ON THE POSITIVE SEMIGROUPS GENERATED BY FLEMING-VIOT TYPE DIFFERENTIAL OPERATORS

. In this paper we study a class of degenerate second-order elliptic diﬀerential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces deﬁned on the d -dimensional hypercube Q d of R d , d ≥ 1. By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted L p -spaces. In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0 , 1]. As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.


(Communicated by Yuri Latushkin)
Abstract. In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the d-dimensional hypercube Q d of R d , d ≥ 1.
By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted L p -spaces.
In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1].
As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.
1. Introduction. The main aim of this paper is to study the class of degenerate second-order elliptic differential operators defined on the d-dimensional hypercube Q d of R d , d ≥ 1, by for every u ∈ C 2 (Q d ) and x = (x 1 , . . . , x d ) ∈ Q d , where a 1 , . . . , a d , b 1 , . . . , b d ∈ R, a i > −1 and b i > −1 for all i = 1, . . . , d.
In this paper we give some further contributions to this research area by heavily using techniques arising from approximation theory. The methods we employ allow to show that these operators generate positive semigroups both in the space of all continuous functions and in weighted L p -spaces with respect to the Jacobi weights determined by the coefficients a i and b i , i = 1, . . . , d.
In addition, we disclose several qualitative properties of the generated semigroups.
As a first step we introduce a sequence of polynomial type positive linear operators which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1]. Among other things, we show that these operators constitute an approximation process for continuous functions as well as for weighted L p -functions.
By using the Trotter-Schnabl approximation theorem, we show that their closures generate positive semigroups which, in turn, are approximated by iterates of the above mentioned positive operators.
As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we describe their asymptotic behaviour by also evaluating the rate of convergence. Finally, we show that they preserve the class of Hölder continuous functions and the one of those continuous functions which are convex with respect to each variable. 2. Preliminaries and notation. We begin by recalling some basic notions about invariant measures which will play a key role within the whole paper. For more details on such a subject, and on its relationship with ergodic theory and asymptotic formulae, we refer the interested reader to [9,16].
Let X be a compact Hausdorff space and let B X be the σ-algebra of all Borel subsets of X; we denote by M + (X) (resp., M + 1 (X)) the cone of all regular Borel measures on X (resp., the cone of all regular Borel probability measures on X).
If µ ∈ M + (X) and 1 ≤ p < +∞, let us denote by L p (X, µ) the space of all (the equivalence classes of) Borel measurable real-valued functions on X which are µ-integrable in the p th power. The space L p (X, µ) is endowed with the natural norm As usual, the symbol C(X) indicates the linear space of all continuous real-valued functions on X. C(X) will be endowed with the uniform norm · ∞ , with respect to which it is a Banach space.
A Markov operator T on C(X) is a positive linear operator T : C(X) → C(X) such that T (1) = 1, where the symbol 1 stands for the constant function of constant value 1 on X.
By the Riesz representation theorem, for every x ∈ X there exists µ x ∈ M + 1 (X) such that By applying the Hölder inequality to each µ x , it follows that (1) It is well-known that every Markov operator T on C(X) admits at least one invariant probability measure, i.e., a measure µ ∈ M + 1 (X) such that (see [16,Section 5.1,p. 178]). Accordingly, on account of (1), if µ is an invariant measure for T , then for every f ∈ C(X) and p ∈ [1, +∞[, hence, T extends to a unique bounded linear operator T p : L p (X, µ) → L p (X, µ) such that T p ≤ 1. Furthermore, T p is a positive operator, since C(X) is a sublattice of L p (X, µ); moreover, if 1 ≤ p < q < +∞, then T p = T q on L q (X, µ).
If X is a compact subset of R d , d ≥ 1, the symbol C 2 (X) stands for the space of all real-valued continuous functions on X which are twice-continuously differentiable on the interior of X and whose partial derivatives of order ≤ 2 can be continuously extended to X. For u ∈ C 2 (X) and i, j = 1, . . . , d, we shall continue to denote by ∂u ∂xi and ∂ 2 u ∂xi∂xj the continuous extensions to X of ∂u ∂xi and ∂ 2 u ∂xi∂xj . If A is a differential operator on C 2 (X), a measure µ ∈ M + 1 (X) is said to be infinitesimally invariant for A if, for every u ∈ C 2 (X), In what follows, we shall also fix some additional notation. Let γ = (γ 1 , . . . , γ d ) ∈ R d , d ≥ 1. If x = (x 1 , . . . , x d ) ∈ R d , x i > 0 for every i = 1, . . . , d, we set If γ i ≥ 0 and x i ≥ 0 for every i = 1, . . . , d, then the power x γ is similarly defined as above assuming 0 0 := 1 by definition.
We also set 0 d := (0, . . . , 0), for every n ≥ 1, n d := (n, . . . , n) and All the results of this paper concern function spaces defined on the d-dimensional In particular, we consider the space and, for M > 0, where · 1 is the norm on R d defined by More generally, given 0 < α ≤ 1, we shall denote by Lip(M, α) the subset of all Hölder continuous functions on Q d with exponent α and constant M , i.e., those Finally, we denote by P m the linear subspace generated by the polynomials on Q d of degree ≤ m.

3.
A generalization of Bernstein-Durrmeyer operators with Jacobi weights on the hypercube. In this section we introduce and study a sequence of positive linear operators acting on weighted L p -spaces. These operators map weighted L p -functions into polynomials on Q d and generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1] (see [8,9,12,13,19,21]).
Although we are mainly interested in the role which they play in the approximation of the semigroups we shall investigate in the subsequent section, it seems that these operators also have an interest on their own as an approximation process for continuous functions as well as for weighted L p -functions.
From now on fix a = (a 1 , . . . , the absolutely continuous measure with respect to the Borel-Lebesgue measure λ d on Q d with density the normalized Jacobi weight Moreover, for every n ≥ 1, consider the operator M n : where, for every n ≥ 1 and Γ(u) (u ≥ 0) being the classical Euler Gamma function. Clearly, the restriction of each M n to C(Q d ) is a Markov operator on C(Q d ).
In order to discuss the main properties of the operators M n , we briefly examine the case d = 1, i.e., the classical Bernstein-Durrmeyer operators with Jacobi weights on [0, 1] (see [12]). where For every n ≥ 1, consider the positive linear operator D n,a,b : In particular, if m, n ≥ 1 and e m (t) = t m , t ∈ [0, 1], it is easy to prove that D n,a,b (e m ) = Γ(n + a + b + 2) Γ(m + n + a + b + 2) (a + 1 + ne 1 ) · · · (a + m + ne 1 ), i.e., D n,a,b (e m ) is a polynomial of degree at most m.
is a polynomial of degree at most m, since it is well-known that the Bernstein operators map polynomials into polynomials of the same degree.
Next we discuss some approximation properties of the operators M n in the spaces (6) is an invariant measure for the operators M n , n ≥ 1, on L 1 (Q d , µ a,b ) and, in particular, for their restrictions to C(Q d ).
This can be easily verified because, for every n ≥ 1 and We also remark that each M n is a contraction from ). By using the convexity of the function |t| p (t ∈ R) and the integral Jensen inequality, we get indeed that, if f ∈ L p (Q d , µ a,b ), then |M n (f )| p ≤ M n (|f | p ) and hence From this remark in particular it follows that each restriction M n | L p (Q d ,µ a,b ) coincides with the extension of M n | C(Q d ) to L p (Q d , µ a,b ) as discussed in Section 2.
Proof. In order to prove statement (a) we shall use the Korovkin type theorem due to Volkov (see, e.g., [5, (4.4.22), p. 245]), from which it follows that i is a Korovkin set in C(Q d ). Therefore, it is enough to verify the approximation formula only for these d + 2 functions.
Obviously, M n (1) = 1 for every n ≥ 1. Taking (12) and (16) into account, for every x = (x 1 , . . . , x d ) ∈ Q d , n ≥ 1 and i = 1, . . . , d, we get Analogously, from (13) it follows that so that lim n→∞ M n (pr 2 i ) = pr 2 i uniformly in Q d , and this completes the proof of (a). As regards statement (b), since C(Q d ) is dense in L p (Q d , µ a,b ) (see, e.g., [11, Lemma 26.2 and Theorem 29.14]) and since, on account of part (a), lim n→∞ M n (f )= f in L p (Q d , µ a,b ) for every f ∈ C(Q d ), it is enough to show that the sequence (M n ) n≥1 is equibounded from L p (Q d , µ a,b ) into L p (Q d , µ a,b ). This, indeed, is a consequence of (18).
Finally, statement (c) is a direct consequence of the previous formulas and [4, Theorem 3.3 and formula (4.3)].

Remark 1.
As already pointed out in the previous proof, , µ a,b ) and, in addition, · p ≤ · ∞ on C(Q d ). Therefore, taking the Weierstrass-Stone theorem into account, the subalgebra of all (restrictions of) polynomials on Q d is dense in L p (Q d , µ a,b ) for the norm · p . Theorem 3.1, part (b), furnishes indeed a constructive method showing how each function f ∈ L p (Q d , µ a,b ) can be approximated by a sequence of polynomials with respect to · p . Now we present some shape preserving properties of the operators M n . First of all, we prove that they preserve the Lipschitz-continuity. To this end it is useful to evaluate the partial derivatives of M n (f ) (f ∈ C(Q d )). We point out, indeed, that, for every n ≥ 1 and for every family (α k ) 0≤k≤n ∈ R n+1 and x ∈ [0, 1], one has . . .
where, for every i = 1, . . . d, the vector v i is given by (2). The next result shows the behaviour of the operators M n on the Lipschitzcontinuous functions (see (3) and (4)). where In particular, where N n := M max 1≤i≤d n n + a i + b i + 2 ≤ M 1 + ω n .
Thanks to Theorem 3.2, it is possible to obtain some further information about the preservation of the Hölder continuity by the operators M n .
To this end, consider the usual modulus of continuity Ω(f, δ), defined, for every bounded function f : Q d → R and δ > 0, by The next result is a direct consequence of Theorem 3.2 and [5, Corollary 6.1.20].
Moreover, if f ∈ Lip(M, α) for some M > 0 and 0 < α ≤ 1 (cf. (5)), then, for every n ≥ 1, We proceed to investigate whether the operators M n preserve convexity. First of all we consider the case d = 1 and, thus, we shall refer to operators (9).
Since f is convex with respect to each variable, ϕ is convex in [0, 1] too and this, together with formula (25) in Proposition 2, completes the proof. 4. The positive semigroups generated by Fleming-Viot type differential operators on the hypercube. After the necessary preliminaries of the previous sections, we finally proceed to look more closely at the degenerate second-order elliptic differential operator defined by for every u ∈ C 2 (Q d ) and Operators similar to (26) have been already studied in several papers (see, e.g., [3], [6,Section 5.8], [7,9,14,17] and the references therein).
The difficulties in studying operators (26) lie in the fact that they degenerate on the boundary of Q d , which is not smooth because of the presence of sides and corners.
In this section, we will show that operator (26) is the pregenerator of a Markov semigroup on C(Q d ) and of a positive contraction semigroup in L p (Q d , µ a,b ); moreover, both these semigroups are obtained as a limits of suitable iterates of the operators M n we studied in Section 3.
First of all we prove that operator (26) is related to operators M n through an asymptotic formula.
For a given x ∈ Q d , we denote by Ψ x ∈ C(Q d ) the function defined by and by d x ∈ C(Q d ) the function defined by where · 2 is the Euclidian norm in R d .
Since, for every y = (y 1 , . . . , y d ) ∈ Q d , and i = 1, . . . , d, and Therefore, considering the measure µ a,b ∈ M + 1 (Q d ) having as density the function defined by (6), then i.e., µ a,b is an infinitesimally invariant measure for the operator A.
Proof. According to [6,Theorem 1.5.2], in order to prove (31), we have to show that, for every i, j = 1, . . . , d, the following conditions hold true: We start by verifying condition (a). For any i = 1, . . . , d and x = (x 1 , . . . , x d ) ∈ Q d , according to (12), so that we get the required assertion.
The next result shows that the operator (A, C 2 (Q d )) pregenerates a Markov semigroup (T (t)) t≥0 on C(Q d ); moreover, a representation formula for such semigroup, involving suitable iterates of the operators M n , is also provided. By means of such a representation formula, we shall deduce some preservation properties of the semigroup itself and we shall describe its asymptotic behaviour.
Theorem 4.2. The differential operator (A, C 2 (Q d )) defined by (26) is closable and its closure (B, D(B)) generates a Markov semigroup (T (t)) t≥0 on C(Q d ) such that, if f ∈ C(Q d ), t ≥ 0 and (k(n)) n≥1 is a sequence of positive integers satisfying lim n→∞ k(n)/n = t, then where each M k(n) n denotes the iterate of M n of order k(n). Moreover, P ∞ := The next result shows the semigroup (T (t)) t≥0 can be extended to L p (Q d , µ a,b ), (p ∈ [1, +∞[), where µ a,b is the Borel probability measure introduced in Theorem 4.2. Moreover, a representation formula similar to (33), as well as the asymptotic behaviour of the extended semigroup (see (34)), can be established.
Theorem 4.4. For every 1 ≤ p < +∞, the semigroup (T (t)) t≥0 on C(Q d ) (see Theorem 4.2) extends to a unique positive contraction semigroup (T p (t)) t≥0 on L p (Q d , µ a,b ), whose generator is an extension of (B, D(B)) to L p (Q d , µ a,b ) and P ∞ is a core for it.
Moreover, if f ∈ L p (Q d , µ a,b ) and (k(n)) n≥1 is a sequence of positive integers satisfying lim n→∞ k(n)/n = t, then Finally, if f ∈ L p (Q d , µ a,b ) and n ≥ 1, in L p (Q d , µ a,b ).
Proof. For every t ≥ 0, denote by T p (t) the unique extension of T (t) to L p (Q d , µ a,b ) as explained in Section 2. The operator T p (t) is a positive linear contraction on L p (Q d , µ a,b ). Since C(Q d ) is dense in L p (Q d , µ a,b ) and · p ≤ · ∞ on C(Q d ), it is easily seen that (T p (t)) t≥0 is a strongly continuous semigroup on L p (Q d , µ a,b ), its generator is an extension of (B, D(B)) to L p (Q d , µ a,b ) and P ∞ is a core for it, by virtue of Theorem 4.2.
Finally, formulas (39) and (40) can be obtained from (33) and (34), taking again into account that C(Q d ) is dense in L p (Q d , µ a,b ) and that, as remarked before Theorem 3.1, each restriction M n | L p (Q d ,µ a,b ) coincides with the continuous extension of M n | C(Q d ) to L p (Q d , µ a,b ).
Aknowledgements. The authors thank the referee for the careful reading of the manuscript and for his/her useful remarks.