin the hospital. [The argument of the hospital staff is that untilthe patient returns to the surgery, he or she is formally undertreatment.]

Assume that the patients arrive at the surgeries in independentPoisson streams of a fixed intensity λ. Assume further that the reg-istration, preliminary diagnosis and communication with the MDCtakes a negligible time. Next, assume that the hospitals assignedto different patients are IID and equiprobable. Finally, let the

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treatment times of different patients also be IID and independentof the hospitals assigned (as a result of the traffic conditions andthe variability of the cases). Then under condition (1.6) the sta-tionary waiting time of a patient converges as N →∞ to a solutionof (1.5). Under condition (1.7) the queues in the system increaseindefinitely.

Example 2. Reaction-diffusion equation

∂u

∂t=

1

2∆u+ (u2 − u), u(x, 0) = u0(x) (2.1)

with travelling waves solutions u = f((x, e)− ct), e ∈ Sd, c ≥√

2.McKean’s formula

u(x, t) = E[ ∏

L∈L(t)

u0(x+XL(t))]. (2.2)

Systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov)equation (d = 1 for simplicity)

∂uk

∂t= −βk

∂uk

∂x+

1

2σ2

k

∂2uk

∂x2+ λk

(φk(u1, . . . , uK)− uk

), (2.3)

uk(x, 0) = u(k)0 (x), k = 1, . . . , K.

Say, φ1(u1, u2, u3) = 12u2

1u32u3 + 1

3u1u

82u

53 + 1

6u2

1u72u

113 , etc.

Extension of McKean’s formula

uk(x, t) = Ek

[ ∏L∈Lk(t)

u(jL(t))0 (x+XL(t))

]. (2.4)

Consider a K ×K matrix

(B(a)

)j,k

= E(j, k)λk

λk + (βk + c)a− σ2k

2a2

(2.5)

where E(j, k) is the mean number of type k offspring produced by asingle type j particle. Let κ(a) be the Perron-Frobenius eigenvalue

of(B(a)

)for a > 0 small enough.

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Then ∃a > 0 such that κ(a) < 1 ⇒ ∃ a travelling wave solutionwith the velocity c.

In example (2.1) ∃a > 0 such that 2/(1+ ca− a2

2) < 1 equivalent to

2/(1 + c2 − c2

2) < 1, i.e. c >

√2.

Example 3. Harmonic analysis on R2 versus H2

Fourier transform

f(λω) =∫R2f(x)e−iλ(x,ω)dx (3.1)

versusf(λ,w) =

∫B2f(z)e(−iλ+1)πho

w (z)dz (3.2)

where ω ∈ S1, w ∈ W = ∂B2 and πhow (z) = 〈z, w〉 is a hyperbolic

distance from the origin O to the horocycle projection hw(z) (withsign; to be taken negative if O lies inside the horocycle).Inverse Fourier transform

f(x) =1

(2π)2

∫S1

∫R+

f(λω)eiλ(x,ω)λdλdω (3.3)

versus

f(z) =1

4π

∫R

∫Wf(λ,w)λ tanh

(πλ

2

)e(iλ+1)πho

w (z)dλdw, (3.4)

dw is normalized by∫

dw = 1.The Plancherel formula∫

R2|f(x)|2dx =

1

(2π)2

∫S1

∫R+

|f(λω)|2λdλdω (3.5)

versusThe mapping f → f extends to an isometry of L2(B2) onto

L2(R+ ×W,

1

2πλ tanh

(πλ

2

)dλdw

), (3.6)

i.e. the Harish-Chandra c−function is specified by∫B2|f(z)|2dz =

1

2π2

∫R|f(λ)|2|c(λ)|−2dλ, (3.7)

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c(λ) =1√π

Γ(

12iλ)

Γ(

12(iλ+ 1)

) ,|c(λ)|−2 = c(λ)−1c(−λ)−1 =

πλ

2tanh

(πλ

2

).

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Example 4. Hyperbolic branching diffusionThe d-dimensional Lobachevsky space Hd can be defined as the inte-rior of the unit ball in Rd endowed with the Klein metric. (For d = 2,a different model is usually considered, with a Poincare metric, where H2 isrepresented as an interior of a unit complex disc or a half-plane. However, theKlein and Poincare models are isometric.) The Klein metric is compatiblewith the usual Euclidean topology (but not with the usual Euclidean metric)in Rd, and the unit sphere in Rd represents the absolute, i.e. theinfinitely-distant boundary ∂Hd of Hd; we endow it with the standardEuclidean topology. A geodesic line across Hd is a (directed) chord γjoining two points w,w′ ∈ Hd (we say that γ is issued from w and entersw′); it is ‘canonically’ isometric to the (directed) straight line R endowedwith the usual Euclidean metric.

More precisely, under the canonical isometry, (a) the origin onthe line R corresponds to the horospheric projection πho(O) of the

centre O ∈ Hd, and (b) the hyperbolic distance between points

z, z′ ∈ γ equals the Euclidean distance between their images on R.A family of geodesics issued from a given point w ∈ ∂Hd (entering

w′ ∈ ∂Hd) forms a geodesic sheaf G−w with an exit pole at w (respectively,a geodesic sheaf G+

w′ with an entrance pole at w′). A hypersurface Sw ⊂ Hd

orthogonal to the geodesics from the sheaf G−w is called a horosphere(with a pole at w); in the Klein metric it is the punctured boundary of anellipsoid touching absolute ∂Hd at w; in a natural ‘stereographic’ projection,a horosphere Sw is isometric to Rd−1. The stereographic isometry is naturallyextended to the punctured absolute ∂Hd \ w; this generates a so-calledstereographic metric χw on ∂Hd (with a pole at w). Clearly, this metricdepends on the choice of the reference point w ∈ ∂Hd. (In the Poincaremodel of Lobachevsky plane where H2 is represented by an upper half-planeC+, the absolute is the real line R ⊂ C, and with w = ∞, χw is preciselythe Euclidean metric on R.)

The Hausdorff dimension h(Λ) of the limiting set Λ ⊆ ∂Hd forbranching diffusion is calculated in a stereographic metric χw? forsome (chosen) point w? ∈ ∂Hd, all their assertions hold for any choice of w?.Moreover, all results remain true for the so-called angular metricsϕz on the absolute ∂Hd depending on the choice of a reference point z ∈ Hd;in this metric, the distance ϕz(w,w

′), between points w,w′ ∈ ∂Hd is equal tothe (Euclidean) angle between the geodesics γ and γ′ issued from w and w′,

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respectively, and passing through z. (In the Poincare model of Lobachevskyplane where H2 is represented by an open unit disk, the absolute is the unitcircle S1 ∈ C, and with z = 0, ϕz is precisely the Euclidean metric on S1.)

To define a branching diffusion process (BDP) Ξt, one specifiesa law of individual motion and a law of fission. Assume that the in-dividual motion is a homogeneous diffusion on Hd, and our analysisis related to spectral properties of the Laplace–Beltrami operator

∆ =∂2

∂ρ2+ (d− 1) coth(ρ)

∂

∂ρ+

1

sinh2(ρ)∆θ (4.1)

where the Euclidean distance r = tanh(ρ). However, the fission(branching) mechanism is completely general. It includes a fis-sion rate λ(z) and offspring number probabilities pk(z) varying withz ∈ Hd in an arbitrary (measurable) fashion. The upper and lower boundsfor Hausdorff dimension are obtained in terms of the behaviour of λ(z) andpk(z) at and near points of ‘extremum’. More precisely, consider the localfission potential (LFP)

V (z) = 2λ(z)(κ(z)− 1), (4.2)

where κ(z) is the mean number of offspring produced at point z:

κ(z) =∑k≥0

kpk(z). (4.3)

In the case of constant fission mechanism Lalley and Sellke (1997) found theHausdorff dimension of limiting set Λ

h(Λ) =[d− 1− ((d− 1)2 − 8V + )1/2

]/2. (4.4)

Generally, the Hausdorff dimension of limiting set Λ can be bounded aboveand below in terms related to points of maximum and minimum of V (z).There are two types of local maximum of V (z): on ∂Hd (the boundary) andinside Hd (away from the boundary), and they are in a ‘competition’, wherethe ‘prize’ is the maximal value of h and the (closely related) dichotomousoutcome transience vs recurrence.

For any set K ⊂ Hd, define the function ψK(x) on Hd as the P x proba-bility of the event that at least one offspring of the process Ξi will ever visitK. The BDP Ξt is called K− recurrent if ψK(x) ≡ 1. The BDP Ξt is

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called recurrent if Ξt is K− recurrent for all sets K with non-emptyinterior. The BDP Ξt is called K− transient if it is not K− recur-rent; that is ψK(x) < 1 for some set K with non-empty interior andsome x.

An intuitive point of view that the maximum at the absolute is more‘important’ than that inside Hd is often correct, but not always. A compli-cated case is where the maximal value V +

b of V (z) in ∂Hd lies below, whilethe maximal value V +

in = supz∈Hd V (z) lies above, the lowest eigen-value of−1

2∆. Here, the probabilistic intuition is not sufficient, and detailed analytic

calculations are needed, whose result defines the answer. The minimal val-ues of V (z) appear in the lower estimate for Hausdorff dimension and do noteffect transience vs recurrence outcome.

Suppose that V (z) satisfies the inequality

sup[V (z) : z ∈ Hd] ≤ (d− 1)2

8. (4.5)

Consider the maximum on the absolute

V +b = max[V (z) : z ∈ ∂Hd]

and suppose that the point w+ ∈ ∂Hd where value V + is attained is a point

of continuity of V on Hd. Then Hausdorff dimension h(Λ) is bounded with

probability 1 by the expression:

h+ =[d− 1− ((d− 1)2 − 8V +)1/2

]/2 (4.6)

and take with positive probability the values in (h+− ε, h+),∀ε > 0, if V +b ≤

(d−1)2

8, and h(Λ) = d − 1 if V + > (d−1)2

8. Surprisingly or not, h(Λ) does not

depend on the precise value of the global maximum sup[V (z) : z ∈ Hd]. In

other words, even if we have the situation where

V +b < sup[V (z) : z ∈ Hd] ≤ (d− 1)2

8,

the Hausdorff dimension is essentially determined by V +b and not by a larger

value sup[V (z) : z ∈ Hd]. However, if

V +b ≤ (d− 1)2

8< sup[V (z) : z ∈ Hd], (4.7)

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the situation becomes more delicate. The cases A and B are possible:A: h(Λ) is bounded by (4.5), orB: with positive probability h(Λ) = d−1, and set Λ coincides with the wholeabsolute ∂Hd.

Sufficient conditions for cases A and B involve the Bessel-type func-tions J 1

ν and J 2ν , of the first and the second kind, respectively, with (ap-

propriately chosen) index ν. Here, J 1ν (y) = J

(1)iν (iy) and J 2

ν (y) = J(2)iν (iy),

y ∈ R, where J (1)ν and J (2)

ν are the standard Bessel functions, i.e. J 1ν and J 2

ν

are two solutions to the Bessel-type equation

y2J ′′ν + yJ ′

ν + (ν2 − y2)Jν = 0, (4.8)

with a specified asymptotics at y ∼ 0: J 1ν (y) ∼ yiν and J 2

ν (y) ∼ y−iν . BothJ 1

ν and J 2ν are oscillating functions of the (real) argument y.

Namely, we measure how ‘massive’ maxima of LFP V (z) are by assessingtheir relative positions, heights and breadths in terms of sets O(1)

ν and O(2)ν of

(real) zeros, of J 1ν (y) and J 2

ν (y), respectively . More precisely, we state ourresults in terms of existence of specific ‘upper’ and ‘lower’ processes havingadditional symmetry properties.

Define a locally homogeneous horospheric (LHH) pair (λ, P )where λ = λ(z) is a non-negative function and P = P (z) is a collectionof probabilities pk(z), k ≥ 2, which are constant between finitely many horo-spheres with a fixed pole w. More precisely, let S(1)

w , S(2)w , . . ., S(2`−1)

w , S(2`)w

be a collection of distinct horospheres with the pole at w ∈ ∂H such thatS(i+1)

w lies inside the horoball B(S(i)w ) bounded by S(i)

w , i = 1, . . ., 2l− 1. The‘layer’ between S(2j−1)

w and S(2j)w is denoted by L(j)

w , j = 1, . . ., `. Finally, weset

Lw = ∪`j=1L

(j)w , L(0)

w = H \ Lw .

We call (λ, P ) an LHH pair with horospheric layers L(1)w , . . ., L(`)

w

with pole at w ∈ ∂H, if (i) λ(z) takes a constant value λ(j) for z ∈ L(j)w and a

constant value λ(0) for z ∈ L(0)w , (ii) P (z) coincides with a given probability

distribution P (j) for z ∈ L(j)w and with a given probability distribution P (0)

for z ∈ L(0)w .

Next, an LHH pair λ(z), P (z) with layers L(1)w , . . ., L(`)

w is called a majorantfor a pair (λ(z), P (z)) if λ(z) ≤ λ(z) and P (z) P (z) ∀ z ∈ H. In other

words, λ(z) ≤ λ(j)

and P (z) P(j)

for z ∈ L(j)w , j = 1, . . . , `, and λ(z) ≤ λ

(0)

and P (z) P(0)

for z ∈ L(0)w . Here is the usual stochastic ordering between

distributions.

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Theorem 4.1. Let (4.6) holds and the pair (λ(z), P (z)) admits

an LHH majorant (λ(z), P (z)), with layers L(1)

w , . . ., L(`)

w and L(0)

w =

H \(∪`

j=1L(j)

w

)with pole at w ∈ ∂H. Set κ(z) =

∑k≥1

kpk(z). Let λ(j)

= λ(z)

and P (z) = P(j)

, with κ(z) = κ(j) :=∑k≥1

kp(j) for z ∈ L(j)

w , j = 0, 1, . . . , l.

SetV

(j)= 2λ

(j)(κ(j) − 1

), j = 0, . . . , `,

and assume that

V(0) ≤ (d− 1)2

8, and V

(1), . . . , V

(`)>

(d− 1)2

8.

Set

ν(j) =

√V

(j) − (d− 1)2

8, j = 1, . . . , `.

Next, choose a geodesic γ ∈ G−w with pole at w and consider the canonicalco-ordinate on γ. Let I(j) be the interval on the straight line R correspondingto the intersection of γ and L(j)

w . Assume that ∀ j = 1, . . ., ` the Bessel-type fuctions J 1

ν(j)(y) and J 2ν(j)(y) have no zero in I(j).

Then ∀ z0 ∈ H process Θz0 is transient, and Case A hold true.

Next define a geodesic cone Gw,B with pole at w ∈ ∂H and base D ⊂∂H \ w where D is an open (d− 1)-dimensional cube (in the stereographicmetric). Cone Gw,D is formed by all geodesics γ from the sheaf G−w whichcross ∂H at points w ∈ D.

Then a GHP (with pole at w and base D) is a subset Π ⊂ H betweentwo distinct horospheres S(1)

w and S(2)w with pole at w ∈ ∂H which is the

intersection of the corresponding horospeheric layer L(1)w and a geodesic cone

Gw,D.For a GHP- pair (λ(z), P (z)) (for a GHP Π) the function λ(z) and the

probability distribution P (z) remain constant on Π and the complement H\Π.That is, λ(z) = λin and P (z) = Pin for z ∈ Π, and λ(z) = λout and P (z) =Pout for z ∈ H \Π, where λin and λout are two given constants while Pin andPout are two given probability distributions.

Finally, a GHP pair (λ(z), P (z)) is called a GHP-minorant for a pair(λ(z), P (z)) if λ(z) ≤ λ(z) and P (z) P (z) ∀ z ∈ H. In other words,λin ≤ λ(z) and P in P (z) for z ∈ Π, and λout ≤ λ(z) and P out P (z) for

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z ∈ H \ Π. For definiteness, we say that (λ(z), P (z)) is a GHP-minorantover GHP Π with pole at w.

Theorem 4.2. Let (4.6) holds and the pair (λ(z), P (z)) admits aGHP minorant (λ(z), P (z)) over GHP Π with pole at w ∈ ∂H. Set

κ in =∑kkp

k(z) and V in = 2λ in (κ in − 1) and assume that V in >

(d− 1)2

8.

Set

ν in =

√V in −

(d− 1)2

8.

Next, choose a geodesic γ ∈ G−w with γ ∩Π 6= ∅ and consider the standard

horospheric co-ordinate on γ. Let I(1) be the interval on the straight line Rcorresponding to the intersection of γ and Π. Assume that each of theBessel-type fuctions J 1

ν in(y) and J 2

ν in(y) has at least two zeros in I(1).

Then ∀ z0 ∈ H, bound P z0

(Λz0 = ∂H

)> 0. Moreover, if P z0

(Λz0 =

∂H)

= 1 for some z0 ∈ H, then process Θz0 is recurrent and Pz0

(Λz0 =

∂H)

= 1 ∀ z0 ∈ H.

Example 5. Symmetric spaces of rank 1

Let K ⊂ G is a compact Lie sub-group of Lie group G and E =G/K. Consider the direct decomposition of Lie algebra g of G intoLie algebra k of K and its orthogonal complement ξ with respect ofKilling form of g

κ(x, y) = Tr(ad(x)ad(y)

). (5.1)

In particular, for two-dimensional Lobachevsky’s plane

H2 = SL(2,R)/SO(2)

For Lobachevsky’s space of dimension d Lie-group G is the group SO(d, 1)of real (d+1)× (d+1)- matrices of determinant one preserving the quadratic

formd∑

i=1

x2i − x2

d+1.

Rank of a symmetric space is the dimension of a Cartan sub-algebra (the maximal abelian sub-algebra) in ξ. A space X is calledtwo-point homogeneous if for any two-points pairs v, w and v′, w′ with d(v, w) =

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d(v′, w′), there exists an isometry φ such that φ(v) = v′, φ(w) = w′. All rank1 symmetric spaces are two-points homogeneous.

Tits’s classification for symmetric spaces (s.s.)E = G/K of rank 1

Real hyperbolic spaces Hd(R), d = 2, 3, . . ., p = d− 1, q = 0Complex hyperbolic spaces Hd(C), d = 4, 6, . . ., p = d− 2, q = 1Quaternion hyperbolic spaces Hd(H), d = 8, 12, . . ., p = d− 3, q = 3Cayley hyperbolic space H16(Cay), p = 8, q = 7

In geodesic polar coordinates, the Laplace-Beltrami operatorhas the form

LE =∂2

∂r2+

1

S(r)

dS

dr

∂

∂r+ LY

where V (r) =∫ r0 S(u)du and LY is the angular Laplace-Beltrami operator.

Moreover,S(r) = Ωp+q+12

−qc−p−q sinhp(cr) sinhq(2cr)

and c = (2p + 8q)−1/2. Eigenvalues of Y → [H, [H, Y ]] are 0, α(H)2, and

possibly(

12α(H)

)2.

This fact allows to test for transience/recurrence:

FR(r) =S2(r)V (r)

(∫ rR S(u)V (u)du)2

, (5.2)

a(R) = infr>R

FR(r), a(R) = lim supr→∞

FR(r). (5.3)

Then a(R) ≥ 4 for some R > 0 ⇒ RECURRENCE.a(R) ⇒ TRANSIENCE. If ∃a = limr→∞ FR(r) then a ≥ 4 ≡ TRAN-SIENCE.

For complex hyperbolic spaces G is SU(d, 1), the group of com-plex (d+ 1)× (d+ 1)- matrices of of determinant one preserving the

quadratic formd∑

i=1

|zi|2 − |zd+1|2.

For a s.s E of rank one, Cartan algebra A is R. Furthermore,Weyl’s group W consists of two elements, its action on A has theform a 7→ ±a, a ∈ A, and Weyl’s chamber D is a half-line. Wechoose a co-ordinate t on A so that D coincides with non-negativehalf-axis R+. Cartan group A is a one-parameter subgroup a(t),

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t ∈ R of G. Manifold A is one-dimensional and coincides with ageodesic γ0 of space E passing through x0. We scale the parameter ton A so that the Riemannian distance between x0 and y0(t) = a(t)x0

obeysρ (x0, y0(t)) = |t|. (5.4)

We choose a direction on γ0 consistent with the sign of t so thatt gives the Cartesian co-ordinate of the point y0(t) on geodesic γ0.The motions from A take γ0 to itself and perform a parallel trans-lation (in the sense of metric ρ) along γ0. These motions are calledshifts along geodesic γ0. The action of Weyl’s group is reduced tothe change of sign of the Cartesian co-ordinate of point γ0. Thisfollow directly from the Cartan and Iwasawa decompositions,

G = KAK, G = NAK. (5.5)

For the rank 1 E any directed geodesic γ on space E may betaken to γ0 by a motion from G so that a given point x1 ∈ γ will betaken to x0. Therefore, the above construction may be performedfor any directed geodesic of E and any point x0 on it to definethe horospheric projection πγ(x) of any point x ∈ E on any directedgeodesic γ. Next, we fix an origin x0 on a geodesic γ and denoteby tγ(x) the Cartesian co-ordinate of the projection πγ(x).

Let Kx be the stationary subgroup of x ∈ E. The orbits of Kx arecalled the group spheres with the centres at x. For the rank onespace E the group spheres coincide with the Riemannian spheresin metric ρ.

Note the following property of the orbits of nilpotent groupN . Let O be an horosphere and let y0 = a(t0)x0 be the point ofintersection of O with geodesic γ0. Choose an arbitrary t > t0 andconsider a sphere St in E centred at y = a(t)x0 and passing throughy0. Horosphere O is the limiting surface for spheres St as t → ∞.Horosphere O is situated outside any sphere St, t > t0.

Choose on a geodesic γ with a fixed origin an arbitrary pointwith Cartesian co-ordinate t0. Call an horoball Bt0 (= Bt0,γ) the setof points y ∈ E for which tγ(y) ≥ t0. Any Riemannian ball centredat a point s ∈ γ with the Cartesian co-ordinate t > t0 and of radiust− t0 is contained in Bt0.

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For the rank 1 s.s.’s the system of the positive roots Σ+, containsat most two distinct roots, of the form α and 2α; their multiplicitiesare denoted by p and q, respectively. In particular, the case q = 0corresponds precisely with Lobachevsky spaces. Hence, for a rank1 space E the linear functional

λ =1

2

∑α∈Σ+

α =(p

2+ q

)α. (5.6)

Hyperbolic branching process (HBP) and its horospheric pro-jection. Let Θ = (Θn, n = 0, 1, ..., ) be a discrete-time HBP onE, of branching two, and let θ be the corresponding single-particleMarkov process on E. Denote by Ξ = (Ξn, n = 0, 1, ..., ) the horo-spheric projection of process Θ on a geodesic γ0 passing through x0

(recall that x0 ∈ E is a point whose stationary subgroup coincideswith K). Process Ξ is a branching Markov process on γ0 invariantunder the shifts along γ0. If J is the random variable representingthe size of jump in Ξ, then

Φ(θ) = EeθJ . (5.7)

We assume that Φ(θ) < ∞ in a neighbourhood of zero. It fol-lows from the Harish-Chandra functional equation that the mea-sure eθ0tGJ (dt) on R is even, where θ0 is the positive solution ofequation

Φ(2θ0) = 1. (5.8)

In the case of horobolic projection of hyperbolic BM θ0 = p2

+ q.Observe that the drift EJ < 0.

Theorem 5.1. The condition

infθ>0

Φ(θ) ≤ 1

κ(5.9)

is necessary and sufficient for process Θ to spend a finite time inany horoball, and hence in any compact.

An HBP satisfying condition (5.9) is called subcritical.

The absolute of a rank one s.s.: compactification of space E.All geodesic in E are partitioned into equivalence classes that are

15

called nil-sheafs; they consists of geodesics the distance betweenwhich vanishes as one moves in the positive direction. The pointsof the absolute ∂E are identified with the nil-sheafs. Furthermore,given x0 ∈ E, for each nil-sheaf G there exists a unique geodesicγ ∈ G passing through x0; this will determine a ‘natural’ topologyon ∂E. This topology does not depend on the choice of x0. On thecontrary, the natural metric on ∂E compatible with this topologydepends on x0: it is exactly the angular metric ϕx0 mentioned inSection 1.

More precisely, it is convenient to introduce a metric ϕx0 on the‘extended’ space E = E ∪ ∂E which is compatible with metric ϕx0

on ∂E. On E, metric ϕx0 coincides with ρ, the original Riemannianmetric. The distance between points G,G′ ∈ ∂E equals ϕx0(G, bfG

′),i.e., the angle between the geodesics γ and γ′ passing through x0 andbelonging to nil-sheafs G and G′, respectively. Finally, the distance

between G ∈ ∂E and x ∈ E can be equated with1

ρ(x, x0)plus the

angle between the geodesic passing through x0 and belonging to Gand the geodesic passing through both x0 and x. In this metric, Eis homeomorphic to a Euclidean ball and ∂E to the correspondingEuclidean sphere. Observe that the map (g, x) 7→ gx, g ∈ G, x ∈ E,maps G× E continuously on E.

A motion g ∈ G transforms a nil-sheaf to another nil-sheaf.Thereby the action of group G is transfered on the absolute ∂E.Note that for any x ∈ E the stationary group Kx acts on ∂E tran-sitively, because any two geodesics issued from point x may betransformed to each other by a motion from Kx. Hence, we intro-duce, for each x ∈ E, a (Borel) probability measure νx on ∂E whichis invariant under the motions g ∈ Kx: measure νx is the image ofthe normalized Haar measure on Kx.

Let G0 be a nil-sheaf containing geodesic γ0. If the motion g ∈ Gis such that gγ0 ∈ G0 then g transforms G0 to itself.

Proposition 5.1. The existence of the limit

limt→∞

a(t)−1ga(t)

is necessary and sufficient for motion g to transform nil-sheaf G0 toitself. Here the one-parameter group a(t), t ∈ R represents theshifts along geodesic γ0.

16

Proposition 5.2. Element g ∈ G belongs to nilpotent group N iff

limt→∞

a(t)−1ga(t) = e, (5.10)

where e is the unit of group G.Group P = UN transforms nil-sheaf G0 to itself.

6. Metrisation of the absolute in EThere is a family of ‘natural’ metrics on ∂E formed by the so-

called angular metrics. Given a point ζ ∈ E, the angular distanceϕζ(G

′,G′′) between points G′, G′′ ∈ ∂E ‘viewed’ from ζ equals theangle between the geodesics γ ′, γ ′′ passing through ζ and enteringG′ and G′′, respectively. The angular metrics ϕζ are equivalent fordifferent ζ ∈ E (for any ζ ′, ζ ′′ ∈ E the ratio ϕζ′(G′,G′′)/ϕζ′′(G′,G′′) isbounded from both sides by positive constants uniformly in G′,G′′

∈ ∂E; the constants depend on ζ ′, ζ ′′). Thus, the value of the HDdoes not depend on the choice of the ‘viewpoint’ ζ ∈ E.

More precisely, it is convenient to introduce a metric ϕξ0 on the

‘extended’ space E = E ∪ ∂E which is compatible with metric ϕξ0

on ∂E. On E, metric ϕξ0 coincides with ρ, the original Riemannianmetric. The distance between points G,G′ ∈ ∂E equals ϕξ0(G,G′),i.e., the angle between the geodesics γ and γ ′ passing through ξ0 andbelonging to G- and G′-sheaf, respectively. Finally, the distancebetween G ∈ ∂E and ζ ∈ E can be equated with 1/ρ(ζ, ξ0) plusthe angle between the geodesic passing through ξ0 and belongingto G and the geodesic passing through both ξ0 and ζ. In thismetric, E is homeomorphic to a closed Euclidean ball and ∂E to thecorresponding Euclidean sphere. Observe that the map (g, ζ) 7→ g∗ζ,g ∈ G, ζ ∈ E, maps G× E continuously onto E.

In addition, we need a class of homothetic metrics ψ which arenot equivalent to the angular one. To this end, we introduce ametric on nilpotent group N

r(n) =(‖u‖4 + c2

)1/4. (6.1)

Lemma 6.1. Nilpotent group N is formed by complex (d + 1) ×

17

(d+ 1) matrices n(= n(u, c)) of the form1 0 u1 −u1

0 1 u2 −u2

−u1 1− ||u||2/2 + ci/2 ||u||2/2− ci/2−u1 1− ||u||2/2 + ci/2 1 + ||u||2/2− ci/2

The left corner is represented by Id−1, the unit (d − 1) × (d − 1)

matrix, u stands for the complex vector

u1...

ud−1

∈ Cd−1, of norm

||u||2 =d−1∑j=1

ujuj and c ∈ R is a real number.

Next, we use the fact that group N acts simply-transitively on∂E. This means that ∀ Γ ∈ ∂E \ G0, the G-sheaf is uniquelyrepresented as G = nG−

0 where n ∈ N . Thus, given two points onthe absolute, G1 = n1G

−0 and G2 = n2G

−0 , where ni ∈ N , i = 1, 2, we

setψG0(G1,G2) = r(n−1

1 n2). (6.2)

Theorem 6.1. Formulas (6.1) and (6.2) define a metric ψG0 on∂E \ G0. This metric is invariant under the action of group N

ψG0(G1,G2) = ψG0(nG1, nG2), n ∈ N, (6.3)

and obeys the property that

ψG0(G1(s),G2(s)) = e−sψG0(G1,G2), (6.4)

where G1(s) = a(s)−1n1a(s)G−0 , G2(s) = a(s)−1n2a(s)G

−0 , and G1 =

n1G−0 , G2 = n2G

−0 . ψG0 is called a homothetic metric (with the pole

at G0).

The proof is based on the following four properties of functionr:

10. r(n) ≥ 0, r(n) = 0 iff n = e, the unit of N .20. r(n−1) = r(n).30. r(n1n2) ≤ r(n1) + r(n2).

18

40. The ratio r(n(s))/r(n) = e−s is constant in n ∈ N .Among these properties it is only 30 that requires a (simple)

analytical proof. Consider a two-dimensional (complex) vector

b(n) =(‖u‖2, c

), (6.5)

then r(n) = ||b(n)||1/2. Vectors b(n1), b(n2) and b(n1n2) are, obviously,related with the equality

b(n1n2) = b(n1) + b(n2)

+(2Re 〈u1,u2〉, Im 〈u1,u2〉

).

(6.6)

To check 30, write

(r(n1n2))2 = ||b(n1n2)|| ≤ ||b(n1)||+ ||b(n2)||

+[(

2Re 〈u1,u2〉)2

+(Im 〈u1,u2〉

)2]1/2

.(6.7)

On the other hand,

[r(n1) + r(n2)]2 ≥ ||b(n1)||+ ||b(n2)||+ 2‖u1‖‖u2‖. (6.8)

Therefore, it suffices to check that(2Re〈u1,u2〉

)2+(Im〈u1,u2〉

)2≤ 4‖u1‖2‖u2‖2. (6.9)

which is straightforward.

The metric ψG0 is related to the element a′ of algebra A gener-ating group a(s), s ∈ R, and to the minimal root α from Σ+.

Properties (6.3) and (6.4) imply that if γ is an arbitrary geodesicfrom G0-sheaf and aγ(s) is a shift along γ then

ψG0(aγ(s)G1, aγ(s)G2) = e−sψG0(G1,G2). (6.10)

Observe that metric ψG0 and the angular metrics are non-equivalenteven on the set ∂E \ O where O is an open neighbourhood of point

G0 on the absolute. In fact, either the ratioψG0(G,G1)

φξ(G,G1)or its inverse

φξ(G,G1)

ψG0(G,G1)is unbounded as G → G1. However, it turns out that

19

ψG0 and ϕx0 are related with an inequality that implies a similarinequality for the HD’s.

Furthermore, two homothetic metrics ψΓ′ and ψ

Γ′′ on ∂E are notequivalent. However, they are equivalent outside a (Euclidean)neighbourhood of a circle passing through Γ′ and Γ′′ which is theorbit of the one-parameter subgroup N0

Γ′ congugated with N0 =

(u, c) : u = 0 (equivalently, the orbit of the one-parameter sub-group N0

Γ′′ congugated with N0 = (u, c) : u = 0).The (limited) affinity between H2d and Ed can be scrutinised

at the level of final formulas for the processes of homogeneousbranching diffusion (HBD) which leads to a spectacular juxtaposi-tion. Recall that the homogeneous diffusions on Ed and H2d havegenerators − ∆Ed/ 2 and − ∆H2d/ 2 where ∆Ed and ∆H2d are the corre-sponding Laplace–Beltrami operators. Let the fission rate be λ > 0and the mean number of offspring κ > 1 in both processes, and setV = 2λ(κ−1) (the fission potential). Then the Hausdorff dimensionshEd and hH2d are given by

hEd =

d−

√d2 − V , if 0 ≤ V ≤ d2,

2d− 1, if V > d2,

hH2d =

2d− 1

2−√

(2d− 1)2

4− V , if 0 ≤ V ≤ (2d− 1)2

4,

2d− 1, if V >(2d− 1)2

4

.

Comparing these formulas, we see that the HBD on the complexhyperbolic space Ed becomes recurrent (i.e., prolific) at the criticalvalue V cr

Ed = d2 which is larger than the corresponding critical valueV crH2d = (2d− 1)2/4. Also, the Hausdorff dimension hEd attains at V cr

Ed

the value d whereas hH2d attains at V crH2d the lower value d− 1/2.

REFERENCES

1. McKean H.P. Applications of Brownian motion to the equa-tions of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl.Math., 28 (1975), 323-331Correction to the above. Commun. Pure Appl. Math., 29 (1976),553-554

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2. Karpelevich F.I., Kelbert M.Ya. and Suhov Yu.M. The higher-order Lindley equations. Stochastic Processes and their Appl., 53(1994), 65-96

3. Karpelevich F.I., Kelbert M.Ya. and Suhov Yu.M. The bound-edness of branching Markov processes. In: The Dynkin Festschrift.Markov Processes and Their Applications (ed M. Freidlin). Progressin Probability, Vol. 34, Boston: Birkhauser, 1994, 143–153

4. Kelbert M. and Suhov Y.M. The Markov branching randomwalk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov)equations. Commun. Math. Phys., 167 (1995), 607-634

5. Lalley S. and Sellke T. Hyperbolic branching Brownian mo-tion. Prob. Theor. Rel. Fields, 108 (1997), 171–192

6. Karpelevich F.I., Pechersky E.A. and Suhov Y.M. A phasetransition for hyperbolic branching processes. Commun. Math.Phys., 195 (1998), 627 - 642

7. Grigoryan A. and Kelbert M. Range of fluctuation of Brow-nian motion on a complete Riemannian manifold. Ann. Probab.,26 (1998), n.1, 78-111

8. Grigoryan A. and Kelbert M. On Hardy-Littlewood typeinequality for Brownian motion on Riemannian manifold. Journ.London Math. Society, 62(2) (2000), 625-639

9. Grigoryan A. and Kelbert M. Asymptotic separation forindependent trajectories of Markov processes, Prob. Theor. Rel.Fields, 119 (2001), 31–69

10. Grigoryan A. and Kelbert M. Recurrence and transienceof branching diffusion processes on Riemannian manifolds. Ann.Probab., 31 (2003), n.1, 244–284

11. Kelbert M. and Suhov Y.M. Branching diffusion on Hd

with variable fission: the Hausdorff dimension of the limiting set.Probab. Theory Appl., 51 (2006), n.1 (The Sinai’s Festschrift)

12. Kelbert M. and Suhov Y.M. Branching diffusion on Hd

with variable fission: a general approach Probab. Theory Appl.,52 (2007)

13. Sazonov I. Kelbert M. and Gravenor M.B. Approxima-tion for epidemic speed in a one-dimensional lattice of SIR models.Physical Review E ( 2007)

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