A stochastic diffusion process for Lochners generalized Dirichlet distribution

A sto chastic diffusion p ro cess fo r Lo chner’s generalized Dirichlet distribution LA-UR 13-2 1573 Accept ed for publ ication in Journ al of Mat hematical Phys ics, Se ptember 12, 2 013 J. Bak osi, J.R. Ris to rcelli { jbakosi,jr rj } @lanl.gov Los Alamos National Lab o ratory , Los Alamos, NM 87545, U SA Abstrac t The metho d of p o ten tial so lutions of F okker-Planck equations is used to develop a transp ort equation for the joint probability of N sto c hastic v ar iables with Lo c hner’s generalized Dirichlet distribution [ 1 ] as its asymptotic so lution. Individual samples of a discre te ensemble, obtained from the sy stem of sto ch a stic differential equations, equiv a le n t to the F okk er -Planck equation developed here, sa tisfy a unit-s um constraint at all times and ensure a b ounded sample space, similarly to the pro cess develop ed in [ 2 ] for the Dirichlet distribution. Consequently , t he generalized Dirichlet diffusio n proc e ss may b e use d to represent realiz ations of a fluctuating ensemble of N v a riables sub ject to a conser v ation principle. Compared to the Dir ichlet distribution and pr o cess, the additional para meters o f the generalized Diric hlet distribution allow a more general class of ph ys ical pro cesses to b e mo deled with a more gener al cov a riance matr ix . Keyw ords: F okker-Planck eq ua tion; Stochastic diffusion; Generalized Dirichlet distribution 1 Intro duction W e dev elop a F okke r -Planck equation whose statistica lly sta tionary solution (i. e. in v arian t) is Lo c h ner’s generaliz ed Dirichlet distribution [ 1 , 3 , 4 ]. The (standard ) Dirichlet distrib u tion [ 5 , 6 , 7 ] has b een used to represent a set of non-negativ e fluctuating v ariables sub ject to a unit-sum requiremen t in a v ariet y of fields, including ev olutionary theory [ 8 ], Ba yesia n statistics [ 9 ], geology [ 10 , 11 ], forensics [ 12 ], econometrics [ 13 ], turbu len t com- bustion [ 14 ], and p opulation biology [ 15 ]. F oll owing the metho d of p oten tial solutions, applied in [ 2 ], we derive a sys te m of coupled sto c hastic differen tial equ ations (SDE) whose (Wiener-pro cess) diffusion terms are nonlinearly coupled and whose inv arian t is Lo c hner’s generalized Dirichlet dis- tribution. The standard Diric hlet distribution can only r epresen t non -p ositiv e co v ariances [ 6 ], whic h limits its applicatio n to a sp ecific class of p rocesses. The stochastic system whose inv arian t is the general- ized Diric hlet distrib u tion allo ws for a more general class of physic al pro cesses with a more general co v ariance matrix. The pro cess ma y b e stationary or non-stationary , not limited to non-p ositiv e co v ariances, and satisfies the unit-sum requir emen t at all times, n ec essary for v ariables that ob ey a conserv ation p rinciple. 2 Preview of results The generalize d Diric hlet distribution for a set of scalars 0 ≤ Y i , i = 1 , . . . , K , P K i =1 Y i ≤ 1, and parameters, α i > 0, β i > 0, as giv en b y Lo c hner [ 1 ] reads G ( Y , α , β ) = K Y i =1 Γ( α i + β i ) Γ( α i )Γ( β i ) Y α i − 1 i Y γ i i with Y i = 1 − i X k =1 Y k , (1) 1 and γ i = β i − α i +1 − β i +1 for i = 1 , . . . , K − 1, and γ K = β K − 1. He re Γ( · ) d enote s the gamma function. W e der ive the sto chastic diffu s io n pr ocess, go verning the scalars, Y i , d Y i ( t ) = U i 2    b i h S i Y K − (1 − S i ) Y i i + Y i Y K K − 1 X j = i c ij Y j    d t + p κ i Y i Y K U i d W i ( t ) , i = 1 , . . . , K, (2) where d W i ( t ) is an isotropic vec tor-v alued Wiener pr ocess with indep end ent in cr ements [ 16 ] and U i = Q K − i j =1 Y − 1 K − j . W e sho w that the statistica lly stationary solution of the coup led system of nonlinear stochastic d ifferen tial equations in ( 2 ) is the generalized Diric h let d istribution, Eq. ( 1 ), pro vided th e coefficients, b i > 0, κ i > 0, 0 < S i < 1, and c ij , with c ij = 0 for i > j , i, j = 1 , . . . , K − 1, satisfy α i = b i κ i S i , i = 1 , . . . , K, (3) 1 − γ i = c 1 i κ 1 = · · · = c ii κ i , i = 1 , . . . , K − 1 , (4) 1 + γ K = b 1 κ 1 (1 − S 1 ) = · · · = b K κ K (1 − S K ) . (5) The r estrict ion on the coefficients ens ure reflection to w ards the in terior of the sample space, w hic h together with the sp ecification Y N = Y K ensures N X i =1 Y i = 1 . (6) Indeed, if for example Y 1 = 0, the diffusion in Eq. ( 2 ) is zero and th e d rift is strictly p ositiv e, while if Y 1 = 1, the d iffusion is zero (as Y K U 1 → 0) and the drift is strictly negativ e. 3 Development of the diffusion p ro cess The diffusion pro cess, Eq. ( 2 ), is dev elop ed by the method of p oten tial solutions. Th e steps b elo w closely follo w the metho dolog y in tro duced in [ 2 ], u s ed to d eriv e a diffusion pro cess for the Diric hlet distribution. W e start from th e Itˆ o diffusion p rocess [ 16 ] for the sto c h astic v ector, Y i , d Y i ( t ) = a i ( Y )d t + K X j =1 b ij ( Y )d W j ( t ) , i = 1 , . . . , K, (7) with drift, a i ( Y ), d iffusion, b ij ( Y ), and the isotropic vecto r-v alued Wiener pro cess, d W j ( t ). Using standard m et h o ds giv en in [ 16 ] the equiv alen t F okk er-Planc k equation go verning th e join t pr ob a- bilit y , F ( Y , t ), derived from Eq. ( 7 ), is ∂ F ∂ t = − K X i =1 ∂ ∂ Y i  a i ( Y ) F  + 1 2 K X i =1 K X j =1 ∂ 2 ∂ Y i ∂ Y j  B ij ( Y ) F  , B ij = K X k =1 b ik b k j . (8) As the drift and d iffusion co efficients are time-homogeneous, a i ( Y , t ) = a i ( Y ) and B ij ( Y , t ) = B ij ( Y ), Eq. ( 7 ) is a statistically stationary pro cess and the solution of Eq. ( 8 ) conv erges to a stationary d istribution [ 16 ], Sec. 6.2.2. Our task is to sp ecify the fun cti onal forms of a i ( Y ) and b ij ( Y ) so that the stationary solution of Eq. ( 8 ) is G ( Y ), d efined by Eq. ( 1 ). 2 A p oten tial solution of Eq. ( 8 ) exists if ∂ ln F ∂ Y j = K X i =1 B − 1 ij 2 a i − K X k =1 ∂ B ik ∂ Y k ! ≡ − ∂ φ ∂ Y j , j = 1 , . . . , K, (9) is satisfied, [ 16 ] Sec. 6.2.2. Since the left hand side of Eq. ( 9 ) is a gradient , the expr ession on the right m ust also b e a gradient and can therefore b e obtained from a scalar p oten tial denoted b y φ ( Y ). Th is puts a constrain t on the p ossible c hoices of a i and B ij and on th e p otenti al, as φ, ij = φ, j i m ust also b e satisfied. Th e p oten tial solution is F ( Y ) = exp[ − φ ( Y )] . (10) No w fu nctio n al forms of a i ( Y ) and B ij ( Y ) that satisfy Eq. ( 9 ), with F ( Y ) ≡ G ( Y ) are sought . The mathematical constraints on the sp ecification of a i and B ij are as follo ws: 1. B ij m ust b e symmetric p ositiv e semi-definite. T his is to ensure that • the square-ro ot of B ij (e.g. the Cholesky-decomp ositi on, b ij ) exists, required b y the corresp ondence of the sto c hastic equation ( 7 ) and the F okk er-Planck equation ( 8 ), • Eq. ( 7 ) represents a diffusion, and • det( B ij ) 6 = 0, r equ ired b y the existence of the in verse in Eq. ( 9 ). 2. F or a p oten tial s ol u tion to exist Eq. ( 9 ) m us t b e satisfied. With F ( Y ) ≡ G ( Y ) Eq. ( 10 ) sh o ws th at the scalar p oten tial must b e − φ ( Y ) = K X i =1 ( α i − 1) ln Y i + K X i =1 γ i ln Y i . (11) It is straigh tforward to ve rif y that the sp ecifications a i ( Y ) = U i 2    b i h S i Y K − (1 − S i ) Y i i + Y i Y K K − 1 X j = i c ij Y j    , (12) B ij ( Y ) = ( κ i Y i Y K U i for i = j, 0 for i 6 = j, (13) satisfy the ab o v e mathematical constraints, 1. and 2. Here U i = Q K − i j =1 Y − 1 K − j , where an empt y pro duct is assumed to b e unity , while an emp ty sum is zero. In addition to the co efficien ts b i > 0, κ i > 0, and 0 < S i < 1, gov ernin g the Diric h let diffus ion pro cess [ 2 ], the drift no w has the additional (not all indep endent) on es, denoted by c ij , with c ij = 0 for i > j , i, j = 1 , . . . , K − 1. Substituting Eqs. ( 11 – 13 ) in to Eq. ( 9 ) yields a system with the same functions on b oth sid es with different co efficien ts, yielding the corresp ondence b et w een the p arameters of th e generalized Diric hlet d istribution, Eq. ( 1 ), and the F okke r-Planck equation ( 8 ) with Eqs. ( 12 – 13 ) as α i = b i κ i S i , i = 1 , . . . , K, (14) 1 − γ i = c 1 i κ 1 = · · · = c ii κ i , i = 1 , . . . , K − 1 , (15) 1 + γ K = b 1 κ 1 (1 − S 1 ) = · · · = b K κ K (1 − S K ) . (16) 3 The ab o v e result is arrived at indu cti vely based on the sp ecial case of K = 3 in App endix A. If Eqs. ( 14 – 16 ) hold, the stationary solution of the F okk er-Planck equation ( 8 ) with drift ( 12 ) and diffusion ( 13 ) is the generalized Diric hlet distribution, Eq. ( 1 ). T he same method olo gy w as app lie d to the Dirichlet case in [ 2 ]. Eqs. ( 14 – 16 ) sp ecify the corresp ondence b et w een the coefficient s of the sto c hastic system ( 7 ) with drift ( 12 ) and diffusion ( 13 ) and the generalized Diric hlet distribution, Eq. ( 1 ). With γ i = β i − α i +1 − β i +1 , i = 1 , . . . , K − 1, and γ K = β K − 1, the corresp ond en ce b et w een ( α i , β i ) and ( b i , S i , κ i , c ij ) is also complete. No te that Eqs. ( 12 – 13 ) are one p ossible wa y of sp ecifying drift and diffu s io n to arriv e at a generalized Dirichlet distribution; other functional forms may b e p ossible. It is str ai ghtforw ard to verify , that setting c 1 i /κ i = · · · = c ii /κ i = 1 for i = 1 , . . . , K − 1, i.e., γ 1 = · · · = γ K − 1 = 0, in Eqs. ( 12 ) and ( 13 ) yields the same system in Eq. ( 9 ) as w ith a i and B ij sp ecified f or the (standard ) Dirichlet distribu tion, see App endix A f or K = 3. The sh ape of the generalized Diric h let distribution, Eq. ( 1 ), is determined by the 2 K co effici ents, α i , β i . E qs. ( 14 – 16 ) sho w that in the s toc hastic system, different com binations of b i , S i , κ i , and c ij ma y yield the same α i , β i and that not all of b i , S i , κ i , and c ij ma y b e c h ose n indep endentl y to mak e the inv arian t generalized Diric hlet. In other w ords , a unique set of SDE co effici ents alw a ys corresp onds to a uniqu e set of distr ibution parameters, but the con v erse is not true: a set of d istr ibution parameters do n ot u niquely d ete rm ine all th e S DE co efficie nts, for a giv en sp ecific asymp tot ic generalized Diric hlet distribu tion. 4 Prop erties of Dirichlet distributions It is useful to sho w ho w the generalized Dirichlet distrib ution, E q . ( 1 ), reduces to stand ard Diric hlet, and their un iv ariate case, the b eta distrib ution. 4.1 Densit y functions Setting γ 1 = · · · = γ K − 1 = 0 in Eq. ( 1 ) yields the (standard) Diric hlet distribu tio n D ( Y , ω ) = Γ  P N i =1 ω i  Q N i =1 Γ( ω i ) N Y i =1 Y ω i − 1 i , (17) with ω i = α i , i = 1 , . . . , K = N − 1, ω N = β K , and Y N = 1 − P K j =1 Y j . In the un iv ariate case, K = N − 1 = 1, Y = ( Y 1 , Y 2 ) = ( Y , 1 − Y ), b oth G and D yield the b eta distribu tio n B ( Y , α, β ) = Γ( α + β ) Γ( α )Γ( β ) Y α − 1 (1 − Y ) β − 1 , (18) with ω 1 = α and ω 2 = β . G , D , and B are zero outside the K -dimensional generalized triangle; th e sample spaces are b ounded. Compared to D , there are K − 1 additional parameters in G for a s et of K scalars. 4.2 Moments All moments of the generalized Diric hlet distribution, Eq. ( 1 ), can b e ob tained from α i and β i of whic h the first t wo are [ 3 , 4 ] h Y i i = Z Y i G ( Y )d Y = α i α i + β i i − 1 Y j =1 β j α j + β j , (19) h y i y j i = h ( Y i − h Y i i )( Y j − h Y j i ) i =          h Y i i  α i + 1 α i + β i + 1 M i − 1 − h Y i i  for i = j, h Y j i  α i α i + β i + 1 M i − 1 − h Y i i  for i 6 = j, (20) 4 i, j = 1 , . . . , K, where M i − 1 = Q i − 1 k =1 ( β k + 1) / ( α k + β k + 1). Setting γ 1 = · · · = γ K − 1 = 0, with ω i = α i , i = 1 , . . . , K = N − 1, ω N = β K , in Eqs. ( 19 – 20 ) red uces to the firs t t wo momen ts of the Diric hlet distribution, h Y i i = ω i ω , (21) h y i y j i =        ω i ( ω − ω i ) ω 2 ( ω + 1) for i = j, − ω i ω j ω 2 ( ω + 1) for i 6 = j, (22) i, j = 1 , . . . , K, where ω = P N j =1 ω j . Eq. ( 20 ) shows that in the generalized Diric hlet distribu tio n Y 1 is alwa ys negativ ely correlated with the other scalars. Ho wev er, Y j and Y m can b e p ositiv ely corr elated for j, m > 1, see also [ 1 ]. According to W ong [ 4 ], “If ther e exists some m > j such that Y j and Y m ar e p ositively (ne g atively) c orr elate d, then Y j wil l b e p ositively (ne g atively) c orr elate d with Y n for al l n > j .” Th is can b e seen from Eq. ( 20 ): the sign of h y m y j i is indep endent of j , so the sign of h y m y j i , m > j w ill imply the signs of all h y n y j i , n > j . This is in con trast with the Diric hlet distribution, Eq. ( 17 ), whose co v ariances are alwa ys non-p ositiv e as can b e seen from Eq. ( 22 ). In the univ ariate case, K = N − 1 = 1, Y = ( Y 1 , Y 2 ) = ( Y , 1 − Y ), the first tw o momen ts of b oth the generalized and the standard Diric hlet distributions, Eqs. ( 19 – 20 ) and Eqs. ( 21 – 22 ), resp ectiv ely , red uce to the momen ts of the b eta distribution, with ω 1 = α and ω 2 = β , h Y i = α α + β , (23) h y 2 i = αβ ( α + β ) 2 ( α + β + 1) . (24) 5 Relation to other diffusion pro cesses It also useful to relate the generalized Diric hlet pro cess, Eq. ( 2 ), to other multiv ariate diffusion pro cesses with linear drift and quadratic diffusion. Setting c 1 i /κ i = · · · = c ii /κ i = 1 for i = 1 , . . . , K − 1, in Eq. ( 2 ) yields d Y i ( t ) = b i 2  S i Y N − (1 − S i ) Y i  d t + p κ i Y i Y N d W i ( t ) , i = 1 , . . . , K = N − 1 , (25) with Y N = 1 − P N − 1 j =1 Y j whose in v arian t is the (standard ) Diric hlet d istribution, Eq. ( 17 ). E q. ( 25 ) is discussed in [ 2 ]. Another diffusion pro cess whose in v arian t is also Diric hlet is the m ultiv ariate W right-F isher p rocess [ 15 ], d Y i ( t ) = 1 2 ( ω i − ω Y i )d t + K X j =1 q Y i ( δ ij − Y j )d W ij ( t ) , i = 1 , . . . , K = N − 1 , (26) where δ ij is Kronec k er’s delta. An other pro cess similar to Eqs. ( 2 ), ( 25 ), and ( 26 ) is the m ultiv ariate Jacobi pro cess, used in econometrics, d Y i ( t ) = a ( Y i − π i )d t + p cY i d W i ( t ) − N − 1 X j =1 Y i p cY j d W j ( t ) , i = 1 , . . . , N (27) 5 of Gourieroux & J asia k [ 13 ] with a < 0, c > 0, π α > 0, and P N j =1 π j = 1. In the un iv ariate case, K = N − 1 = 1, Y = ( Y 1 , Y 2 ) = ( Y , 1 − Y ), the generalized Diric h let , Diric hlet, W right- Fisher, and Jacobi diffusions, Eqs. ( 2 ), ( 25 ), ( 26 ), ( 27 ), resp ectiv ely , all redu ce to d Y ( t ) = b 2 ( S − Y )d t + p κY (1 − Y )d W ( t ) , (28) see also [ 17 ], whose in v arian t is the b eta distribution, whic h b elongs to the family of Pe arson diffusions, discuss ed in detail by F orman & Sorens en [ 18 ]. 6 Summa ry F ollo wing the d ev elopment in [ 2 ] w e started with a multiv ariate distribu tio n for a set of sto c hastic v ariables that s atisfies a conserv ation pr inciple in which all v ariables sum to un it y . Applying the constrain ts on the existence of p otent ial solutions of F okke r-Planck equations, we derived a system of s to c hastic differential equ ations ( 2 ) whose joint distr ibution in the statistically stationary state is Lo c hn er’s generalized Dirichlet distrib ution, Eq. ( 1 ). Eq. ( 2 ) is a generalization of the Dirichlet diffusion pro cess dev elop ed in [ 2 ]. Compared to the standard Diric hlet p rocess, the generalized diffusion allo ws f or r epresen ting a more general class of stochastic pro cesses with a more general co v ariance matrix. The pro cess ma y b e stationary or non-stationary , not limited to non-p ositiv e co v ariances, and satisfies the un it- su m r equiremen t, Eq. ( 6 ), at all times, n ec essary for v ariables that ob ey a conserv ation p rinciple. References [1] R. H. Lo c hner. A Generalized Dirichlet Distribution in Ba y esian Life Testing. Journal of the R oyal Statistic al So cie ty. 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Sc andinavian Journal of Statistics , 35:438– 465, 2008. [19] P . E. Klo eden and E. P la ten. N umeric al Solution of Sto chastic Differ ential E quations . Spr inger, Berlin, 1999. 7 App endix A: In d u c tive p ro of of Eqs. ( 14 – 16 ) based on K = 3 Eqs. ( 14 – 16 ) are no w arrive d at for K = 3, yielding th e corresp ondence of the generalized Diric h let distribution, Eq. ( 1 ), and its sto c hastic pr ocess, Eq. ( 2 ), for K = 3. The pr ocedure generalizes to arbitrary K > 3. F rom Eq. ( 11 ) the scalar p oten tial for K = 3 is − φ ( Y 1 , Y 2 , Y 3 ) = ( α 1 − 1) ln Y 1 + ( α 2 − 1) ln Y 2 + ( α 3 − 1) ln Y 3 + γ 1 ln(1 − Y 1 ) + γ 2 ln(1 − Y 1 − Y 2 ) + γ 3 ln(1 − Y 1 − Y 2 − Y 3 ) . (29) F rom Eqs. ( 12 – 13 ) the d r ift and diffusion for K = 3 are a 1 = b 1 / 2 (1 − Y 1 )(1 − Y 1 − Y 2 ) h S 1 (1 − Y 1 − Y 2 − Y 3 ) − (1 − S 1 ) Y 1 i + Y 1 (1 − Y 1 − Y 2 − Y 3 ) (1 − Y 1 )(1 − Y 1 − Y 2 )  c 11 / 2 1 − Y 1 + c 12 / 2 1 − Y 1 − Y 2  , (30) a 2 = b 2 / 2 1 − Y 1 − Y 2 h S 2 (1 − Y 1 − Y 2 − Y 3 ) − (1 − S 2 ) Y 2 i + c 22 2 · Y 2 (1 − Y 1 − Y 2 − Y 3 ) (1 − Y 1 − Y 2 ) 2 , (31 ) a 3 = b 3 2 h S 3 (1 − Y 1 − Y 2 − Y 3 ) − (1 − S 3 ) Y 3 i , (32) B 11 = κ 1 Y 1 (1 − Y 1 − Y 2 − Y 3 ) (1 − Y 1 )(1 − Y 1 − Y 2 ) , (33) B 22 = κ 2 Y 2 (1 − Y 1 − Y 2 − Y 3 ) 1 − Y 1 − Y 2 , (34) B 33 = κ 3 Y 3 (1 − Y 1 − Y 2 − Y 3 ) , (35) B 12 = B 23 = B 13 = 0 , (36) Substituting Eqs. ( 29 – 36 ) in to Eq. ( 9 ) for K = 3 yields α 1 − 1 Y 1 − γ 1 1 − Y 1 − γ 2 1 − Y 1 − Y 2 − γ 3 1 − Y 1 − Y 2 − Y 3 = =  b 1 κ 1 S 1 − 1  1 Y 1 +  c 11 κ 1 − 1  1 1 − Y 1 +  c 12 κ 1 − 1  1 1 − Y 1 − Y 2 +  1 − b 1 κ 1 (1 − S 1 )  1 1 − Y 1 − Y 2 − Y 3 , (37) α 2 − 1 Y 2 − γ 2 1 − Y 1 − Y 2 − γ 3 1 − Y 1 − Y 2 − Y 3 = =  b 2 κ 2 S 2 − 1  1 Y 2 +  c 22 κ 2 − 1  1 1 − Y 1 − Y 2 +  1 − b 2 κ 2 (1 − S 2 )  1 1 − Y 1 − Y 2 − Y 3 , (38) α 3 − 1 Y 3 − γ 3 1 − Y 1 − Y 2 − Y 3 =  b 3 κ 3 S 3 − 1  1 Y 3 +  1 − b 3 κ 3 (1 − S 3 )  1 1 − Y 1 − Y 2 − Y 3 , (39) 8 whic h sh o ws th at if α 1 = b 1 κ 1 S 1 , (40) α 2 = b 2 κ 2 S 2 , (41) α 3 = b 3 κ 3 S 3 , (42) 1 − γ 1 = c 11 κ 1 , (43) 1 − γ 2 = c 12 κ 1 = c 22 κ 2 , (44) 1 + γ 3 = b 1 κ 1 (1 − S 1 ) = b 2 κ 2 (1 − S 2 ) = b 3 κ 3 (1 − S 3 ) , (45) all hold, the inv arian t of Eq. ( 2 ) is Eq. ( 1 ) for K = 3, G ( Y 1 , Y 2 , Y 3 , α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ) = Γ( α 1 + β 1 )Γ( α 2 + β 2 )Γ( α 3 + β 3 ) Γ( α 1 )Γ( β 1 )Γ( α 2 )Γ( β 2 )Γ( α 3 )Γ( β 3 ) × (46) × Y α 1 − 1 1 Y α 2 − 1 2 Y α 3 − 1 3 (1 − Y 1 ) γ 1 (1 − Y 1 − Y 2 ) γ 2 (1 − Y 1 − Y 2 − Y 3 ) γ 3 , with γ 1 = β 1 − α 2 − β 2 , γ 2 = β 2 − α 3 − β 3 , γ 3 = β 3 − 1 . (47) Eqs. ( 40 – 45 ) giv e the corresp ondence b et w een the co efficien ts of the sto c hastic sys te m, Eq. ( 2 ), and its in v arian t, Eq. ( 1 ), for K = 3. With Eq. ( 47 ) the corresp ondence b et we en the parameters of the join t probabilit y d ensit y fu nction (PDF), ( α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ), and th e co efficien ts of the sto c hastic system, ( b 1 , b 2 , b 3 , S 1 , S 2 , S 3 , κ 1 , κ 2 , κ 3 , c 11 , c 12 , c 22 ), is also giv en. It is straigh tforward to v erify that setting γ 1 = γ 2 = 0 in Eq. ( 46 ) yields the Diric hlet distrib u- tion, Eq. ( 17 ), for K = 3 ( N = 4), D ( Y 1 , Y 2 , Y 3 , ω 1 , ω 2 , ω 3 , ω 4 ) = Γ( ω 1 + ω 2 + ω 3 + ω 4 ) Γ( ω 1 )Γ( ω 2 )Γ( ω 3 )Γ( ω 4 ) Y ω 1 − 1 1 Y ω 2 − 1 2 Y ω 3 − 1 3 (1 − Y 1 − Y 2 − Y 3 ) ω 4 − 1 (48) with ω 1 = α 1 , ω 2 = α 2 , ω 3 = α 3 , ω 4 = β 3 . (49) Similarly , setting c 11 /κ 1 = c 12 /κ 1 = c 22 /κ 2 = 1 in Eqs. ( 37 – 38 ) r educes to th e system corresp onding that of the Diric h let case [ 2 ]. App endix B: Numerical simulation: The effec t of the extra co efficient for K = 2 Numerical sim ulations are used to demonstrate the effect of the extra co efficien t, c 11 , compared to the standard Diric hlet case, give n in [ 2 ]. The time-ev olution of an ens em ble of 10 , 000 p articles h as b een n umerically computed b y in te- grating the system ( 7 ), with drift and diffusion ( 12 – 13 ), for K = 2, i.e., ( Y 1 , Y 2 , Y 3 = 1 − Y 1 − Y 2 ), d Y 1 = b 1 / 2 1 − Y 1  S 1 Y 3 − (1 − S 1 ) Y 1  d t + Y 1 Y 3 1 − Y 1 · c 11 / 2 1 − Y 1 d t + r κ 1 Y 1 Y 3 1 − Y 1 d W 1 , (50 ) d Y 2 = b 2 2  S 2 Y 3 − (1 − S 2 ) Y 2  d t + p κ 2 Y 2 Y 3 d W 2 , (51) Y 3 = 1 − Y 1 − Y 2 . (52) 9 T ab . 1: Co efficien ts of Eqs. ( 50 – 5 2 ) and asymptotic moments for three simulatio n cases. Asymptotic moments for K = 2, see Eqs. ( 19 – 20 ) h Y 1 i = α 1 α 1 + β 1 h Y 2 i = α 2 α 2 + β 2 · α 1 α 1 + β 1  y 2 1  = h Y 1 i  α 1 + 1 α 1 + β 1 + 1 − h Y 1 i   y 2 2  = h Y 2 i  α 2 + 1 α 2 + β 2 + 1 · α 1 + 1 α 1 + β 1 + 1 − h Y 2 i  h y 1 y 2 i = h Y 2 i  α 1 α 1 + β 1 + 1 − h Y 1 i  Diric hlet SDE co efficien ts (common to all cases) b 1 = 1 / 10 S 1 = 5 / 8 κ 1 = 1 / 80 b 2 = 3 / 2 S 2 = 2 / 5 κ 2 = 3 / 10 Generalized Diric hlet SDE co efficien ts c 11 = κ 11 = 1 / 80 c 11 = − 1 / 80 c 11 = − 1 / 4 (case 1) (case 2) (case 3) PDF parameters fr om th e SDE co efficien ts, see Eqs. ( 14 – 16 ) α 1 = b 1 κ 1 S 1 α 2 = b 2 κ 2 S 2 1 − γ 1 = c 11 κ 1 1 + γ 2 = b 1 κ 1 (1 − S 1 ) = b 2 κ 2 (1 − S 2 ) β 2 = 1 + γ 2 = b 1 κ 1 (1 − S 1 ) = b 2 κ 2 (1 − S 2 ) β 1 = γ 1 + α 2 + β 2 = 1 − c 11 κ 1 + α 2 + β 2 SDE asymptotic momen ts for cases 1, 2, 3 c 11 = 1 80 α 1 = 5 α 2 = 2 β 2 = 3 β 1 = 5 h Y 1 i = 1 2 h Y 2 i = 1 5  y 2 1  = 1 44  y 2 2  = 4 275 h y 1 y 2 i = − 1 110 c 11 = − 1 80 α 1 = 5 α 2 = 2 β 2 = 3 β 1 = 7 h Y 1 i = 5 12 h Y 2 i = 7 30  y 2 1  = 35 1872  y 2 2  = 609 35100 h y 1 y 2 i = − 35 4680 c 11 = − 1 4 α 1 = 5 α 2 = 2 β 2 = 3 β 1 = 26 h Y 1 i = 5 31 h Y 2 i = 52 155  y 2 1  = 65 15376  y 2 2  = 11141 38440 0 h y 1 y 2 i = − 13 7688 In Eqs. ( 50 – 51 ) d W 1 and d W 2 are indep end ent Wiener pr ocesses, sampled fr om Gaussian streams of rand om num b ers with mean h d W i i = 0 and co v ariance h d W i d W j i = δ ij d t . Eqs. ( 50 – 52 ) were adv anced in time with the Euler-Maruy ama sc heme [ 19 ] with time step ∆ t = 0 . 025. T he co efficie nts of the stochastic sy s te m ( 50 – 52 ), the corresp onding parameters and the first t wo moments of th e asymptotic generalized Diric hlet d istributions for K = 2 are sho wn in T able 1 . T hree d ifferent cases w ere s imulated. Here th e initial condition of ( Y 1 , Y 2 ) ≡ 0 was used. Th e in itial PDF in all cases is the same: all samples are zero and the PDF is therefore n ot Diric hlet nor Generalized Diric hlet, see also [ 2 ] for nonzero but different non-Diric h let initial conditions. Our motiv ation is t w o-fold: (1) to sho w that the solution app roac hes the in v ariant, and (2) to sho w ho w the new additional parameter in the generalized Diric hlet SDE affects th e dynamics. Had the initial conditions coincided with the giv en in v arian t, the PDF (and its statistics) w ould not hav e c hanged in time – as has b een 10 0 0 0 20 20 0 0 20 0 40 40 40 60 60 20 20 60 20 80 80 40 40 80 40 100 100 60 60 100 60 120 120 80 80 120 80 140 140 100 100 140 100 120 120 120 Time Time 140 140 Time 140 Time Time Time 0 0 0 −0.02 −0.02 −0.02 0.1 0.1 0.1 0.2 0.2 0.2 −0.01 −0.01 −0.01 0.3 0.3 0.3 0 0 0 0.4 0.4 0.4 0.01 0.01 0.01 0.5 0.5 0.5 0.02 0.02 0.02 0.6 0.6 0.6 0.03 0.03 0.03 Means Means Means 0.04 0.04 0.04 Covariances Covariances Covariances = 1/2 = 5/12 = 5/31 = 1/5 = 7/30 = 52/155 = 35/1872 = 1/44 = 65/15376 c11 = 1/80 c11 = −1/80 c11 = −1/4 = 609/35100 = 4/275 = 11141/384400 = −35/4680 = −1/110 = −13/7688 c11 = −1/80 c11 = 1/80 c11 = −1/4 Fig. 1: T ime evolutio n of th e first t w o moments of Eqs. ( 50 – 52 ). First row: c 11 = κ 1 = 1 / 80 (standard Diric hlet, see also [ 2 ]), second ro w: c 11 = − 1 / 80, th ird r o w: c 11 = − 1 / 4. demonstrated mathematically . Th e SDE co efficie nts in the three simulati ons only d iffer in the extra generalized Diric hlet co efficien t, c 11 , otherwise, the setup corresp onds to the example in [ 2 ]. In the fir st sim ulation c 11 = κ 1 = 1 / 80, i.e., c 11 is not a free co efficien t and is c hosen to yield an asymptotic solution that is a (standard ) Diric hlet, the same as in [ 2 ]. In the second and th ird sim ulations c 11 are f reely c hosen and th us yield generalized Diric h let solutions. Figure 1 s h o ws the ev olutions of the firs t tw o momen ts in time for the three cases. 11