Multiple Poisson-Dirichlet diffusions on generalized Kingman simplices

Multiple P oisson–Diric hlet diffusions on generalized Kingman simplices Cristina Cost antini National Gr oup for Mathematic al A nalysis, Pr ob ability and Applic ations, INdAM, Italy Ma tteo R uggiero Stern Scho ol of Business, NYU Abu Dhabi F ebruary 25, 2026 W e construct a new class of infinite-dimensional diffusions taking v alues in a generalized Kingman simplex. Our mo del describ es the temp oral ev olution of the relative frequencies of infinitely-man y types whic h are lab ele d b y an arbitrary finite num b er of marks or colors, but unlab ele d within each mark. W e start with a finite-dimensional construction whic h extends to W righ t–Fisher diffusions a self-similarit y property kno wn for Dirichlet distributions, and corresp onds to a m ultiple skew-product representation of the W right–Fisher diffusion relativ e to the marks in the p opulation. After ranking decreasingly the frequencies within eac h mark, we iden tify the limit in distribution of the resulting diffusion when the n um b er of t yp es for eac h mark go es to infinity , and describe its infinitesimal operator. The limiting pro cess reduces to a diffusion in the Thoma simplex in the sp ecial case of only t wo marks, whereas the infinitely-man y-neutral-alleles mo del is recov ered when all frequencies ha ve the same mark. The stationary measure of the limit diffusion is sho wn to b e the recen tly introduced multiple Poisson–Diric hlet distribution, whic h extends Kingman’s P oisson–Diric hlet distribution and is the de Finetti represen ting measure for a family of random partitions whose elemen ts are marked. Keyw ords : infinitely-man y-alleles mo del; m ulti-parameter random time c hange; m ultiple random partition; skew-product decomp osition; Thoma simplex; W right– Fisher diffusion. MSC Primary: 60J60. Secondary: 60G10 92D25, 1 In tro duction 1.1 Diffusions pro cesses on infinite-dimensional simplices The P oisson–Dirichlet diffusion, mostly known as infinitely-many-neutr al-al leles mo del , is a diffusion pro cess taking v alues in the space ∇ :=  x ∈ [0 , 1] ∞ : x 1 ≥ x 2 ≥ . . . ≥ 0 , ∞ X i =1 x i ≤ 1  , (1.1) 2 Cost antini and R uggiero sometimes called Kingman simplex, which is the closure (in the product topology) of the infinite- dimensional ordered simplex ∇ :=  x ∈ [0 , 1] ∞ : x 1 ≥ x 2 ≥ . . . ≥ 0 , ∞ X i =1 x i = 1  . (1.2) On a suitable domain, the generator of the Poisson–Diric hlet diffusion is defined by A θ f ( x ) := 1 2 ∞ X i,j =1 x i ( δ ij − x j ) ∂ 2 f ∂ x i ∂ x j ( x ) − θ 2 ∞ X i =1 x i ∂ f ∂ x i ( x ) , (1.3) for x ∈ ∇ , and by the con tinuous extension of the ab ov e function to ∇ , for x ∈ ∇ − ∇ . Here θ > 0 and δ ij is the Kroneck er delta. The contin uous extension of ( 1.3 ) to ∇ is still denoted by A θ f . Ethier and Kurtz (1981) show ed that the closure of A θ generates a F eller semigroup on C ( ∇ ) and the associated pro cess, whic h has contin uous tra jectories, is stationary and rev ersible with resp ect to Kingman’s Poisson–Diric hlet distribution PD( θ ) (Kingman, 1975) ; cf. also F eng (2010) , Theorem 5.2. See Ethier and Kurtz (1993) , Section 9, and F eng (2010) , Section 5.1, for general reviews on this mo del. The family of diffusions with op erator A θ ma y b e though t of as describing the temp oral ev olution of the decreasingly rank ed frequencies of infinitely-many types in an ideally infinite population, sub ject to paren t-indep enden t mutation and random genetic drift. This in terpretations is supported b y constructions obtained through W righ t–Fisher (WF) and Moran type discrete c hains; see, for example Ethier and Kurtz (1981) , Section 3. The mo del is also closely related to a class of Fleming– Viot probability-measure-v alued diffusions (Fleming and Viot, 1979) driven by an ergo dic pure- jump pro cess, sometimes called the lab ele d infinitely-man y-neutral-alleles mo del; cf. Ethier and Kurtz (1986) , Section 10.4. In these mo dels, one keeps trac k not only of the frequencies, but also of the types, b y assigning to eac h frequency a lab el (given by a lo cation in an arbitrary P olish space). It turns out that the op erator A θ describ es the evolution of this Fleming–Viot pro cess once the lab els are remov ed and the associated frequencies are decreasingly ranked, reason for which it is said to b e the unlab ele d v ersion of the (lab eled) infinitely-many-neutral-alleles model; cf. Ethier and Kurtz (1993) , Theorem 9.2.1. P etrov (2009) in tro duced a tw o-parameter extension of the P oisson–Dirichlet diffusion, with generator A α,θ f = A θ f − α 2 ∞ X i =1 ∂ f ∂ x i , 0 ≤ α < 1 , θ ≥ − α. (1.4) The diffusion on ∇ asso ciated to ( 1.4 ) is stationary and reversible with respect to the t w o-parameter P oisson–Dirichlet distribution (Pitman, 1995; Pitman and Y or, 1997) , and w as further inv estigated in Ruggiero and W alker (2009); F eng and Sun (2010); F eng et al. (2011); Ruggiero et al. (2013); Costan tini et al. (2017); F orman et al. (2021; 2022; 2023); Griffiths et al. (2024) . Multiple Poisson–Dirichlet diffusions 3 Boro din and Olshanski (2009) generalized the approach of Ethier and Kurtz (1981) and intro- duced a class of diffusion processes taking v alues in the so-called Thoma simplex T :=  ( x, y ) : x, y ∈ ∇ , ∞ X i =1 ( x i + y i ) ≤ 1  . (1.5) Olshanski (2010) introduced another class of diffusions on T related to Jac k p olynomials (cf. Mac- donald, 2015 , Sec VI.10), whic h reduces to the P oisson–Diric hlet diffusion when a certain parameter go es to zero. See also Olshanski (2018); Korotkikh (2020; 2024) . In this paper w e consider a further direction. W e aim at constructing a diffusion pro cess modeling the temp oral evol ution of the frequencies of infinitely-many t yp es which are lab ele d b y an arbitrary finite n um b er of marks (or colors), but are unlab ele d within eac h mark. The state space of this pro cess is the gener alize d Kingman simplex K H :=  z = ( z 1 , . . . , z H ) : z h ∈ ∇ , H X h =1 ∞ X i =1 z h,i ≤ 1  , (1.6) where each z h ∈ ∇ describes the frequencies of infinitely-many types with mark h . When H = 1, K H clearly reduces to ( 1.1 ) and the process reduces to ( 1.3 ), while, for H = 2, K H reduces to ( 1.5 ) but our pro cess do es not reduce to those in Borodin and Olshanski (2009); Olshanski (2010) . The pro cess we construct is stationary with resp ect to the recen tly introduced multiple Poisson– Dirichlet distribution (Straho v, 2024a;b) (Cf. Definition 1.1 b elow). This distribution extends Kingman’s celebrated P oisson–Dirichlet distribution to a probability measure on  z ∈ K H : H X h =1 ∞ X i =1 z h,i = 1  ⊂ K H and is the de Finetti representing measure for a family of multiple r andom p artitions , i.e., random partitions whose elemen ts are lab eled b y means of finitely many marks. These are also related to a generalization of Chinese r estaur ant pr o c esses (Pitman, 2006) and a refinemen t of the Ewens sam- pling formula (T av ar ´ e and Ewens, 1997) . Cf. (Straho v, 2024a;b) for details and further connections with represen tation theory and harmonic analysis on branching graphs. Multiple random partitions find application in population genetics, when the alleles sampled from an infinite-alleles mo del need to be classified in to a finite n umber of classes. An example is giv en by cystic fibrosis, a respiratory disease caused b y mutations in one gene, the cystic fi- br osis tr ansmembr ane c onductanc e r e gulator (Coleman and Tsongalis, 2009) . The roughly 2000 alterations to this gene that hav e so far been describ ed ha ve a wide range of disease severit y and can be grouped according to their functional defect in to six or sev en classes, cf., e.g., Row e at al. (2005); De Bo eck and Margarida (2016) . Such classification is imp ortant as existing treatments and pharmacotherapies are sp ecific to mutation classes, cf. V eit et al. (2016) . Our construction 4 Cost antini and R uggiero th us provides a model for the temp oral dynamics of the relative frequencies of the alleles when they are classified in to a finite num b er of classes, as in the cystic fibrosis example. Another potential application of our pro cess is to Bay esian inference. A v ery activ e line of researc h in Bay esian statistics is concerned with designing temp oral dynamics with Dirichlet, one- or t wo-parameter Poisson–Diric hlet and related distributions as equilibrium measures. See the recent review (Quin tana et al., 2022) . The distributions of these sto c hastic pro cesses on the resp ective path spaces can b e used as prior distributions . These, together with temp orally correlated data dra wn from the ev olving random measure at discrete times, can b e used for p osterior inference on the tra jectories or for inference on the model parameters, typically with the aid of simulation-based strategies lik e Mark o v Chain Mon te Carlo or Sequen tial Monte Carlo. See, e.g., Caron et al. (2017); Mena and Ruggiero (2016); Ro driguez and ter Horst (2008); Kon Kam King et al. (2021) among man y others. In this p ersp ective, the distribution of the diffusion pro cess we construct is amenable to b e used as a prior distribution for tra jectories from [0 , ∞ ) to K H , together with serially-correlated data dra wn from an infinite p o ol of types classified in to finitely-man y classes. F urthermore, recen tly P apaspiliop oulos et al. (2016); Ascolani et al. (2021; 2023); Dalla Pria et al. (2025) show ed that the priors induced on the path spaces b y certain Fleming–Viot, Dawson–W atanab e and P oisson– Diric hlet diffusions are analytically tractable when treated as hidden Marko v mo dels (Capp´ e et al., 2005) , with noisy data collected at discrete times from the underlying ev olving p opulation. W e exp ect the pro cess w e construct to enjoy similar properties (not explored here) and thus to allo w the deriv ation of closed form expressions in a Ba y esian p osterior analysis. Since it is stationary with resp ect to the m ultiple P oisson-Dirichlet distribution, w e call our pro- cess the multiple Poisson–Dirichlet diffusion . W e construct the m ultiple P oisson–Dirichlet diffusion as the limit of WF diffusions, after the types ha v e b een classified in to H classes, the frequencies of the types in each class ha ve b een decreasingly rank ed and the n umber of types in eac h class is let go to infinit y . Our approach relies on a multiple skew–pr o duct de c omp osition that generalizes the skew–product decomp osition relating WF diffusions and squared Bessel pro cesses via a ran- dom time change (W arren and Y or, 1998; Pal, 2011) , b y employing a multip ar ameter r andom time change (Helm, 1974; Kurtz, 1980) (see Section 2 ). Note that, from a tec hnical p oint of view, the step from random time changes to multiparameter random time changes is not immediate. The m ultiple P oisson–Dirichlet diffusion is first obtained as the limit of m ultiple skew-product decom- p osed WF diffusions, hence in m ultiple sk ew-pro duct decomposed form. See also (Konno and Shiga, 1988; P erkins, 1991; F orman et al., 2023) for sk ew-pro duct decomp ositions of Fleming–Viot and Da wson–W atanab e pro cesses. Next we fully describ e the generator of the multiple P oisson–Dirichlet diffusion in non-decomposed form and its domain (see Section 3 ). Finally we show that m ultiple P oisson–Dirichlet diffusions are stationary with resp ect to m ultiple P oisson-Dirichlet distributions (see Section 4 ). Multiple Poisson–Dirichlet diffusions 5 1.2 Ov erview of the results In this section w e pro vide an informal o v erview of the construction that will be developed in the rest of the paper. Consider an ideally infinite population of individuals with a finite num b er of t yp es, where the t yp es are mark ed. W e assume there are H ≥ 1 possible marks and K ≥ 2 t yp es for eac h mark, and let the pair ( h, i ) identify type i among those with mark h , where i = 1 , . . . , K for eac h h = 1 , . . . , H . The o verall n umber of types is th us H K . W e refrain from using the app ealing terminology “subgroup h ” to av oid misunderstandings on the spatial structure of the system, whic h w e do not consider here. W e let the relative frequencies of types in this system evolv e as an H K -type WF diffusion with generator B K f ( z 1 , 1 , . . . , z H,K ) = 1 2 H X h,k =1 K X i,j =1 z h,i ( δ h,k δ i,j − z k,j ) ∂ 2 f ∂ z h,i ∂ z kj + 1 2 H X h =1 K X i =1  θ h K − | θ | z h,i  ∂ f ∂ z h,i , (1.7) where θ 1 , . . . , θ H > 0, θ := ( θ 1 , . . . , θ H ) and | θ | := P H h =1 θ h . In a classical discrete construction of a WF diffusion through a scaling limit of WF Marko v chains (cf., e.g., Ethier and Kurtz (1986) , Section 10.2; Etheridge (2009) , Section 4.1) this would correspond to assuming tw o evolutionary mec hanisms acting on the p opulation. The first is a mean-field random resampling, whereby tw o individuals are c hosen at random from the curren t population, one of the t wo is remov ed and the other generates a clone, whic h giv es rise to the second-order term in B K . The second is given b y m utation, whereb y mutation to a type with mark h is parent-independent, i.e., it does not dep end on the previous t yp e of the mutan t, and o ccurs at rate θ h / 2 K . The m utation mec hanism leads to the first order term which identifies the drift of eac h comp onent. Here w e are allowing for self-m utations for notational simplicity , and assume absence of selection throughout. When H = 1 in ( 1.7 ), Ethier and Kurtz (1981) show ed that, ranking the frequencies in decreasing order and letting K go to infinity , one obtains the Poisson–Diric hlet diffusion with op erator ( 1.3 ). In this pap er w e carry out a similar op eration for a fixed, arbitrary H > 1, when for each mark h = 1 , . . . , H , the frequencies Z K h,i , i = 1 , . . . , K , are ranked decreasingly and K is let go to infinity . After some attempts with the classical approach of Ethier and Kurtz (1981) and Boro din and Olshanski (2009) , based on the Hille–Y osida theorem, an approac h whic h turned out to b e fruitful is that suggested b y the self-similarit y prop erty of the Dirichlet distribution, whic h is well known to b e the rev ersible, stationary distribution of a WF diffusion. This property , recalled in detail in Prop osition 2.1 b elow, informally amoun ts to the fact that one can group the random Dirichlet- distributed frequencies according to an arbitrary partition of the indices, and b oth the sums for eac h group, and the renormalized frequencies within each group, are still Dirichlet distributed with appropriate parameters. 6 Cost antini and R uggiero If Z K :=  Z K 1 , . . . , Z K H  is the diffusion with generator B K , with Z K h := ( Z K h, 1 , . . . , Z K h,K ), w e th us consider the ( H − 1)-dimensional pro cess  K X i =1 Z K 1 ,i ( · ) , . . . , K X i =1 Z K H,i ( · )  whic h describ es the ev olution of the total masses asso ciated to eac h of the H marks, and, for eac h h = 1 , . . . , H , the ( K − 1)-dimensional pro cess  Z K h, 1 ( · ) P K i =1 Z K h,i ( · ) , . . . , Z K h,K ( · ) P K i =1 Z K h,i ( · )  , whic h describes the evolution of the relative frequencies of the t yp es with mark h . It turns out that, formally , the pro cess of mark masses and the H processes of t yp e frequencies within eac h mark ha ve join t generator A K f ( w , x ) = 1 2 H X h,k =1 w h ( δ h,k − w k ) ∂ 2 f ∂ w h ∂ w k + 1 2 H X h =1  θ h (1 − w h ) − ( | θ | − θ h ) w h  ∂ f ∂ w h + H X h =1 1 w h  1 2 K X i,j =1 x h,i ( δ i,j − x h,j ) ∂ 2 f ∂ x h,i ∂ x h,j + θ h 2 K X i =1  1 K − x h,i  ∂ f ∂ x h,i  , where we are denoting b y w h the mass associated to mark h and by x h,i the relative frequency of t yp e i within those with mark h . The first tw o terms in A K corresp ond to the generator of an H -type WF diffusion with parent-independent mutation gov erned b y parameters θ 1 , . . . , θ H , while eac h term in square brack ets identifies a K -type WF diffusion with symmetric m utation occurring at rate θ h / 2 K tow ards each type among those with mark h . The co efficients 1 /w h can b e in terpreted as deriving from a multip ar ameter r andom time change . This leads to the results of Section 2 , informally describ ed as follows. When θ 1 , . . . , θ H ≥ 1, if P K i =1 Z K h,i (0) > 0 for all h = 1 , . . . , H , and  K X i =1 Z K 1 ,i (0) , . . . , K X i =1 Z K H,i (0)  , Z K 1 (0) P K i =1 Z K 1 ,i (0) , . . . , Z K H (0) P K i =1 Z K H,i (0) are m utually indep endent, it holds that Z K :=  Z K 1 , . . . , Z K H  d =  W K 1 X K 1 , . . . , W K H X K H  , (1.8) where W K is an H -t yp e WF diffusion with mutation parameters θ 1 , . . . , θ H and X K h is defined for eac h h = 1 , . . . , H as X K h ( t ) := X K h  Z t 0 1 W K ( s ) ds  , (1.9) Multiple Poisson–Dirichlet diffusions 7 where X K h is a K -type WF diffusion with symmetric m utation with rates θ h / 2 K and W K , X K 1 , . . . , X K h are indep endent. Eq. ( 1.9 ) expresses the fact that changes in the relativ e frequencies of the t yp es with mark h o ccur on the time scale R t 0  1 /W K h ( s )  ds . Heuristically , this may b e related to the fact that, under random resampling, at time s a parent will b e c hosen from the t yp es with mark h with probability W K h ( s ), hence we ha ve to w ait on a verage for a (1 /W K h ( s ))s . amoun t of time until a paren t with mark h is chosen. The pro cess driv en b y A K admits in terpretation in hierarchical terms, whereby at the top la y er there is a WF diffusion which gov erns the masses asso ciated to the marks, and at the low er la y er there are the relative frequencies within each mark, randomly ev olving according to another WF diffusion each: conditionally on a sample path for the random masses, the H pro cesses for each mark would b e indep endent WF diffusions, albeit on differen t time scales determined b y the evolution of their random masses. The representation ( 1.8 ) allo ws us to pro v e, under a mild assumption on the initial distributions, that the process obtained from Z K =  Z K 1 , . . . , Z K H  b y ranking decreasingly the t yp e frequencies for eac h mark, conv erges in distribution, as K go es to infinit y , to a Marko v pro cess Z . No w, since the generator of Z K is B K in ( 1.7 ), it is natural to conjecture as the generator of the limit pro cess Z the op erator ˆ B f ( z 1 . . . , z H ) := 1 2 H X h,k =1 ∞ X i,j =1 z h,i ( δ h,k δ i,j − z k,j ) ∂ 2 f ∂ z h,i ∂ z k,j − | θ | 2 H X h =1 ∞ X i =1 z h,i ∂ f ∂ z h,i . Ho wev er, it turns out that the generator of Z has the ab o ve form only for a sub class of functions in its domain. In fact, the functions in this domain are such that f ( z ) := f 0 ( | z 1 | , . . . , | z H | ) F ( z ) , | z h | := X i ≥ 1 z h,i , (1.10) where f 0 and F ( z ) hav e a sp ecific form, and the limit op erator is B f = ˆ B f + 1 2 H X h =1 θ h ∂ f 0 ∂ | z h | F . All this will b e fully detailed in Section 3 . Note that w e could isolate the indep endent dynamics within each mark and highligh t the inter- action among the marks b y writing, for f as in ( 1.10 ), B f = H X h =1 B h f − 1 2 X 1 ≤ k  = h ≤ H ∞ X i,j =1 z h,i z k,j ∂ 2 f ∂ z h,i ∂ z k,j , where B h f := 1 2 ∞ X i,j =1 z h,i ( δ ij − z h,j ) ∂ 2 f ∂ z h,i ∂ z h,j − | θ | 2 ∞ X i =1 z h,i ∂ f ∂ z h,i + θ h 2 ∂ f 0 ∂ | z h | F . 8 Cost antini and R uggiero In Section 4 we inv estigate stationary distributions. It is useful to recall here that Kingman’s PD( θ ) distribution on ∇ , besides being the rev ersible measure of ( 1.3 ), is also the de Finetti measure in Kingman’s representation theorem for random partitions, whose marginal law is the celebrated Ew ens sampling formula. See Kingman (1978); T av ar´ e and Ew ens (1997); Crane (2016); T a v ar´ e (2021) ; see also the discussion in Section 2 of Griffiths et al. (2024) . Recen tly , Strahov (2024a;b) extended suc h represen tation to m ultiple partition structures, extending the Ew ens sampling for- m ula to the case of a partition where the random multiplicities are associated to one of a finite n umber of marks. In this case, the representing measure in de Finetti’s theorem is a probabilit y distribution on ( 1.6 ), recalled in the follo wing definition. Definition 1.1. (Multiple Poisson–Diric hlet distribution). Let υ = ( υ 1 , . . . , υ H ) ∼ Dir H ( θ 1 , . . . , θ H ) b e a Diric hlet distributed v ector, and let ξ h ∼ PD( θ h ), h = 1 , . . . , H , be indep enden t ∇ -v alued random v ariables, independent of υ . Then ζ := ( ζ 1 , . . . , ζ H ) ∈ K H , where ζ h := υ h ξ h = ( υ h ξ h, 1 , υ h ξ h, 2 , . . . ), is said to ha v e multiple Poisson–Dirichlet distribution PD( θ ) with parameters θ := ( θ 1 , . . . , θ H ). □ It turns out that the m ultiple P oisson–Dirichlet distribution is a stationary distribution for the pro cess Z constructed in Section 3 , whic h therefore we name multiple Poisson–Dirichlet diffusion . As a corollary , w e provide a construction of the m ultiple Poisson–Diric hlet distribution analogous to Kingman’s construction of PD( θ ) as a limit in distribution of ranked symmetric Diric hlet random frequencies. 1.3 Notation W e highligh t here t w o notational conv entions that will b e used throughout. W e use ro w v ector notation as in ( 1.8 ). W e also define, for a closed set D , the space of t wice-differen tiable functions on D as C 2 ( D ) := { f ∈ C ( D ) : ∃ ˜ f ∈ C 2 ( R d ) , ˜ f | D = f } , where C ( D ) is the set of con tinuous functions on D . 2 Multiple sk ew-pro duct decomp osition for W righ t–Fisher diffu- sions W e start by recalling a w ell kno wn self-similarit y property for Dirichlet distributions. Recall that a Diric hlet distribution Dir H ( θ 1 , . . . , θ H ) has densit y on the usual H probabilit y simplex ∆ H :=  w ∈ [0 , 1] H : H X h =1 w h = 1  , (2.1) Multiple Poisson–Dirichlet diffusions 9 with resp ect to the ( H − 1)-dimensional Leb esgue measure, given by Γ( P H h =1 θ h ) Q H h =1 Γ( θ h ) w θ 1 − 1 1 · · · w θ H − 1 H , θ 1 , . . . , θ H > 0 , where w H = 1 − P H − 1 h =1 w h , and Γ( a ) = R ∞ 0 y a − 1 e − y dy . Prop osition 2.1. L et ( ζ 1 , . . . , ζ H K ) ∼ Dir H K  θ 1 K , . . . , θ 1 K , θ 2 K , . . . , θ 2 K , . . . , θ H K , . . . , θ H K  (2.2) wher e e ach θ h is r ep e ate d K times, and define υ := ( υ 1 , . . . , υ H ) and ξ h := ( ξ h, 1 , . . . , ξ h,K ) by υ h := hK X j =( h − 1) K +1 ζ j , ξ h,i := ζ ( h − 1) K + i P hK j =( h − 1) K +1 ζ j , for h = 1 , . . . , H and i = 1 , . . . , K . Then υ ∼ Dir H ( θ 1 , . . . , θ H ) , ξ h ∼ Dir K  θ h K , . . . , θ h K  , h = 1 , . . . , H, wher e υ , ξ 1 , . . . , ξ H ar e indep endent. The c onverse is also true. Pr o of. This is a sp ecial case of a more general result which relies on the construction of Dirichlet distributions through normalization of indep endent gamma random v ariables. See Theorem 1.3.1 in F eng (2010) and Prop osition G.3 in Ghoshal and v an der V aart (2013) . The ab ov e self-similarit y holds for general partitions of the indices in the vector of parameters in ( 2.2 ), and can b e also extended to gamma sub ordinators and Diric hlet random probability measures. See, e.g., F eng (2010) , Theorem 2.23, and Ghoshal and v an der V aart (2013) , Section 4.1.2. In this section w e pro vide an analog of Proposition 2.1 for WF diffusions. In this construction, w e replace ζ 1 , . . . , ζ H K with an H K -type WF diffusion Z 1 , . . . , Z H K with paren t-indep enden t m utation with rates θ 1 2 K , . . . , θ 1 2 K , θ 2 2 K , . . . , θ 2 2 K , . . . , θ H 2 K , . . . , θ H 2 K . W e also replace υ , ξ 1 , . . . , ξ H with W K , X K 1 , . . . , X K H , where W K is an H -type WF diffusion with paren t-indep enden t mutation with rates θ 1 / 2 , . . . , θ H / 2, and W K dep ends on K only by its initial distribution, and eac h X K h is a K -type WF diffusion observed on a random time scale, namely X K h ( t ) := X K h  Z t 0 1 W K h ( s ) ds  , (2.3) where X K h is a K -type WF diffusion with symmetric m utation with rates θ h / (2 K ). Informally , the H K types are marked with H lab els, eac h marking K types; W K driv es the ev olution of the 10 Cost antini and R uggiero masses assigned to eac h group of types with the same mark, while X K h describ es the evolution of the relative frequencies within the group of types with mark h . The app earance of the random time scale ( 2 . 3 ) can b e related to the following heuristic observ ation. As it is w ell known, a WF diffusion mo dels the evolution of the type frequencies in an ideally infinite p opulation under the assumption that, at eac h generation, each individual of the next generation c ho oses its paren t at random from the curren t generation. Then, at eac h generation, a parent will b e chosen from t yp es with mark h with probabilit y W K h and we will ha v e to wait on av erage for an interv al of length 1 /W K h un til a parent with mark h is c hosen. Therefore changes in the relative frequencies of types with mark h will occurr on the time scale R t 0  1 /W K h ( s )  ds . The n um b er of marks H will b e fixed throughout the paper, hence typically dropp ed as a super- script or subscript unless needed for clarit y , whereas the n umber of types with the same mark, K , is k ept fixed in this section but later will b e let diverge to infinity . More precisely , let W K = ( W K 1 , . . . , W K H ) b e a WF diffusion on the H -simplex ( 2.1 ) with gener- ator, for θ = ( θ 1 , . . . , θ H ) and | θ | := P H h =1 θ h , A 0 f ( w ) := 1 2 H X h,k =1 w h ( δ h,k − w k ) ∂ 2 f ∂ w h ∂ w k + 1 2 H X h =1 ( θ h − | θ | w h ) ∂ f ∂ w h , (2.4) acting on the domain D ( A 0 ) := C 2 (∆ H ) (cf. Section 1.3 ), where δ h,k is the Kronec ker delta. See Ethier and Kurtz (1986) , Theorem 8.2.8. The Dir H ( θ 1 , . . . , θ H ) distribution is the stationary and rev ersible measure for ( 2.4 ); see, e.g., F eng (2010) , Theorem 5.1. Let no w ∆ ◦ H :=  w ∈ ∆ H : w h > 0 , h = 1 , . . . , H  . (2.5) The following lem ma identifies the restriction we are going to consider in ( 2.4 ) so that ∆ H \ ∆ ◦ H is an en trance b oundary . Lemma 2.2. L et W K b e a WF diffusion with gener ator ( 2.4 ) . If θ h ≥ 1 for al l h = 1 , . . . , H , then, almost sur ely, W K ( t ) ∈ ∆ ◦ H for al l t > 0 . If, in addition, W K (0) ∈ ∆ ◦ H almost sur ely, then P ( W K ( t ) ∈ ∆ ◦ H , ∀ t ≥ 0) = 1 . Pr o of. When H = 2, it follows from Karlin and T aylor (1981) , Example 8, p. 239, that if θ 1 , θ 2 ≥ 1, b oth 0 and 1 are entrance b oundaries for W K ( t ) ∈ [0 , 1]. F or H > 2, the claim follo ws from reduction to the previous case. More sp ecifically , for every h = 1 , . . . , H , the single comp onent W K h is a WF diffusion on [0 , 1] with mutation parameters θ h and θ ( − h ) := P k  = h θ k . Cf. Dawson (2010) , Section 6.4. Hence, under the assumption on the parameters, θ h , θ ( − h ) ≥ 1 and thus 0 and 1 are en trance boundaries for W K h as w ell. It follows that for no h = 1 , . . . , H , W K h ev er touc hes 0 or 1, hence W K ( t ) ∈ ∆ ◦ H for all t > 0 almost surely . The second claim is no w ob vious. Multiple Poisson–Dirichlet diffusions 11 W e assume henceforth that θ h ≥ 1 , h = 1 , . . . , H , so, b y also assuming an initial condition W K (0) ∈ ∆ ◦ H almost surely , W K is a diffusion with state space ∆ ◦ H . Then W K has generator ( 2 . 4 ) on the domain { f : f = ˜ f   ∆ ◦ H , ˜ f ∈ C 2 (∆ H ) } . Since the set of polynomials of H v ariables is a core for the operator ( 2.4 ), W K is also the unique solution of the martingale problem for A 0 with domain D ( A 0 ) := { w p 1 1 w p 2 2 · · · w p H H , p 1 , . . . , p H ∈ Z , p 1 , . . . , p H ≥ 0 } . (2.6) Let now X K h , h = 1 , . . . , H , b e m utually independent WF diffusions, independent of W K , eac h taking v alues in ∆ K . X K h has generator A K h f ( x ) := 1 2 K X i,j =1 x i ( δ i,j − x j ) ∂ 2 f ∂ x i ∂ x j + θ h 2 K X i =1  1 K − x i  ∂ f ∂ x i , D ( A K h ) := C 2 (∆ K ) . (2.7) Here m utation is symmetric (for simplicity of notation w e are allowing self-m utations) and the rev ersible measure is a Dir K ( θ h /K, . . . , θ h /K ) distribution. F or ∆ ◦ H as in ( 2.5 ), set no w E K := ∆ ◦ H × ∆ H K , (2.8) where ∆ H K denotes the H -fold cartesian pro duct of ∆ K . F or w ∈ ∆ ◦ H , define β 0 ( w ) := 1 , β h ( w ) := 1 w h , h = 1 , . . . , H , (2.9) and define the random time-c hanged pro cesses X K h ( t ) := X K h  Z t 0 β h ( W K ( s )) ds  , h = 1 , . . . , H . (2.10) Consider the op erator A K of the form A K f ( w , x ) := A 0 f ( w , x ) + H X h =1 β h ( w ) A K h f ( w , x ) , (2.11) where w = ( w 1 , . . . , w H ) ∈ ∆ H , x = ( x 1 , . . . , x H ), x h ∈ ∆ K , h = 1 , . . . , H . Here, with a sligh t abuse of notation, we still denote by A 0 the op erator that acts on f as a function of w lik e A 0 in ( 2.4 ), and, for each h = 1 , . . . , H , by A K h the op erator that acts on f as a function of x h lik e A K h in ( 2.7 ). 12 Cost antini and R uggiero W e will consider tw o domains for A K . The first one is defined as D 0 ( A K ) :=  f = H Y h =0 f h : f 0 ∈ D ( A 0 ) , lim w → w 0 f 0 ( w ) = 0 , ∀ w 0 ∈ ∆ H \ ∆ ◦ H , f h ∈ D ( A K h ) , h = 1 , . . . , H ,  , (2.12) for D ( A 0 ) as in ( 2.6 ) and D ( A K h ) as in ( 2.7 ). E.g., for H = 2 w e require that lim w → 0 f 0 ( w ) = lim w → 1 f 0 ( w ) = 0, where w is the mass of one of t w o subgroups. The second domain is defined as D ( A K ) :=  f : f = ˜ f   E K , ˜ f ∈ C 2  E K  , sup ( w,x ) ∈ E K β h ( w h ) | A K h f ( w , x ) | < ∞ , h = 1 , . . . , H  , (2.13) with E K as in ( 2.8 ). Note that D 0 ( A K ) ⊆ D ( A K ) . The follo wing result sho ws that ( W K , X K 1 , . . . , X K H ), with X K h as in ( 2.10 ), solves the martingale problem for A K . Theorem 2.3. L et W K b e a WF diffusion with gener ator given by ( 2 . 4 ) and ( 2 . 6 ) . F or h = 1 , . . . , H and X K h WF diffusions with gener ator ( 2.7 ) , let X K h b e as in ( 2.10 ) . Then ( W K , X K 1 , . . . , X K H ) is a solution of the martingale pr oblem for ( A K , D ( A K )) as in ( 2.11 ) - ( 2.13 ) , and henc e of the martingale pr oblem for ( A K , D 0 ( A K )) with D 0 ( A K ) as in ( 2.12 ) . Pr o of. Let X K h , h > H , b e indep enden t WF diffusions with generator ( 2 . 7 ), indep endent of ( W K , X K 1 , . . . , X K H ). Giv en 0 < δ ≤ 1, define the functions β δ 0 ( w ) := 1 , β δ h ( w h ) := 1 w h ∨ δ , h = 1 , . . . , H , β δ h ( w h ) := 0 , h > H , and let X K,δ 0 ( t ) := W K ( t ) , X K,δ h ( t ) := X K h  Z t 0 β δ h ( X K,δ 0 )( s )) ds  , h ≥ 1 . Define also A K,δ f ( w , x ) := A 0 f ( w , x ) + H X h =1 β δ h ( w h ) A K h f ( w , x ) , with domain D ( A K,δ ) :=  f : f = ˜ f   E K , ˜ f ∈ C 2  E K   . The v ector X K,δ := ( X K,δ 0 , X K,δ 1 , . . . , X K,δ H , X K,δ H +1 , . . . ) is a solution of the system (6.2.1) in Ethier and Kurtz (1986) , with the corresp onding notation Y 0 := W K and Y h := X K h for h ≥ 1 and Multiple Poisson–Dirichlet diffusions 13 where w e hav e applied the trivial time-change determined b y β δ 0 to W K (the subsequen t citations in this proof will b e from the same source). It is immediate that X K,δ is the path wise unique solution. Therefore, by Theorem 6.2.2 (cf. also eqn. (6.2.7) and following definitions), w e hav e that τ δ ( t ) := ( τ δ 0 ( t ) , τ δ 1 ( t ) , τ δ 2 ( t ) , . . . ), defined b y τ δ h ( t ) := Z t 0 β δ h ( X K,δ 0 ( s )) ds, is an  F u  u ∈ [0 , ∞ ) ∞ -stopping time for every t ≥ 0. This trivially implies that X K,δ is a nonan tici- pating solution (cf. (6.2.15) and the follo wing discussion). Then, by Theorem 6.2.8, H Y j =0 f j ( X K,δ j ( s )) − Z t 0 H X h =0 β δ h ( X K,δ 0 ( s )) Y j ∈{ 0 ,...,H } ,j  = h f j ( X K,δ j ( s )) A K h f h ( X K,δ h ( s )) ds is a martingale for ev ery f 0 ∈ D ( A 0 ) and f h ∈ D ( A K h ), h = 1 , . . . , H . Therefore f  X K,δ 0 ( t ) , . . . , X K,δ H ( t )  − Z t 0 A K,δ f  X K,δ 0 ( s ) , . . . , X K,δ H ( s )  ds is a martingale for ev ery f that is a linear com bination of pro ducts as ab ov e. Since polynomials are dense in C 2  E K  (in the norm ∥ f ∥ := P | λ |≤ 2 sup E K | D λ f ( w , x 1 , x 2 , . . . ., x H ) | , for λ a multi-index), it is also a martingale for ev ery f ∈ D ( A K,δ ). Let now { δ n } n ≥ 1 ⊂ R + b e a sequence decreasing to 0. As n → ∞ , for eac h h = 1 , . . . , H the sequence β δ n h ( W K ( · )) conv erges uniformly ov er compact time interv als to β h ( W K ( · )), almost surely . Then also ( W K , X K,δ n 1 , . . . , X K,δ n H ) con verges uniformly o ver compact time in terv als to ( W K , X K 1 , . . . , X K H ), almost surely , and for every f ∈ D ( A K ), we hav e, as n → ∞ , the con vergence A K,δ n f ( W K ( · ) , X K,δ n 1 ( · ) , . . . , X K,δ n H ( · )) → A K f ( W K ( · ) , X K 1 ( · ) , . . . , X K H ( · )) , in the b ounded and p oin twise sense, almost surely . Therefore, the martingale f ( W K ( t ) , X K,δ n 1 ( t ) , . . . , X K,δ n H ( t )) − Z t 0 A K,δ n f ( W K ( s ) , X K,δ n 1 ( s ) , . . . , X K,δ n H ( s )) ds con verges boundedly and p oint wise, almost surely , to f ( W K ( t ) , X K 1 ( t ) , . . . , X K H ( t )) − Z t 0 A K f ( W K ( s ) , X K 1 ( s ) , . . . , X K H ( s )) ds and the limit is a martingale. The following result adds uniqueness for the solution in Theorem 2.3 . Denote by P ( G ) the space of probabilit y distributions on a generic P olish space G . W e will also use the notation for pro duct measures × m n =1 µ n := µ 1 × · · · × µ m . 14 Cost antini and R uggiero Theorem 2.4. F or every µ ∈ P ( E K ) of the form µ = × H h =0 µ h , ther e is at most one solution of the martingale pr oblem for ( A K , D 0 ( A K )) with initial distribution µ , and henc e of the martingale pr oblem for ( A K , D ( A K )) with initial distribution µ . Pr o of. Let ( W K , X K 1 , . . . , X K H ) b e a solution of the martingale problem for ( A K , D 0 ( A K )) with initial distribution µ , whose existence is guaran teed by Theorem 2.3 . F or h > H , µ h ∈ P (∆ K ), X K h ( t ) := X K h (0), t ≥ 0, where X K h (0) has distribution µ h and  X K h (0)  h>H are m utually indep en- den t and indep endent of ( W K , X K 1 , . . . , X K H ). Set also X K 0 := W K . Then X K := ( X K 0 , X K 1 , . . . , X K H , X K H +1 , . . . ) is a solution of the martingale problem for the op erator A K ∞ f ( w , x ) := A 0 f ( w , x ) + X h ≥ 1 β h ( w ) A K h f ( w , x ) , with A 0 , A K h as in ( 2.11 ) and β h as in ( 2 . 9 ) for h ≤ H , β h ≡ 0 for h > H , with domain D ( A K ∞ ) :=  f = f 0 Y h ∈ I f h : f 0 ∈ D ( A 0 ) , lim w → w 0 f 0 ( w ) = 0 , ∀ w 0 ∈ ∆ H \ ∆ ◦ H , f h ∈ D ( A K h ) , h ∈ I , I ⊆ { 1 , 2 , . . . } , I finite  . Define no w α ( w ) :=  H Y h =1 w h  − 1 , w ∈ ∆ ◦ H , and let η be the sto chastic pro cess defined pathwise by Z η ( t ) 0 α ( X K 0 ( s )) ds = t, t ≥ 0 . Then, almost surely , Z T 0 α ( X K 0 ( s )) ds < ∞ , ∀ T ≥ 0 , so that lim t →∞ η ( t ) = ∞ almost surely and the assumptions of Prop osition 6.2.10 of Ethier and Kurtz (1986) are satisfied. Let X K,α ( t ) := X K ( η ( t )) , t ≥ 0 . Then η ( t ) = Z t 0 1 α ( X K,α 0 ( s )) ds and f ( X K,α ( t )) − Z t 0 1 α ( X K,α 0 ( s )) A K ∞ f ( X K,α ( s )) ds (2.14) Multiple Poisson–Dirichlet diffusions 15 is a martingale for ev ery f ∈ D ( A K ∞ ). Let A K ∞ ,b f ( w , x ) := A 0 f ( w , x ) + X h ≥ 1 A K h f ( w , x ) , with domain D  A K ∞ ,b  :=  f = f 0 Y h ∈ I f h : f 0 ∈ D ( A 0 ) , f h ∈ D ( A K h ) , h ∈ I , I ⊆ { 1 , 2 , . . . } , I finite  . Ev ery f ∈ D ( A K ∞ ,b ) can b e approximated p oint wise and b oundedly by functions { f n } ⊆ D ( A K ∞ ) in such a w ay that { α − 1 A K ∞ f n } n ≥ 1 con verges p oin t wise and b oundedly to α − 1 A K ∞ ,b f . F or instance, for f = f 0 Q h ∈ I f h one can tak e f n := f n 0 Q h ∈ I f h , with f n 0 ( w ) := f 0 ( w ) χ ( nw 1 ) · · · χ ( nw H ) and where χ : R → [0 , 1] a smooth, nondecreasing function such that χ ( w h ) = 0 for w h ≤ 0, χ ( w h ) = 1 for w h ≥ 1. It follows that ( 2.14 ) is a martingale for every f ∈ D ( A K ∞ ,b ). Since α − 1 and α − 1 β h , for h ≥ 1, are b ounded functions, w e can apply Theorem 6.2.8 of Ethier and Kurtz (1986) (cfr. also page 314) to obtain that there is a v ersion ˜ X K,α of X K,α that is a nonan ticipating solution of ˜ X K,α 0 ( t ) = ˜ W K  Z t 0 1 α ( ˜ X K,α 0 ( s )) ds  , ˜ X K,α h ( t ) = ˜ X K h  Z t 0 β h ( ˜ X K,α 0 ( s )) α ( ˜ X K,α 0 ( s )) ds  , h ≥ 1 , where ˜ W K and  ˜ X K h  h ≥ 1 are indep endent pro cesses with generators A 0 and  A K h  h ≥ 1 , and initial distributions µ 0 and { µ h } h ≥ 1 . By Prop osition 6.2.10 of Ethier and Kurtz (1986) this yields that there is a v ersion ˜ X K of X K that is a nonan ticipating solution of ˜ X K 0 ( t ) = ˜ W K ( t ) , ˜ X K h ( t ) = ˜ X K h  Z t 0 β h ( ˜ X K 0 ( s )) ds  , h ≥ 1 . As this equation has a path wise unique solution, we can conclude that the solution of the martingale problem for the op erator A K with initial distribution × h ≥ 0 µ h is unique in distribution, whic h yields the assertion. Let ∆ H K b e defined as in ( 2.1 ) with H replaced by H K . Recall that w e view all v ectors as ro w v ectors. F or z ∈ ∆ H K , let z = ( z 1 , . . . , z H ) , z h = ( z h, 1 , . . . , z h,K ) , z h,i := z ( h − 1) K + i , (2.15) and ∆ H, ◦ H K :=  ( z 1 , . . . , z H ) ∈ ∆ H K : K X i =1 z h,i > 0 , h = 1 , . . . , H  , (2.16) 16 Cost antini and R uggiero and define the bijectiv e map S : E K → ∆ H, ◦ H K ( E K as in ( 2.8 )) as S ( w , x ) := w x = ( w 1 x 1 , . . . , w H x H ) , w h x h = ( w h x h, 1 , . . . , w h x h,K ) . (2.17) Then S − 1 is giv en by S − 1 ( z ) =  K X i =1 z i , . . . , K X i =1 z H,i  , z 1 P K i =1 z 1 ,i , . . . , z H P K i =1 z H,i ! . S and S − 1 are con tinuous. Let B K b e the generator of an H K -t yp e WF diffusion. In view of ( 2.15 ), w e can write B K as B K f ( z 1 , 1 , . . . , z H,K ) = 1 2 H X h,k =1 K X i,j =1 z h,i ( δ h,k δ i,j − z k,j ) ∂ 2 f ∂ z h,i ∂ z kj − 1 2 H X h =1 K X i =1  θ h K − | θ | z h,i  ∂ f ∂ z h,i , D ( B K ) = C 2 (∆ H K ) . (2.18) As is well known, the martingale problem for B K is well p osed (see, for instance, Theorem 8.2.8 in Ethier and Kurtz (1986) ). Next, w e formalize the connection b etw een ( 2.11 ) and ( 2.18 ). W e first sho w a preliminary result, used in the subsequent theorem. Lemma 2.5. L et A K b e as in ( 2.11 ) and B K as in ( 2.18 ) . Then, for f ∈ D ( B K ) and S as in ( 2.17 ) , f ◦ S ∈ D ( A K ) and A K ( f ◦ S ) = ( B K f ) ◦ S. Pr o of. The pro of consists in a long but straightforw ard computation, and is therefore omitted. W e are now ready to state the main result of this section. By leveraging on the previous Lemma, the following Theorem sho ws that a solution of the martingale problem for B K can b e represen ted as a multiparameter random time c hange of H indep endent K -type WF diffusions with generators ( 2.7 ), each w eighted b y the corresp onding comp onent of an H -type WF diffusion with generator ( 2 . 4 ), which also determines the random time rescaling. This amoun ts to a self-similarit y prop erty for WF diffusions, analogous to that of Proposition 2.1 , and pro vides a multiple skew-pr o duct r epr esentation of the WF diffusion Z K . Theorem 2.6. L et ( Z K 1 , . . . , Z K H ) b e the solution of the martingale pr oblem for B K as in ( 2 . 18 ) , with initial state ( Z K 1 (0) , . . . , Z K H (0)) ∈ ∆ H, ◦ H K . L et W K h (0) := K X i =1 Z K h,i (0) , W K (0) = ( W K 1 (0) , . . . , W K H (0)) , (2.19) Multiple Poisson–Dirichlet diffusions 17 and assume that W K (0) , Z K 1 (0) W K 1 (0) , . . . , Z K H (0) W K H (0) are m utually indep endent . (2.20) Then  Z K 1 ( · ) , . . . , Z K H ( · )  d =  W K 1 ( · ) X K 1 ( · ) , . . . , W K H ( · ) X K H ( · )  , (2.21) or, e quivalently,  K X i =1 Z K 1 ,i ( · ) , . . . , K X i =1 Z K h,i ( · )  , Z K 1 ( · ) P K i =1 Z K 1 ,i ( · ) , . . . , Z K H ( · ) P K i =1 Z K h,i ( · ) ! d =   W K 1 ( · ) , . . . , W K H ( · )  , X K 1 ( · ) , . . . , X K H ( · )  , wher e, for h = 1 , . . . , H , X K h ( t ) = X K h ( R t 0 β h ( W K ( s )) ds ) is as in ( 2.10 ) , and W K , X K 1 , . . . , X K H ar e mutual ly indep endent with gener ators ( 2.4 ) - ( 2.6 ) and ( 2.7 ) and initial c onditions W K (0) , X K 1 (0) := Z K 1 (0) /W K 1 (0) , . . . , X K H (0) := Z K H (0) /W K H (0) . Pr o of. By Lemma 2.5 , for f ∈ D ( B K ), we ha v e f ◦ S ∈ D ( A K ) and A K ( f ◦ S ) = B K f ◦ S . Therefore, with X K h defined b y ( 2 . 10 ) for each h = 1 , . . . , H , f  W K 1 ( t ) X K 1 ( t ) , . . . , W K H ( t ) X K H ( t )  − Z t 0 B K f  W K 1 ( s ) X K 1 ( s ) , . . . , W K H ( s ) X K H ( s )  ds is a martingale, hence the process  W K 1 ( · ) X K 1 ( · ) , . . . , W K H ( · ) X K H ( · )  is a solution of the martingale problem for B K with initial condition ( Z K 1 (0) , . . . , Z K H (0)). By virtue of the uniqueness of the martingale problem for B K , the latter equals ( Z K 1 , . . . , Z K H ) in distribution. The second assertion follo ws immediately from the fact that S is bijectiv e and S − 1 is con tinuous. In the next section w e will show that suc h a representation carries o ver to the infinite-dimensional setting, when the n umber of types for each mark is let go to infinity . 3 Multiple P oisson–Diric hlet diffusions In this section we consider the problem of letting K in ( 2.21 ) go to infinity , after ranking decreasingly the comp onents in eac h random time-changed WF diffusion X K h ( · ) as in ( 2.3 ). F or ∆ ◦ H as in ( 2.5 ), set E := ∆ ◦ H × ∇ H , E ◦ := ∆ ◦ H × ∇ H , (3.1) 18 Cost antini and R uggiero where ∇ H and ∇ H are the H -fold cartesian pro ducts of ∇ as in ( 1.1 ) and of ∇ as in ( 1.2 ), resp ectiv ely . With a slight abuse of notation, we will still denote by S the con tinuous map from E to K H , where K H is the generalized Kingman simplex in ( 1.6 ), defined as S ( w , x ) := ( w 1 x 1 , . . . , w H x H ) , w h x h = ( w h x h, 1 , w h x h, 2 , . . . ) , x h ∈ ∇ . (3.2) Let K ◦ H :=  z ∈ K H : H X h =1 ∞ X i =1 z h,i = 1 , 0 < ∞ X i =1 z h,i , h = 1 , . . . , H  . (3.3) Then S − 1  K ◦ H  = E ◦ , (3.4) and S is bijective from E ◦ (cf. ( 3.1 )) to K ◦ H . Lemma 3.1. The maps z → P ∞ i =1 z h,i , h = 1 , . . . , H , ar e c ontinuous on K ◦ H . The map S − 1 : K ◦ H → E ◦ is c ontinuous. Pr o of. The map z → P ∞ i =1 z h,i is not contin uous on K H , but the follo wing argumen t shows that it is contin uous on K ◦ H . Let { z n } b e a sequence of p oints in K ◦ H con verging to a p oint z ∈ K ◦ H , and set b h := ∞ X i =1 z h,i , b n h := ∞ X i =1 z n h,i . W e hav e b h ≤ lim inf n →∞ b n h , h = 1 , . . . , H , 1 = H X h =1 b h = H X h =1 b n h . Then b H = 1 − H − 1 X h =1 b h ≥ 1 − lim inf n →∞ H X h =1 b n h ≥ lim sup n →∞ b n H , so that b H = lim n →∞ b n H . No w let us show that, for every h , 2 ≤ h ≤ H , b l = lim n →∞ b n l ∀ l : h ≤ l ≤ H = ⇒ b l = lim n →∞ b n l ∀ l : h − 1 ≤ l ≤ H . In fact the ab o ve yields h − 1 X l =1 b l = 1 − H X l = h b l = lim n →∞  1 − H X l = h b n l  = lim n →∞ h − 1 X l =1 b n l , Multiple Poisson–Dirichlet diffusions 19 and hence b h − 1 = h − 1 X l =1 b l − h − 2 X l =1 b l ≥ lim n →∞ h − 1 X l =1 b n l − lim inf n →∞ h − 2 X l =1 b n l ≥ lim sup n →∞ b n h − 1 . The second assertion is no w obvious. Let ∆ c K b e the corner of the h yp ercub e [0 , 1] K defined as ∆ c K :=    z ∈ [0 , 1] K : K X j =1 z j ≤ 1    , and consider the con tinuous map ρ K : ∆ c K → ∇ , defined as ρ K ( x ) := ( x (1) , x (2) , . . . x ( K ) , 0 , 0 , . . . ) (3.5) where x (1) ≥ x (2) ≥ , . . . ≥ x ( K ) are the descending order statistics of the co ordinates of x h ∈ ∆ c K . Defining, for z ∈ ∆ H, ◦ H K , ρ K ( H ) ( z ) := ( ρ K ( z 1 ) , ρ K ( z 2 ) , ...ρ K ( z H )) ∈ K ◦ H , (3.6) and, for ( w , x ) ∈ E K , ρ K ( H ) ( w , x ) := ( w , ρ K ( x 1 ) , . . . , ρ K ( x H )) ∈ E ◦ , (3.7) w e can write, recalling that S denotes both the map in finite dimension ( 2.17 ) and the analogous map in infinite dimension ( 3.2 ), for ( w, x ) ∈ E K , S  ρ K ( H ) ( w , x )  = ρ K ( H )  S ( w , x )  , (3.8) and, for z ∈ ∆ H, ◦ H K , ρ K ( H ) ( S − 1 ( z )) = S − 1 ( ρ K ( H ) ( z )) . (3.9) Let now Z K := ( Z K 1 , . . . , Z K H ) b e the solution (unique in distribution) of the martingale problem for B K as in ( 2.18 ), with initial condition satisfying ( 2.19 )-( 2.20 ). By Theorem 2.6 , w e hav e Z K d = S  W K , X K  , X K = ( X K 1 , . . . , X K H ) with X K h as in ( 2.10 ). Then, b y ( 3 . 8 ), ρ K ( H ) ( Z K ) d = S  ρ K ( H ) ( W K , X K )  . (3.10) Ethier and Kurtz (1981) show ed that ρ K ( X K h ) conv erges in distribution, as K → ∞ , to a Poisson– Diric hlet diffusion with generator ( 1.3 ) with θ replaced by θ h and domain the algebra generated by the functions φ 1 ( x ) ≡ 1 , φ m ( x ) := ∞ X i =1 x m i , m ≥ 2 , (3.11) pro vided the initial distributions con verge. The follo wing theorem sho ws that this allo ws to identify the limit in distribution for ( 3.10 ). 20 Cost antini and R uggiero Theorem 3.2. L et Z K b e the (unique in distribution) solution of the martingale pr oblem for B K as in ( 2.18 ) and let W K (0) b e as in ( 2.19 ) . Assume W K (0) and Z K (0) satisfy ( 2.20 ) and that, as K → ∞ ,  W K (0) , ρ K ( H )  Z K (0)  d − →  W (0) , Z (0)  , (3.12) with W (0) ∈ ∆ ◦ H almost sur ely, ∆ ◦ H as in ( 2.5 ) . L et W := ( W 1 , . . . , W H ) b e the WF diffusion with gener ator ( 2.4 ) and initial c ondition W (0) , β h b e as in ( 2.9 ) for h = 1 , . . . , H , and let X 1 , . . . , X H b e indep endent Poisson–Dirichlet diffusions with gener ators ( 1.3 ) , indep endent of W , e ach with p ar ameter θ h and with initial c onditions Z h (0) /W h (0) (note that ( 2.19 ) - ( 2.20 ) and ( 3.12 ) imply that W (0) , Z 1 (0) /W 1 (0) , . . . , Z H (0) /W H (0) ar e indep endent). Then ρ K ( H )  Z K  d − → Z wher e Z h ( · ) := W h ( · ) X h ( · ) , X h ( t ) := X h  R t 0 β h ( W ( s )) ds  , h = 1 , . . . , H , with β h as in ( 2.9 ) , and Z ( t ) ∈ K ◦ H , with K ◦ H as in ( 3.3 ) , for al l t > 0 , almost sur ely. In p articular, if Z (0) ∈ K ◦ H almost sur ely, then Z ( t ) ∈ K ◦ H for al l t ≥ 0 , almost sur ely. In addition W (0) 1 Z (0) ∈ K ◦ H =  X i ≥ 1 Z 1 i (0) , . . . , X i ≥ 1 Z h,i (0)  1 Z (0) ∈ K ◦ H , or, e quivalently,  W (0) , X 1 (0) , . . . , X H (0)  1 Z (0) ∈ K ◦ H = S − 1  Z (0)  1 Z (0) ∈ K ◦ H . (3.13) Pr o of. Let W K , X K 1 , . . . , X K H b e as in Theorem 2.6 . By virtue of the Skorohod Representa- tion Theorem, we can supp ose that, almost surely ,  W K , ρ K ( X K 1 ) , . . . , ρ K ( X K H )  con verges to ( W , X 1 , . . . , X H ) uniformly ov er compact time in terv als. Since, almost surely , W ( t ) ∈ ∆ ◦ H for all t ≥ 0, it follows that ρ K  X K h ( R t 0 β h ( W K ( s )) ds )  con verges almost surely to X h ( R t 0 β h ( W ( s )) ds ) uniformly ov er compact time in terv als, for h = 1 , . . . , H . The first assertion then follows from the fact that the map  w ( · ) , x ( · )  ∈ C E ◦ [0 , ∞ ) → S ( w ( · ) , x ( · )) ∈ C ∇ ◦ ( H ) [0 , ∞ ) is contin uous with resp ect to uniform conv ergence on compact time interv als. The second assertion follo ws from Lemma 3.1 . Let now A h := A θ h , where A θ h is as in ( 1.3 ) with θ replaced b y θ h . Note that the algebra generated b y the functions ( 3.11 ) is also the linear space generated b y the functions Φ m h ( z h ) := Y q ≥ 1 φ m h q ( z h ) , m h = ( m h 1 , m h 2 , m h 3 , . . . ) ∈ M , M :=  m ∈ N ∞ : m 1 ≥ m 2 ≥ · · · , m l = 1 for l ≥ r + 1 for some r ∈ N  , (3.14) Multiple Poisson–Dirichlet diffusions 21 (recall that φ 1 ≡ 1). Thus, for ev ery initial distribution on ∇ , the Mark ov pro cess with generator A h with domain the algebra generated by the functions ( 3.11 ) is also the unique solution of the martingale problem for A h with domain D ( A h ) :=  Φ m h , m h ∈ M  , (3.15) where Φ m and M are as in ( 3.14 ). Therefore in the sequel we will supp ose the domain of A h to b e giv en by ( 3.15 ). Define the op erator A : D ( A ) ⊆ C b ( E ) → C b ( E ) as A f ( w , x ) := A 0 f ( w , x ) + H X h =1 β h ( w ) A h f ( w , x ) , (3.16) where, with a slight abuse of notation, we still denote by A 0 the op erator that acts on f as a function of w lik e A 0 in ( 2.4 ) and, for eac h h = 1 , . . . , H , b y A h the operator that acts on f as a function of x h lik e A h . Let also D ( A ) :=  f = H Y h =0 f h , f 0 ∈ D ( A 0 ) , lim w → w 0 f 0 ( w ) = 0 , ∀ w 0 ∈ ∆ H \ ∆ ◦ H , f h ∈ D ( A h ) , h = 1 , . . . , H  , (3.17) with A 0 as in ( 2.4 )-( 2.6 ). Theorem 3.3. L et W, X 1 , . . . , X H b e as in The or em 3.2 . Then ( W, X 1 , . . . , X H ) is the unique (in distribution) solution of the martingale pr oblem for A as in ( 3.17 ) - ( 3.16 ) with initial distribution given by the law of  W (0) , X 1 (0) , . . . , X H (0)  . Pr o of. The assertion follo ws b y the same arguments used in the pro ofs of Theorems 2.3 and 2.4 . Remark 3.4. Both f and A f can b e extended contin uously to functions in C ( E ), where E is the closure of E , that is E = ∆ H × ∇ H . □ The follo wing result identifies the generator of the pro cess Z in Theorem 3.2 as a comp osition of the op erator ( 3.16 )-( 3.17 ) with the map S − 1 and sho ws that it is the limit of the op erators B K in ( 2.18 ), after the appropriate reordering of the test function arguments. Theorem 3.5. L et Z b e the limiting pr o c ess in The or em 3.2 , with initial c ondition Z (0) ∈ K ◦ H almost sur ely, K ◦ H as in ( 3.3 ) . Then Z is a solution of the martingale pr oblem for the op er ator B : D ( B ) ⊆ C b ( K ◦ H ) → C b ( K ◦ H ) define d as B f ( z ) := A g ( S − 1 ( z )) , z ∈ K ◦ H , D ( B ) :=  f ∈ C b ( K ◦ H ) : f ◦ S = g   E ◦ , g ∈ D ( A )  , (3.18) 22 Cost antini and R uggiero wher e A is as in ( 3.16 ) - ( 3.17 ) and the definition of B is wel l p ose d b e c ause the density of E ◦ in E implies that, for every f ∈ C b ( K ◦ H ) , ther e is at most one g ∈ D ( A ) such that f ◦ S = g   E ◦ . F or every µ ∈ P ( E ◦ ) of the form µ = × H h =0 µ h , the solution Z of the martingale pr oblem for B , with initial distribution µ ◦ S − 1 , is unique in distribution. Mor e over, for every f ∈ D ( B ) , we have f ◦ ρ K ( H ) ∈ D ( B K ) , with B K as in ( 2.18 ) , and sup z ∈ ∆ H, ◦ H K    B K ( f ◦ ρ K ( H ) )( z ) − B f ( ρ K ( H ) ( z ))    → 0 , (3.19) as K → ∞ . Pr o of. F or the first assertion, the pro of is analogous to that of Theorem 2.6 . The second assertion follo ws b y verifying that S − 1 ( Z ( · )) is a solution of the martingale problem for A defined b y ( 3.17 )- ( 3.16 ) and b y Theorem 3.3 . Let us now turn to ( 3.19 ). By a slight abuse of notation, in the sequel we denote b y f ◦ S the function in D ( A ) that is the unique con tinuous extension of f ◦ S to E . F rom ( 3.18 ), we hav e B f ( z ) = A ( f ◦ S )( S − 1 ( z )) , ∀ z ∈ K ◦ H . On the other hand, for f ∈ D ( B ), f ◦ ρ K ( H ) ∈ D ( B K ), and, b y Lemma 2.5 , B K ( f ◦ ρ K ( H ) )( S ( w , x )) = A K ( f ◦ ρ K ( H ) ◦ S )( w , x ) , ∀ ( w , x ) ∈ E K , or equiv alently , taking in to account that S (as in ( 2.17 )) is bijectiv e b et ween E K and ∆ H, ◦ H K , B K ( f ◦ ρ K ( H ) )( z ) = A K ( f ◦ ρ K ( H ) ◦ S )( S − 1 ( z )) , ∀ z ∈ ∆ H, ◦ H K . and hence, b y ( 3.8 ), B K ( f ◦ ρ K ( H ) )( z ) = A K ( f ◦ S ◦ ρ K ( H ) )( S − 1 ( z )) . Then, b y ( 3.9 ), for all z ∈ ∆ H, ◦ H K ,    B K ( f ◦ ρ K ( H ) )( z ) − B f ( ρ K ( H ) ( z ))    =    A K ( f ◦ S ◦ ρ K ( H ) )( S − 1 ( z )) − A ( f ◦ S )( S − 1 ( ρ K ( H ) ( z )))    =    A K (( f ◦ S ) ◦ ρ K ( H ) )( S − 1 ( z )) − A ( f ◦ S )( ρ K ( H ) ( S − 1 ( z )))    from whic h sup z ∈ ∆ H, ◦ H K    B K ( f ◦ ρ K ( H ) )( z ) − B f ( ρ K ( H ) ( z ))    = sup z ∈ ∆ H, ◦ H K    A K (( f ◦ S ) ◦ ρ K ( H ) )( S − 1 ( z )) − A ( f ◦ S )( ρ K ( H ) ( S − 1 ( z )))    = sup ( w,x ) ∈ E K    A K (( f ◦ S ) ◦ ρ K ( H ) )( w , x ) − A ( f ◦ S )( ρ K ( H ) ( w , x ))    Multiple Poisson–Dirichlet diffusions 23 whic h go es to zero as K → ∞ by Theorem 2.5 in Ethier and Kurtz (1981) , taking in to account that, for f 0 ∈ D ( A 0 ), sup ∆ ◦ H | f 0 ( w ) β h ( w ) | ≤ 1 for all h = 1 , . . . , H . The next theorem, which concludes this section, provides an explicit description of B and D ( B ) in Theorem 3.5 . Theorem 3.6. L et B and D ( B ) b e as in The or em 3.5 . Then D ( B ) =  H Y h =1 | z h | m h 0 Φ m h ( z h ) , m h ∈ M , m h 0 ≥ 1 − X q ≥ 1 m h q 1 m h q ≥ 2 , h = 1 , . . . , H  , (3.20) wher e | z h | := P i ≥ 1 z h,i , and Φ m , M ar e given by ( 3.14 ) , and B f ( z ) = 1 2 H X h,k =1 ∞ X i,j =1 z h,i ( δ h,k δ i,j − z k,j ) ∂ 2 f ∂ z h,i ∂ z kj − H X h =1 | θ | 2 ∞ X i =1 z h,i ∂ f ∂ z h,i + 1 2 H X h =1 θ h m h 0 | z h | − 1 f ( z ) . (3.21) Pr o of. Since S is bijectiv e b et ween E ◦ and K ◦ H , a function f belongs to D ( B ) in ( 3.18 ) if and only if f ( z ) = g ( S − 1 ( z )) , ∀ z ∈ K ◦ H , for some g ∈ D ( A ) . Since g belongs to D ( A ) if and only if g ( w , x ) = w p 1 1 · · · w p H H H Y h =1 Φ m h ( x h ) , where, for eac h h = 1 , . . . , H , p h ≥ 1, m h ∈ M , Φ m and M are giv en b y ( 3.14 ), f b elongs to D ( B ) if and only if it has the form f ( z ) = | z 1 | p 1 · · · | z H | p H H Y h =1 Φ m h  z h | z h |  = | z 1 | m 1 0 · · · | z H | m H 0 H Y h =1 Φ m h ( z h ) , with m h 0 = p h − P q ≥ 1 m h q 1 m h q ≥ 2 , h = 1 , . . . , H , which gives ( 3.20 ). The explicit form of B can be found b y taking adv antage of ( 3.19 ). Let B K b e as in ( 2.18 ), ρ K ( H ) as in ( 3.6 ), and define, for f ∈ D ( B ), ˜ B f as the righ t-hand side of ( 3.21 ). Using the fact that ∂ ∂ z h,i  | z h | m h 0 Φ m h ( z h )  = m h 0 ( | z h | ) m h 0 − 1 Φ m h ( z h ) + ( | z h | ) m h 0 X q ≥ 1 1 m h q ≥ 2 m h q z m h q − 1 hi Y l  = q φ m h l ( z h ) , 24 Cost antini and R uggiero and that ρ K ( H ) ( z ) i = 0 for i ≥ K + 1, we find, for z ∈ ∆ H, ◦ H K ,    B K ( f ◦ ρ K ( H ) )( z ) − ˜ B f ( ρ K ( H ) ( z ))    = H X h =1    K X i =1 θ h K ∂ f ∂ z h,i ( z ) − θ h m h 0 | z h | − 1 f ( z )    = H X h =1 θ h      K X i =1 1 K  m h 0 | z h | m h 0 − 1 Φ m h ( z h ) + | z h | m h 0 X q ≥ 1 1 m h q ≥ 2 m h q z m h q − 1 h,i Y l  = q φ m h l ( z h )  − m h 0 | z h | m h 0 − 1 Φ m h ( z h )     Y k  = h | z k | m k 0 Φ m k ( z k ) = 1 K H X h =1 θ h | z h | m h 0 X q ≥ 1 1 m h q ≥ 2 m h q K X i =1 z m h q − 1 h,i Y l  = q φ m h l ( z h ) Y k  = h | z k | m k 0 Φ m k ( z k ) . No w, for m ≥ 2, K X i =1 z m − 1 h,i ≤ | z h | m − 1 , φ m ( z h ) ≤ | z h | m , whic h implies, in particular, taking also into account that φ 1 ≡ 1 and the definition of D ( B ), | z k | m k 0 Φ m k ( z k ) ≤ | z k | m k 0 + P l ≥ 1 1 m k l ≥ 2 m k l ≤ | z k | ≤ 1 , ∀ k = 1 , . . . , H . Therefore, k eeping in mind the definition of D ( B ),    B K ( f ◦ ρ K ( H ) )( z ) − ˜ B f ( ρ K ( H ) ( z ))    ≤ 1 K H X h =1 θ h X q ≥ 1 1 m h q ≥ 2 m h q | z h | m h 0 + P l : m h l ≥ 2 m h l − 1 ≤ 1 K H X h =1 θ h X q ≥ 1 1 m h q ≥ 2 m h q , Since θ h and m h q , for all h, q , are fixed, and m h ∈ M has at most finitely-many co ordinates greater than 2, the previous implies sup z ∈ ∆ H, ◦ H K    B K ( f ◦ ρ K ( H ) )( z ) − ˜ B f ( ρ K ( H ) ( z ))    → 0 , as K → ∞ , (3.22) Moreo ver, for ev ery z ∈ K ◦ H , there exists a sequence { z K } suc h that z K ∈ ∆ H, ◦ H K for all K ≥ 1 and ρ K ( H ) ( z K ) → z . F or example, one can tak e z K = ( z K 1 , . . . , z K H ) as z K h, 1 := z h, 1 + X j ≥ K +1 z h,j , z K h,i := z h,i , 2 ≤ i ≤ K. Multiple Poisson–Dirichlet diffusions 25 Then, for ev ery z ∈ K ◦ H , w e can write    B f ( z ) − ˜ B f ( z )    ≤    B f ( z ) − B f ( ρ K ( H ) ( z K ))    +    B f ( ρ K ( H ) ( z K )) − ˜ B f ( ρ K ( H ) ( z K ))    +    ˜ B f ( ρ K ( H ) ( z K )) − ˜ B f ( z )    , (3.23) As K → ∞ , the first and the third terms on the right-hand side of ( 3.23 ) conv erge to zero b ecause B f and ˜ B f are contin uous on K ◦ H , while the second term go es to zero by ( 3.22 ) and ( 3.19 ). Hence B f ( z ) = ˜ B f ( z ) for all z ∈ K ◦ H . Remark 3.7. Note that ( 3.22 ) holds only for the functions of D ( ˜ B ) := D ( B ), and without Theorems 3.2 and 3.5 it w ould not be clear what D ( B ), should be. Even supp osing D ( ˜ B ) is known, ( 3.22 ) is not sufficien t to conclude that ρ K ( H )  Z K  con verges, as K → ∞ , to some Mark o v process Z , b ecause it is not clear a priori that ˜ B is the generator of a Mark ov pro cess. □ Remark 3.8. K ◦ H is dense in K H giv en b y ( 1.6 ), but B f cannot in general b e extended to a con tinuous function on K H , even when f can b e extended to a contin uous function on K H . In fact there are sequences { z n } ⊆ K ◦ H suc h that z n → z ∈ K H \ K ◦ H and  S − 1 ( z n )  has differen t limit p oin ts, hence S − 1 ( · ) in ( 3.18 ) is not con tinuous on K H . F or instance, let H = 2 and consider z n 1 i :=  1 2 i +2 + [1 + ( − 1) n ] 1 8 n , 1 ≤ i ≤ n, 1 2 i +2 , i ≥ n + 1 , z n 2 i :=  1 2 i +1 + [1 + ( − 1) n +1 ] 1 8 n , 1 ≤ i ≤ n, 1 2 i +1 , i ≥ n + 1 . Then  S − 1 ( z n )  has t wo differen t limit p oints, namely  1 4 , 3 4 , x 1 , x 2  , x 1 i = 1 2 i , x 2 i = 2 3 1 2 i , and  1 2 , 1 2 , ˜ x 1 , ˜ x 2  , ˜ x 1 i = 1 2 i +1 , ˜ x 2 i = 1 2 i . The function f ( z 1 , z 2 ) := ∞ X i =1 z 3 1 i ∞ X i =1 z 3 2 i , b elongs to D ( B ) with g ( w 1 , w 2 , x 1 , x 2 ) := w 3 1 w 3 2 P ∞ i =1 x 3 1 i P ∞ i =1 x 3 2 i . A g is con tinuous on K H , there- fore  B f ( z n )  =  A g ( S − 1 ( z n )  has limit points A g ( 1 4 , 3 4 , x 1 , x 2 ) and A g ( 1 2 , 1 2 , ˜ x 1 , ˜ x 2 ), which are differen t, for generic v alues of θ 1 and θ 2 . □ 4 Stationary distributions F or E K in ( 2.8 ), define the probabilit y measure µ ∗ K on E K as µ ∗ K := × H h =0 µ ∗ K h , µ ∗ K 0 = µ ∗ 0 := Dir H ( θ 1 , . . . , θ H )   ∆ ◦ H , µ ∗ K h := Dir K  θ h K , . . . , θ h K  . (4.1) The follo wing result shows that µ ∗ K is the stationary distribution of ( W, X K 1 , . . . , X K H ) in Theorem 2.3 . 26 Cost antini and R uggiero Theorem 4.1. L et A K b e as in ( 2.11 ) . with D 0 ( A K ) as in ( 2.12 ) . F or every f ∈ D 0 ( A K ) , we have Z E K A K f dµ ∗ K = 0 . (4.2) In p articular, the solution of the martingale pr oblem for A K , with initial distribution µ ∗ K as in ( 4.1 ) , is a stationary pr o c ess. Pr o of. F or A 0 as in ( 2.4 ) and A K h as in ( 2.7 ), the stationarity of a WF diffusion with resp ect to the Diric hlet distribution (cf. Lemma 4.1 in Ethier and Kurtz (1981) ) yields Z ∆ ◦ H A 0 f 0 dµ ∗ 0 = 0 , Z ∆ K A K h f h dµ ∗ K h = 0 , ∀ f 0 ∈ D ( A 0 ) , f h ∈ D ( A K h ) , h = 1 , . . . , H , whereb y the first assertion now follo ws through a simple computation by using ( 2.11 )-( 2.12 ) in the left-hand side of ( 4.2 ). F or the second claim, let ˆ C ( E K ) :=  f ∈ C  E K  : lim w → w 0 | sup x ∈ ∆ H K f ( w , x ) | = 0 , ∀ w 0 ∈ ∆ H \ ∆ ◦ H  . The linear span of D 0 ( A K ) is an algebra and is dense in ˆ C ( E K ). In addition, A K satisfies the p ositiv e maxim um principle because the martingale problem for A K with initial condition a Dirac probabilit y measure has a solution by Theorem 2.3 . (Indeed, let f ∈ D 0 ( A K ) and f ( w ∗ , x ∗ ) = sup ( w,x ) ∈ E K f ( w , x ) ≥ 0, ( w ∗ , x ∗ ) ∈ E K . Suppose, b y contradiction, that A K f ( w ∗ , x ∗ ) > 0 and let U ∗ b e an op en neighborho o d of ( w ∗ , x ∗ ), in the relative top ology on E K , such that A K f ( w , x ) > 0 for ( w , x ) ∈ U ∗ . Let ( W K ( · ) , X K 1 ( · ) , . . . , X K H ( · )) b e a solution of the martingale problem for A K with initial condition ( w ∗ , x ∗ ) and τ ∗ b e the first exit time of ( W K ( · ) , X K 1 ( · ) , . . . , X K H ( · )) from U ∗ . Then, for ev ery t > 0, 0 ≥ E  f ( W K ( t ∧ τ ∗ ) , X K 1 ( t ∧ τ ∗ ) , . . . , X K H ( t ∧ τ ∗ )  − f (( w ∗ , x ∗ ) = E  Z t ∧ τ ∗ 0 A K f ( W K ( s ) , X K 1 ( s ) , . . . , X K H ( s )) ds  > 0 , whic h, of course, is a contradiction.) Then the second assertion follows from Theorem 4.9.17 of Ethier and Kurtz (1986) together with Theorem 2.4 . Corollary 4.2. L et B K b e as in ( 2.18 ) , and define ν ∗ K := Dir H K  θ 1 K , . . . , θ 1 K , θ 2 K , . . . , θ 2 K , . . . , θ H K , . . . , θ H K      ∆ H, ◦ H K (4.3) wher e e ach θ h is r ep e ate d K times. The solution of the martingale pr oblem for B K with initial distribution ν ∗ K is stationary. Multiple Poisson–Dirichlet diffusions 27 Pr o of. By Lemma 2.1 , ν ∗ K = µ ∗ K ◦ S − 1 , where µ ∗ K is giv en by ( 4.1 ). Then, by Theorem 2.6 , the solution of the martingale problem for ( 2.18 ) with initial distribution ν ∗ K is S ( W, X K 1 , . . . , X K H ), where ( W, X K 1 , . . . , X K H ) is the solution of the martingale problem for A K with initial distribution µ ∗ K , and the assertion follo ws by Theorem 4.1 . No w, for E as in ( 3.1 ), define the distribution µ ∗ on E as µ ∗ := × H h =0 µ ∗ h , µ ∗ 0 := Dir H ( θ 1 , . . . , θ H )   ∆ ◦ H , µ ∗ h := PD( θ h ) , (4.4) where PD( θ h ) denotes the Kingman’s P oisson–Diric hlet distribution with parameter θ h . Note that µ ∗ ( E − E ◦ ) = 0 . (4.5) The next result sho ws that µ ∗ is the stationary distribution of A as in ( 3.16 ). Theorem 4.3. L et A b e as in ( 3.16 ) . F or every f ∈ D ( A ) , with D ( A ) as in ( 3.17 ) , and µ ∗ as in ( 4.4 ) , we have Z E A f dµ ∗ = 0 . (4.6) In p articular, the solution of the martingale pr oblem for A , define d by ( 3.16 ) , with initial distribution µ ∗ is a stationary pr o c ess. Pr o of. The fact that R ∆ ◦ H A 0 f 0 dµ ∗ 0 = 0 follo ws from Lemma 4.1 in Ethier and Kurtz (1981) . F ur- thermore, for A h as in ( 1.3 ), it follows from Theorems 4.3 and 4.5 of Ethier and Kurtz (1981) that Z ∇ A h f h dµ ∗ h = 0 , from which the first assertion now follows through a simple computation by using ( 3.16 )-( 3.17 ) in the left-hand side of ( 4.6 ). Let ˆ C ( E ) :=  f ∈ C  E  : lim w → w 0 sup x ∈∇ H | f ( w , x ) | = 0 , ∀ w 0 ∈ ∆ H \ ∆ ◦ H  , with ∇ H as in ( 3.1 ). The linear span of D ( A ) is an algebra and is dense in ˆ C ( E ). In addition, by an argument analogous to that used in the pro of of Theorem 4.1 , A satisfies the p ositive maxim um principle. Then the second assertion follows from Theorem 4.9.17 of Ethier and Kurtz (1986) together with Theorem 3.3 . The following Corollary of Theorem 4.3 identifies the stationary distribution of Z in Theorem 3.2 . Let ν ∗ := µ ∗ ◦ S − 1 , 28 Cost antini and R uggiero where µ ∗ is giv en by ( 4.4 ). Then ν ∗ is the m ultiple P oisson–Dirichlet distribution of Strahov (2024a;b) , cf. Definition 1.1 . ( 3.4 ) and ( 4.5 ) imply ν ∗ ( K H − K ◦ H ) = 0 . (4.7) Corollary 4.4. L et Z b e as in The or em 3.2 with initial distribution ν ∗ .Then Z is a stationary pr o c ess. Pr o of. Let ( W, X ) := ( W, X 1 , . . . , X H ) b e as in Theorem 3.2 , hence, in particular, S ( W , X )(0) = Z (0). ( 3.4 ), ( 3.13 ), the fact that S is bijective b etw een E ◦ and K ◦ H , ( 4.5 ) and ( 4.7 ) yield, for every Borel subset B of E , P  ( W , X )(0) ∈ B  = P  ( W , X )(0) ∈ B  , Z (0) ∈ K ◦ H ) = P  ( W , X )(0) ∈ B ∩ E ◦ , Z (0) ∈ K ◦ H  = P  S − 1 ( Z (0)) ∈ B ∩ E ◦ , Z (0) ∈ K ◦ H  = P  Z (0) ∈ S ( B ∩ E ◦ ) , Z (0) ∈ K ◦ H  = P  Z (0) ∈ S ( B ∩ E ◦ )  = ν ∗  S (( B ∩ E ◦ ))  = µ ∗  S − 1 ( S ( B ∩ E ◦ ))  = µ ∗  B ∩ E ◦  = µ ∗ ( B ) , that is the initial distribution of ( W, X 1 , . . . , X H ) is µ ∗ . The assertion then follows from Theorem 4.3 . In the previous result, ν ∗ is the law induced on K H b y µ ∗ on E through the map S as in ( 3.2 ), and therefore coincides with the multiple P oisson–Dirichlet distribution of Definition 1.1 . W e hav e th us constructed a diffusion pro cess with v alues in the generalized Kingman simplex K H as in ( 1.6 ) with stationary distribution the m ultiple P oisson–Diric hlet distribution PD( | θ | ), with | θ | = ( θ 1 , . . . , θ H ). W e conclude with a construction of the m ultiple Poisson–Diric hlet distribution PD( | θ | ) in the spirit of Kingman’s construction of the PD( θ ) distribution through limits of ranked Dirichlet dis- tributed random frequencies. Recall that ρ K is defined by ( 3.5 ) and ρ K H is defined by ( 3.6 ) and ( 3.7 ). Corollary 4.5. L et ν ∗ K b e as in ( 4.3 ) and ν ∗ b e as in Cor ol lary 4.4 . Then ν ∗ K ◦  ρ K H  − 1 w − → ν ∗ . Pr o of. Let µ ∗ K b e giv en by ( 4.1 ). By Lemma 2.1 and ( 3.8 ), w e hav e ν ∗ K ◦  ρ K H  − 1 = µ ∗ K ◦ S − 1 ◦  ρ K H  − 1 = µ ∗ K ◦  ρ K H  − 1 ◦ S − 1 , where in the first equalit y S : E K → ∆ H ◦ H K , with E K and ∆ H ◦ H K giv en b y ( 2.8 )) and ( 2.16 ) respec- tiv ely , and in the second equalit y S : E → K H , with E and K H giv en b y ( 3.1 ) and ( 1.6 ) resp ectively . F rom Kingman (1975) (cf. also Corollary 4.7 in Ethier and Kurtz (1981) for a more general result) w e hav e that µ ∗ K h ◦ ( ρ K ) − 1 w − → µ ∗ h , where µ ∗ h is a PD( θ h ) distribution as in ( 4.4 ). Therefore µ ∗ K ◦  ρ K ( H )  − 1 w − → µ ∗ . Since ν ∗ = µ ∗ ◦ S − 1 and S : E → K H is con tinuous, the assertion follows. Multiple Poisson–Dirichlet diffusions 29 Ac kno wledgemen ts MR was supp orted by the Europ ean Union – NextGenerationEU, PRIN 2022 PNRR, Grant no. P2022H5WZ9. References Ascolani, F., Lijoi, A. and Ruggier o, M. (2021). Predictive inference with Fleming–Viot- driv en dep endent Diric hlet pro cesses. 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