A Class of Infinite Dimensional Diffusion Processes with Connection to Population Genetics
aa r X i v : . [ m a t h . P R ] N ov A Class of Infinite Dimensional Diffusion Processes withConnection to Population Genetics
Shui FengMcMaster [email protected] Feng-Yu WangBeijing Normal University and Swansea [email protected]
Abstract
Starting from a sequence of independent Wright-Fisher diffusion processes on [0 , ∞ := { x ∈ [0 , N : P i ≥ x i = 1 } with GEM distribution as the reversible measure. Log-Sobolev inequalities areestablished for these diffusions, which lead to the exponential convergence to the correspondingreversible measures in the entropy. Extensions are made to a class of measure-valued processesover an abstract space S . This provides a reasonable alternative to the Fleming-Viot processwhich does not satisfy the log-Sobolev inequality when S is infinite as observed by W. Stannat[13]. Key words:
Poisson-Dirichlet distribution, GEM distribution, Fleming-Viot process, log-Sobolevinequality.
AMS 1991 subject classifications:
Primary: 60F10; Secondary: 92D10.
Population genetics is concerned with the distribution and evolution of gene frequencies in a largepopulation at a particular locus. The infinitely-many-neutral-alleles model describes the evolutionof the gene frequencies under generation independent mutation, and resampling. In statisticalequilibrium the distribution of gene frequencies is the well known Poisson-Dirichlet distributionintroduced by Kingman [8]. When a sample of size n genes is selected from a Poisson-Dirichletpopulation, the distribution of the corresponding allelic partition is given explicitly by the Ewenssampling formula . This provides an important tool in testing neutrality of a population.Let ∆ ∞ = { x = ( x , x , ... ) ∈ [0 , N : ∞ X k =1 x k = 1 } , ∇ = { x = ( x , x , ... ) ∈ [0 , N : x ≥ x ≥ · · · ≥ , ∞ X k =1 x k = 1 } . The Poisson-Dirichlet distribution with parameter θ >
P D ( θ )) is a probability mea-sure Π θ on ∇ . We use P ( θ ) = ( P ( θ ) , P ( θ ) , ... ) to denote the ∇ -valued random variable withdistribution Π θ . The component P k ( θ ) represents the proportion of the k th most frequent alleles.If u is the individual mutation rate and N is the effective population size, then the parameter θ = 4 N u is the population mutation rate. A different way of describing the distribution is throughthe following size-biased sampling. Let U k , k = 1 , , ... , be a sequence of independent, identicallydistributed random variables with common distribution Beta (1 , θ ), and set(1.1) X θ = U , X θn = (1 − U ) · · · (1 − U n − ) U n , n ≥ . Clearly ( X θ , X θ , . . . ) is in space ∆ ∞ . The law of X θ , X θ , ... is called the one parameter GEMdistribution and is denoted by Π gemθ . The descending order of X θ , X θ , ... has distribution Π θ . Thesequence X θk , k = 1 , , ... has the same distribution as the size-biased permutation of Π θ .Let ξ k , k = 1 , ... be a sequence of i.i.d. random variables with common diffusive distribution ν on [0 , ν ( x ) = 0 for every x in [0 , θ,ν = ∞ X k =1 P k ( θ ) δ ξ k . It is known that the law of Θ θ,ν is Dirichlet ( θ, ν ) distribution, and is the reversible distributionof the Fleming-Viot process with mutation operator (cf. [2])(1.3) Af ( x ) = θ Z ( f ( y ) − f ( x )) ν ( dx ) . For 0 ≤ α < , θ > − α , let { V k : k = 1 , , ... } be a sequence of independent random variablessuch that V k is a Beta (1 − α, θ + kα ) random variable for each k . Set(1.4) X θ,α = V , X θ,αn = (1 − V ) · · · (1 − V n − ) V n , n ≥ . The law of X θ,α , X θ,α , ... is called the two-parameter GEM distribution and is denoted by Π gemα,θ .The law of the descending order statistic of X θ,α , X θ,α , ... is called the two-parameter Poisson-Dirichlet distribution (henceforth Π α,θ ) studied thoroughly in Pitman and Yor [12] . The sequence X θ,αk , k = 1 , , ... has the same distribution as the size-biased permutation of Π α,θ . It is shown inPitman [10] that the two-parameter Poisson-Dirichlet distribution is the most general distributionwhose size-biased permutation has the same distribution as the GEM representation (1.4). A two-parameter “Ewens sampling formula” is obtained in [11]. Let Θ θ,α,ν be defined similarly to Θ θ,ν with X θk being replaced by X θ,αk . We call the law of Θ θ,α,ν a Dirichlet ( θ, α, ν ) distribution.2he Poisson-Dirichlet distribution and its two-parameter generalization have many similarstructures including the urn construction in [7] and [3], GEM representation, sampling formula,etc.. But we have not seen a stochastic dynamic model similar to the infinitely-many-neutral-allelesmodel and the Fleming-Viot process developed for the two-parameter Poisson-Dirichlet distributionand Dirichlet ( θ, α, ν ) distribution.As the first result in this paper, we are able to construct a class of reversible infinite dimensionaldiffusion processes, the GEM processes, so that both Π gemθ and its two-parameter generalizationΠ gemα,θ appear as the reversible measures for appropriate parameters.In [13], the log-Sobolev inequality is studied for the Fleming-Viot process with motion givenby (1.3). It turns out that the log-Sobolev inequality holds only when the type space is finite.In the second result of this paper, we will first construct a measure-valued process that has the Dirichlet ( θ, ν ) distribution as reversible measure. Then we will establish the log-Sobolev inequalityfor the process.The rest of the paper is organized as follows. The GEM processes associated with Π gemθ andΠ gemα,θ are introduced in section 2. Section 3 includes the proof of uniqueness and the log-Sobolevinequality of the GEM process. Finally in section 4, the measure-valued process is introduced andthe corresponding log-Sobolev inequality is established. For any i ≥
1, let a i , b i be two strictly positive numbers. We assume that(2.1) inf i b i ≥ . Let X i ( t ) be the unique strong solution of the stochastic differential equation(2.2) dX i ( t ) = ( a i − ( a i + b i ) X i ( t )) dt + p X i ( t )(1 − X i ( t )) dB i ( t ) , X i (0) ∈ [0 , , where { B i ( t ) : i = 1 , , ... } are independent one dimensional Brownian motions. It is known thatthe process X i ( t ) is reversible with reversible measure π a i ,b i = Beta (2 a i , b i ). By direct calculation,the scale function of X i ( · ) is given by s i ( x ) = ( 14 ) a i + b i Z x / dyy a i (1 − y ) b i . By (2.1), we have lim x → s i ( x ) = + ∞ for all i . Thus starting from the interior of [0 , X i ( t ) will not hit the boundary 1 with probability one. Let E = [0 , N . The process X ( t ) = ( X ( t ) , X ( t ) , ... )3s then a E -valued Markov process. Consider the mapΦ : E → ¯∆ ∞ , x = ( x , x , ... ) → ( ϕ ( x ) , ϕ ( x ) , .. )with ϕ ( x ) = x , ϕ n ( x ) = x n (1 − x ) · · · (1 − x n − ) , n ≥ . Clearly Φ is a bijection and the process Y ( t ) = Φ( X ( t )) is thus a Markov process. Let ¯ E :=[0 , N be the closure of E , C ( ¯ E ) denote the set of all continuous function on ¯ E , and C cl ( ¯ E ) be theset of functions in C ( ¯ E ) that have second order continuous derivatives, and depend only on a finitenumber of coordinates. The sets C ( E ) and C cl ( E ) will be the respective restrictions of C ( ¯ E ) and C cl ( ¯ E ) on E . Then the generator of process X ( t ) is given by Lf ( x ) = ∞ X k =1 (cid:8) x k (1 − x k ) ∂ f∂x k + ( a k − ( a k + b k ) x k ) ∂f∂x k (cid:9) , f ∈ C cl ( E ) , and can be extended to C cl ( ¯ E ). The sets B ( E ) and B (∆ ∞ ) are bounded measurable functions on E and ∆ ∞ , respectively.Let a = ( a , a , . . . , ) , b = ( b , b , . . . ), and µ a , b = ∞ Y k =1 π a k ,b k , Ξ a , b = µ a , b ◦ Φ − . Then we have
Theorem 2.1
The processes X ( t ) and Y ( t ) are reversible with respective reversible measures µ a , b and Ξ a , b . Proof:
The reversibility of X ( t ) follows from the reversibility of each X i ( t ). Now for any two f, g in B (∆ ∞ ), the two functions f ◦ Φ , g ◦ Φ are in B ( E ). From the reversibility of X ( t ), we have forany t > Z ∆ ∞ f ( y ) E y [ g ( y ( t ))]Ξ a , b ( d y ) = Z E f (Φ( x )) E x [ g (Φ( x ( t )))] µ a , b ( d x )= Z E g (Φ( x )) E x [ f (Φ( x ( t )))] µ a , b ( d x )= Z ∆ ∞ g ( y ) E y [ f ( y ( t ))]Ξ a , b ( d y ) . Hence Y ( t ) is reversible with reversible measure Ξ a , b . ✷ Remark.
The one parameter GEM distribution Π gemθ corresponds to a i = , b i = θ , and the twoparameter GEM distribution Π gemα,θ corresponds to a i = − α , b i = θ + iα .4 Uniqueness and Poincar´e/Log-Sobolev Inequalities
Let ¯∆ ∞ := { x ∈ [0 , N : ∞ X i =1 x i ≤ } be the closure of space ∆ ∞ in R N under the topology induced by cylindrically continuous functions.The probability Ξ a , b can be extended to the space ¯∆ ∞ . For simplicity, the same notation is usedto denote this extended probability measure.Now, for x ∈ ¯∆ ∞ such that n X i =1 x i < , for all finite n, let L ( x ) = ∞ X i,j =1 a ij ( x ) ∂ ∂x i ∂x j + ∞ X i =1 b i ( x ) ∂∂x i , where a ij ( x ) := x i x j i ∧ j X k =1 ( δ ki (1 − P k − l =1 x l ) − x k )( δ kj (1 − P k − l =1 x l ) − x k ) x k (1 − P kl =1 x l ) ,b i ( x ) := x i i X k =1 ( δ ik (cid:0) − P k − l =1 x l (cid:1) − x k )( a k (cid:0) − P k − l =1 x l (cid:1) − ( a k + b k ) x k ) x k (1 − P kl =1 x l ) . Here and in what follows, we set P i =1 = 0 and Q i =1 = 1 by conventions. By treating as one,the definition of L ( x ) can be extended to all points in ¯∆ ∞ . Through direct calculation one cansee that L is the generator of the GEM process.It follows from direct calculation that(3.1) ∞ X i,j =1 | a ij ( x ) | ≤ , | b i ( x ) | ≤ i X k =1 ( b k x k + a k ) , x ∈ ¯∆ ∞ . Indeed, since 1 − P i − l =1 x l ≥ x i and P ≤ i
0, let α a,b be the largest constant such that for f ∈ C b ([0 , π a,b ( f log f ) ≤ α a,b Z x (1 − x ) f ′ ( x ) π a,b (d x ) + π a,b ( f ) log π a,b ( f )holds. According to [13, Lemma 2.7], we have α a,b ≥ a ∧ b . Moreover, it is easy to see that for a, b > r (1 − r ) d dr + ( a − ( a + b ) r ) ddr on [0 ,
1] has a spectral gap a + b with eigenfunction h ( r ) := a − ( a + b ) r. So, the Poincar´e inequality(3.3) π a,b ( f ) ≤ a + b Z x (1 − x ) f ′ ( x ) π a,b (d x ) + π a,b ( f ) holds. 6et C ∞ cl ([0 , N ) denote the set of all bounded, C ∞ cylindrical functions on [0 , N , and F C ∞ b = { f | ¯∆ ∞ : f ∈ C ∞ cl ([0 , N ) } . Then we have the following theorem.
Theorem 3.1
For any f, g ∈ F C ∞ b , we have (3.4) E ( f, g ) := Ξ a , b (Γ( f, g )) = − Ξ a , b ( f L g ) . Consequently, ( E , F C ∞ b ) is closable in L ( ¯∆ ∞ ; Ξ a , b ) and its closure is a conservative regularDirichlet form, which satisfies the Poincar´e inequality (3.5) Ξ a , b ( f ) ≤ i ≥ ( a i + b i ) E ( f, f ) , f ∈ D ( E ) , Ξ a , b ( f ) = 0 . If moreover inf { a i ∧ b i : i ≥ } > , the log-Sobolev inequality (3.6) Ξ a , b ( f log f ) ≤ β a , b E ( f, f ) , f ∈ D ( E ) , Ξ a , b ( f ) = 1 holds for some β a , b ≥ inf { a i ∧ b i : i ≥ } > . Proof:
For any f, g ∈ F C ∞ b , there exists n ≥ f ( x ) = f ( x , · · · , x n ) , g ( x ) = g ( x , · · · , x n ) , x = ( x , · · · , x n , · · · ) ∈ [0 , N . Let ϕ ( n ) ( x ) = ( ϕ ( x ) , . . . , ϕ n ( x )) , which maps [0 , n on to ∆ n := { x ∈ [0 , n : P ni =1 x i ≤ } . Define L n := n X i =1 x i (1 − x i ) ∂∂x i + n X i =1 ( a i − ( a i + b i ) x i ) ∂∂x i , and π n a , b = n Y i =1 π a i ,b i , Ξ n = π n a , b ◦ ϕ ( n ) − . Then, regarding { Ξ n := π n a , b ◦ ϕ ( n ) − : n ≥ } as probability measures on7∆ ∞ , by letting Ξ n := Ξ n (d x · · · d x n ) × δ (d x n +1 , · · · ), it converges weakly to Ξ a , b . Since L n issymmetric w.r.t. π n a , b we have Z [0 , n n X i =1 x i (1 − x i ) (cid:16) ∂∂x i f ◦ ϕ ( n ) (cid:17)(cid:16) ∂∂x i g ◦ ϕ ( n ) (cid:17) d π n a , b = − Z [0 , n g ◦ ϕ ( n ) L n f ◦ ϕ ( n ) d π n a , b . (3.8)Noting that ϕ i ( x ) = x i i − Y l =1 (1 − x l ) , x i = ϕ i ( x )1 − P i − l =1 ϕ l ( x ) , i ≥ , we have d f ◦ ϕ ( n ) ( x )d x i = X j ≥ i ( δ ij − x i ) ϕ j ( x ) x i (1 − x i ) d f d ϕ j ◦ ϕ ( n ) ( x ) . Therefore, Z [0 , n n X i =1 x i (1 − x i ) (cid:16) ∂∂x i f ◦ ϕ ( n ) (cid:17)(cid:16) ∂∂x i g ◦ ϕ ( n ) (cid:17) d π n a , b = Z [0 , n Γ( f, g ) ◦ ϕ ( n ) d π n a , b = Z ∆ n Γ( f, g )dΞ n . (3.9)By (3.1) and (3.7), we have Γ( f, g ) ∈ C b ( ¯∆ ∞ ) so that the weak convergence of Ξ n to Ξ a , b implies(3.10) lim n →∞ Z ∆ n Γ( f, g )dΞ n = Z ¯∆ ∞ Γ( f, g )dΞ a , b . Similarly, by straightforward calculations we find L n f ◦ ϕ ( n ) ( x ) = ( L f ) ◦ ϕ ( n ) ( x ) . Moreover, (3.1) and (3.7) imply that g L f ∈ C b ( ¯∆ ∞ ) . Thus we arrive atlim n →∞ Z ∆ n g ◦ ϕ ( n ) L n f ◦ ϕ ( n ) d π n a , b = Z ¯∆ ∞ g L f dΞ a , b . Therefore, (3.4) follows by combining this with (3.9) and (3.10). This implies the closability of( E , F C ∞ b ), while the regularity of its closure follows from the compactness of ¯∆ ∞ under the usualmetric 8 ( x , y ) := ∞ X i =1 − i | x i − y i | . Indeed, it is trivial that D ( E ) ∩ C ([0 , N ) ⊃ F C ∞ b which is dense in D ( E ) under E / given by E ( f, f ) = E ( f, f ) + k f k . Moreover, for any F ∈ C ( ¯∆ ∞ ) = C ( ¯∆ ∞ ), by its uniform continuity due to the compactness of thespace, ¯∆ ∞ x F n ( x ) := F ( x , · · · , x n , , , · · · ) , n ≥ F. Since a cylindric con-tinuous function can be uniformly approximated by functions in F C ∞ b under the uniform norm,it follows that F C ∞ b is dense in C ( ¯∆ ∞ ) under the uniform norm. That is, the Dirichlet form( E , D ( E )) is regular.Next, the desired Poincar´e and log-Sobolev inequalities can be deduced from (3.3) and (3.2)respectively. For simplicity, we only prove the latter. By the additivity property of the log-Sobolevinequality (cf. [5]), µ n ( h log h ) ≤ β n a , b Z [0 , n n X i =1 x i (1 − x i ) (cid:16) ∂h∂x i (cid:17) d π n a , b + µ n ( h ) log π n a , b ( h )holds for all h ∈ C b ([0 , n ) , where β n a , b = inf { α a i ,b i : i = 1 , . . . , n } , f ( n ) ( x ) = f ( x , . . . , x n , , . . . ) . Combining this with (3.9), for any f ∈ D , the domain of L , we haveΞ n ( f ( n )2 log f ( n )2 ) ≤ β n a , b Z ∆ n Γ ( n ) ( f, f )dΞ n + Ξ n ( f ( n )2 ) log Ξ n ( f ( n )2 ) . Therefore, as explained above, (3.6) for f ∈ D follows immediately by letting n → ∞ . Hence, theproof is completed since D ( E ) is the closure of D under E / . ✷ We remark that since ( E , D ( E )) is regular, according to [6, 9], ( L, D ) generates a Hunt processwhose semigroup P t is unique in L (Ξ a , b ) . Thus the GEM process constructed in section 2 is theunique Feller process generated by L . Moreover, it is well-known that the log-Sobolev inequality(3.6) implies that P t converges to Ξ a , b exponentially fast in entropy; more precisely (see e.g. [1,Proposition 2.1]), Ξ a , b ( P t f log P t f ) ≤ e − β a , b t Ξ a , b ( f log f ) , f ≥ , Ξ a , b ( f ) = 1 . P t .Thus, according to Theorem 3.1, we have constructed a diffusion process which converges to itsreversible distribution Ξ a , b in entropy exponentially fast. It was shown in Stannat [13] that the log-Sobolev inequality fails to hold for the Fleming-Viotprocess with parent independent mutation when there are infinite number of types. In this section,we will construct a class of measure-valued processes for which the log-Sobolev inequality holdseven when the number of types is infinity.Let us first consider a measure-valued processes on a Polish space S induced by the above con-structed process and a proper Markov process on S N . More precisely, let X t := ( X ( t ) , · · · , X n ( t ) , · · · )be the Markov process on ∆ ∞ associated to ( E , D ( E ), and ξ t := ( ξ ( t ) , · · · , ξ n ( t ) , · · · ) be a Markovprocess on S N , independent of X t . We consider the measure-valued process η t := ∞ X i =1 X i ( t ) δ ξ i ( t ) , where X i can be viewed as the proportion of the i -th family in the population, and ξ i its type orlabel. Then the above process describes the evolution of all (countably many) families on the space S . Let M be the set of all probability measures on S . Then the state space of this process is M := { γ ∈ M : supp γ contains at most countably many points } , which is dense in M under the weak topology.Due to Theorem 3.1, if ξ t converges to its unique invariant probability measure ν on S N , then η t converges to Π := (Ξ a , b × ν ) ◦ ψ − for ψ : ∆ ∞ × S N → M ; ψ ( x , ξ ) := ∞ X i =1 x i δ ξ i . Unfortunately the process η t is in general non-Markovian. So we like to modify the constructionby using Dirichlet forms.Let ν be a probability measure on S N and ( E S N , D ( E S N )) a conservative symmetric Dirichletform on L ( ν ) . We then construct the corresponding quadratic form on L ( M ; Π) as follows: E M ( F, G ) := Z S N E ( F ξ , G ξ ) ν (d ξ ) + Z ∆ ∞ E S N ( F x , G x ) π a,b (d x ) F, G ∈ D ( E M ) := (cid:8) H ∈ L (Π) : H x := H ◦ ψ ( x, · ) ∈ D ( E S N ) for Ξ a , b -a.s. x ,H ξ := H ◦ ψ ( · , ξ ) ∈ D ( E ) for ν -a.s. ξ, such that E M ( H, H ) < ∞ (cid:9) . M , to make the state space complete one may also consider the abovedefined form a symmetric form on L ( M ; Π)(= L ( M ; Π)). Theorem 4.1
Assume there exists α > such that ν ( f log f ) ≤ α E S N ( f, f ) + ν ( f ) log ν ( f ) , f ∈ D ( E S N ) holds, then (4.1) Π( F log F ) ≤ α ∧ β a , b E M ( F, F ) + Π( F ) log Π( F ) , F ∈ D ( E M ) . If moreover D ( E M ) ⊂ L ( M ; Π) is dense, then ( E M , D ( E M )) is a conservative Dirichlet formon L ( M ; Π) so that the associated Markov semigroup P t satisfies (4.2) Π( P t F log P t F ) ≤ Π( F log F )e − ( β a , b ∧ α ) t , t ≥ , F ≥ , Π( F ) = 1 , and ( E M , D ( E M )) is regular provided so is ( E S N , D ( E S N )) and S is compact. Proof:
Let D ( ˜ E ) = (cid:8) ˜ F ∈ L (Ξ a , b × ν ) : ˜ F ( x, · ) ∈ D ( E S N ) for Ξ a , b -a.s. x, ˜ F ( · , ξ ) ∈ D ( E ) for ν -a.s. ξ, such that ˜ E ( ˜ F , ˜ F ) < ∞ (cid:9) , where ˜ E ( ˜ F , ˜ G ) := Z ∆ ∞ E S N ( ˜ F ( x , · ) , ˜ G ( x , · ))Ξ a , b (d x ) + Z S N E ( ˜ F ( · , ξ ) , ˜ G ( · , ξ )) ν (d ξ ) . Then ( ˜ E , D ( ˜ E )) is a symmetric Dirichlet form on L (∆ ∞ × S N ; Ξ a , b × ν ) and (see e.g. [5, Theorem2.3])(4.3) (Ξ a , b × ν )( ˜ F log ˜ F ) ≤ β a , b ∧ α (Ξ a , b × ν )( ˜ F ) , ˜ F ∈ D ( ˜ E ) , (Ξ a , b × ν )( ˜ F ) = 1 . Let ˜ P t be the Markov semigroup associated to ( ˜ E , D ( ˜ E )). Then (4.2) follows from the fact that η t = ψ ( X ( t ) , ξ ( t )) and (4.3) implies (cf. [1, Proposition 2.1])(Ξ a , b × ν )( ˜ P t G log ˜ P t G ) ≤ (Ξ a , b × ν )( G log G )e − β a , b ∧ α ) t for all t ≥ G with (Ξ a , b × ν )( G ) = 1 . Since F ∈ D ( E M ) if and only if F ◦ ψ ∈ D ( ˜ E ), and 11 M ( F, F ) = ˜ E ( F ◦ ψ, F ◦ ψ ) , (4.1) follows from (4.3). By the same reason and noting that ( ˜ E , D ( ˜ E )) is a Dirichlet form, weconclude that ( E M , D ( E M )) is a Dirichlet form provided it is densely defined on L ( M ; Π) . Finally,if S is compact then so is M (under the weak topology). Thus, as explained in the proof of Theorem3.1, for regular ( E S N , D ( E S N )) the set { f ( h· , g i , · · · , h· , g n i ) : n ≥ , f ∈ C b ( R n ) , g i ∈ C ( S ) , ≤ i ≤ n } ⊂ C ( M ) ∩ D ( E M )is dense both in C ( M ))(= C ( M )) under the uniform norm and in D ( E M ) under the Sobolevnorm. ✷ Remark.
Obviously, we have a similar assertion for the Poincar´e inequality: if there exists λ > ν ( f ) ≤ λ E S N ( f, f ) + ν ( f ) , f ∈ D ( E S N )holds, then Π( F ) ≤ λ ∧ inf i ≥ ( a i + b i ) E M ( F, F ) + Π( F ) , F ∈ D ( E M ) . To see that the above theorem applies to a class of measure-valued processes on S , we presentbelow a concrete condition on E S N such that assertions in Theorem 4.1 apply. In particular, it isthe case if E S N is the Dirichlet form of a particle system without interactions. Proposition 4.2
Let ν i be the i -th marginal distribution of ν and for a function g on S let g ( i ) ( ξ ) := g ( ξ i ) , i ≥ . Assume that S := n g ∈ C ( S ) : g ( i ) ∈ D ( E S N ) , sup i ≥ E S N ( g ( i ) , g ( i ) ) < ∞ o is dense in C ( S ) . Then ( E M , D ( E M )) is a symmetric Dirichlet form. Proof:
Under the assumption and the fact that C cl (∆ ∞ ) is dense in L ( M ; Π) , the set S := (cid:8) f ( h· , g i , · · · , h· , g n i ) : n ≥ , f ∈ C b ( R n ) , g i ∈ S , ≤ i ≤ n (cid:9) is dense in L ( M ; Π) . Therefore, by Theorem 4.1 it suffices to show that S ⊂ D ( E M ); that is,for F := f ( h· , g i , · · · , h· , g n i ) ∈ S , one has F ◦ ψ ∈ D ( ˜ E ) . Let F m ( x ) = F (cid:16) m X i =1 x i g ( ξ i ) , · · · , m X i =1 x i g n ( ξ i ) (cid:17) , x ∈ ∆ ∞ , m ≥ . ξ ∈ S N , ∂ x i F ◦ ψ ( · , ξ )( x ) = n X k =1 ∂ k f g k ( ξ i ) , i ≥ F m ∈ D ( E ) and (3.1) yields E ( F m , F m ) ≤ C for some constant C > m ≥ ξ ∈ S N . Thus, F ◦ ψ ( · , ξ ) ∈ D ( E ) for each ξ ∈ S N and(4.4) sup ξ E ( F ◦ ψ ( · , ξ ) , F ◦ ψ ( · , ξ )) ≤ C. On the other hand, since g k ∈ S , ≤ k ≤ n, noting that for any x ∈ ∆ ∞ | F ◦ ψ ( x , ξ ) − F ◦ ψ ( x , ξ ′ ) | ≤ ( n X k =1 k ∂ k f k ∞ ) ∞ X i =1 x i | g k ( ξ i ) − g k ( ξ ′ i ) | , we conclude in the spirit of [9, Proposition I-4.10] that F ◦ ψ ( x , · ) ∈ D ( E S N ) and E S N ( F ◦ ψ ( x , · ) , F ◦ ψ ( x , · )) ≤ C ′ for some C ′ > x . Combining this with (4.4) we obtain F ◦ ψ ∈ D ( ˜ E ) . ✷ Acknowledgement .The research of S. Feng is supported by NSERC of Canada. The researchof F.Y. Wang is supported by NNSFC(10121101), RFDP(20040027009) and the 973-Project of P.R.China.
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