CONTINUOUS SELECTIONS OF BOREL MEASURES, POSITIVE OPERATORS AND DEGENERATE EVOLUTION PROBLEMS

. In this paper we continue the study of a sequence of positive linear operators which we have introduced in [9] and which are associated with a continuous selection of Borel measures on the unit interval. We show that the iterates of these operators converge to a Markov semigroup whose generator is a degenerate second-order elliptic diﬀerential operator on the unit interval. Some qualitative properties of the semigroup, or equivalently, of the solutions of the corresponding degenerate evolution problems, are also investigated.


INTRODUCTION
In the previous paper [9] we have undertaken the study of a new sequence (C n ) n≥1 of positive linear operators acting on the space of Lebesgue functions on the unit interval.We have investigated their approximation and shape preserving properties, presenting some estimates of the rate of convergence by means of suitable moduli of smoothness.
Under additional hypotheses on α, we show that the operator A defined on the subspace is the generator of a Markov semigroup (T (t)) t≥0 on C([0, 1]).Furthermore we prove that for every t ≥ 0, for every sequence (k(n)) n≥1 of positive integers such that k(n)/n → t (n → ∞) and for every f ∈ C([0, 1]), (2) lim uniformly on [0, 1].Thanks to formula (2), we obtain some qualitative properties of the semigroup (T (t)) t≥0 and hence of the solutions of the diffusion equations associated with the operator (A, D M (A)).
In the last part of the paper we discuss a converse problem and we show that, given a differential operator of the form (1) generating a Markov semigroup there exists a continuous selection of Borel measures whose corresponding operators C n represent the semigroup by means of their iterates.

THE OPERATORS C N
In this section we recall the definition and the main properties of the sequence of the operators C n , introduced in [9], whose iterates will be studied in the subsequent sections as we quoted in the Introduction.
As usual, we shall denote by C([0, 1]) the space of all real valued continuous functions on [0, 1] endowed with the sup-norm • ∞ .
A continuous selection of probability Borel measures on [0, 1] is a family (µ x ) 0≤x≤1 of probability Borel measures on [0,1] such that for every f ∈ C([0,1]) As in [9] we shall fix a continuous selection (µ x ) 0≤x≤1 of probability Borel measures on [0, 1] satisfying the following additional assumption: Let (a n ) n≥1 and (b n ) n≥1 be two real sequences such that, for every n ≥ 1, 0 ≤ a n < b n ≤ 1.For every n ≥ 1 consider the positive linear operator where µ n x denotes the tensor product of n copies of µ x .The operator C n is well-defined and maps the space It is worth pointing out that to a given continuous selection (µ x ) 0≤x≤1 of probability Borel measures on [0, 1] it is possible to associate another sequence of positive linear operators, namely the Bernstein-Schnabl operators, which are defined as for every n ≥ 1, f ∈ C([0, 1]) and 0 ≤ x ≤ 1.These operators have been extensively studied (see, e.g., [1], [7] and [10]).
There is a close relationship between the operators C n and B n .In [9, Remark 1.3] we showed that, for a given f ∈ L 1 ([0, 1]), considering the function then, for every n ≥ 1, the operator C n can be written as where the mapping ) is defined by Another formula which relates the operators C n to the operators B n is given in [9,Remark 1.4].It has been useful both for investigating the behaviour of the operators C n on convex functions and for suggesting a possible generalization of our results replacing the interval [0, 1] with an arbitrary interval (not necessarily bounded) or with a convex subset of some locally convex space.
We recall it here: where µ n denote the image measure of the Borel-Lebesgue measure λ 1 under the mapping Some examples of operators C n can be found in [9,Examples 1.5].In particular we point out that, if , then they can be rewritten as and, by taking a n = 0, b n = 1 for each n ≥ 1, they turn into the well-known Kantorovich operators ( [21]; [7, pp. 333-335]).
Another example of operators C n can be obtained by considering the continuous selection (ν λ x ) 0≤x≤1 of the probability Borel measures ν x defined by , where the measure µ x is given by µ In [9, Section 2] we investigated the approximation properties of the operators C n in the space C([0, 1]) and, in some cases, in the space L p ([0, 1]).We also presented several estimates of the rate of convergence by means of suitable moduli of smoothness.Shape preserving properties of these operators were also discussed (see [9,Section 3]).In particular we proved that each operator C n preserves both the class of Hölder continuous functions and the one of convex continuous functions.
We recall here some of these results which will be useful in Section 3. The first one shows that each operator C n preserves the class of Hölder continuous functions.The next one gives information about the preservation of convex functions by the operators C n .For more details see [9,Section 3].
For given M > 0 and 0 Moreover, for any f ∈ C([0, 1]) the symbol ω(f, •) stands for the usual modulus of smoothness of the first order which is defined by Theorem 1.2.Consider the operators C n associated with the continuous selection of probability Borel measures (µ x ) 0≤x≤1 defined by (1.3).Suppose that: (c 1 ) The operator T , given by (1.1), maps continuous convex functions into (continuous) convex functions;

AN ASYMPTOTIC FORMULA
In this section we establish an asymptotic formula for the sequence of the operators C n defined by (1.3) with respect to the uniform norm.
The first result about asymptotic formulae is due to Voronovskaja [24].It states that, considering the sequence (B n ) n≥1 of the classical Bernstein operators defined on the unit interval [17] (see also, e.g., [7, pp. 218 uniformly with respect to x ∈ [0, 1].Such a result shows that for Bernstein operators the convergence cannot be too fast, even if the approximating function is smooth. In order to show an asymptotic formula for the sequence (C n ) n≥1 , we shall use a generalization of Voronovskaja's result due to Mamedov [22] (see also [5,Theorem 1]), which holds for an arbitrary sequence (L n ) n≥1 of positive linear operators acting on C([0, 1]) and which is stated below.
As usual, for every x ∈ [0, 1] the symbol ψ x stands for the function Theorem 2.1.Consider a sequence (L n ) n≥1 of positive linear operators from C([0, 1]) into itself and let α, β and γ be functions defined on for some even positive integer q ≥ 4.
For a proof we refer the reader to [5, Theorem 1] where a more general result for not necessarily compact interval is presented.Now we are in a position to state and prove the main result of this section.
Theorem 2.2.Consider the sequence (C n ) n≥1 of the operators C n defined by (1.3) and assume that the sequence uniformly with respect to x ∈ [0, 1], where T (e 2 )(x) = Proof.We shall apply Theorem 2.1 with q = 4. Observe that condition (i), (ii) and (iii) of the above result are satisfied with γ = 0, 3), (2.4)]).In order to verify condition (iv), we shall explicitly determine the function C n (ψ 4 x ).

MARKOV SEMIGROUPS ASSOCIATED WITH A CLASS OF ONE-DIMENSIONAL DIFFUSION EQUATIONS AND THEIR APPROXIMATION
The main aim of this section is to discuss some one-dimensional diffusion equations on the unit interval by means of the theory of C 0 -semigroups of operators and to represent the relevant solutions (or the corresponding semigroups) by iterates of the operators C n .For more details about the theory of C 0 -semigroups we refer the reader to [16], [20], [23].
From now on we suppose that the family (µ x ) 0≤x≤1 satisfies the following further condition (3.1) Suppose in addition that α is differentiable at 0 and 1 and where Suppose that there exists d := lim n→∞ (a n +b n ) > 0 and consider the differential operator A defined by setting for every u ∈ C 2 (]0, 1[) and continue to denote with A : D M (A) −→ C([0, 1]) the operator defined by setting for every u ∈ D M (A) and x ∈ [0, 1] We recall that a core for a linear operator A : Proof.We introduce the auxiliary operator ( 1) defined on the domain D(B) := D M (A).Thus, B = λA and Then, (B, D(B)) is the generator of a Feller semigroup on C([0, 1]) (see [13, pp. 120-121]).Therefore, since A = λB, the result follows by a well-known result about the generation of the multiplicative perturbation of generators (see [7,Theorem 1.6.11]).Since 1 ∈ D M (A) and A1 = 0, the semigroup is a Markov semigroup.Finally in order to prove that C 2 ([0, 1]) is a core for (A, D M (A)), we use Theorem 2.3 of [13] applied to the operator B defined by (1), obtaining that ) is a core for (B, D(B)) and hence for (A, D(A)).
At this point we are in a position to obtain a result about the approximation of the semigroup above.
The p-th power (p ≥ 1) of the operator ) is defined as x u (x) for every x ∈ [0, 1] and hence, by Theorem 2.2, We finally need the following general result which can be obtained by Trotter's theorem on the approximation of semigroups (see, e.g., [ Theorem 3.2.Let (T (t)) t≥0 be a strongly continuous semigroup on a Banach space E with generator (A, D(A)).Consider a sequence (L n ) n≥1 of bounded linear operators on E and assume that (i) There exists M ≥ 1 and ω ∈ R such that Then for every t ≥ 0, for every sequence By using the representation formula (3.8), it is possible to obtain some properties of the semigroup from the preservation properties of the operators C n (see Theorems 1.1 and 1.2).Proposition 3.4.Under the same assumptions of Theorem 3.3, considering the Markov semigroup (T (t)) t≥0 generated by the operator (A, D M (A)) defined by (3.5), the following statements hold true: (1) If the operator T , defined by (1.1), maps Lip 1 1 into Lip 1 1, then (2) If the operator T , given by (1.1), satisfies the hypotheses of Theorem 1.2, then for every t ≥ 0, T (t) maps continuous convex functions into (continuous) convex functions.
It is possible to obtain a further property which holds for the semigroup (T (t)) t≥0 .We need the following lemma.Then T (t)ϕ = aT (t)e 1 + b and so ϕ ≤ T (t)ϕ.Accordingly By continuity the inequality f ≤ T (t)f can be extended on the whole interval [0, 1] and so the result follows.
Remark 3.7.From the general theory of C 0 -semigroups of operators and from Theorem 3.1 it follows that for every u 0 ∈ D M (A) the following initialboundary differential problem of diffusion type (3.9) has a unique solution given by u(x, t) = T (t)u 0 (x) (0 ≤ x ≤ 1 , t ≥ 0).
Moreover |u(x, t)| ≤ u 0 (0 ≤ x ≤ 1 , t ≥ 0) and u(•, t) is positive for every t ≥ 0 provided that u 0 ≥ 0. Furthermore, by Theorem 3.3, the solutions can be approximate by means of iterates of the operators C n .Finally, Propositions 3.4 and 3.6 give some qualitative properties of them as well.
We point out that, if d ≤ 1, then one can consider the sequences a n = 0 and b n = d (n ≥ 1) instead of the ones given by (3.11).
We also remark that, under the assumptions of Theorem 3.8, the operator T corresponding to the selection (3.10) via formula (1.1) is given by Therefore, if such an operator T maps Lip 1 1 into Lip 1 1 and/or if it satisfies conditions (c 1 ) and (c 2 ), then the semigroup generated by (A, D M (A)) maps Lip M α into itself (M > 0, 0 < α ≤ 1) and/or continuous convex functions into convex functions.
Moreover, if d = 2, property (a) of Proposition 3.6 holds true as well.

2
and the symbol ⊗ denotes the tensor product among measures.Then each operator C n maps continuous convex functions into (continuous) convex functions.Examples of selections of measures satisfying (c 1 ) and (c 2 ) can be found in [10, Examples 2.7].

Theorem 3 . 3 .
6) and (3.7) and from Theorems 3.1 and 3.2 the next result immediately follows.Under the same assumptions of Theorem 3.1, considering the Markov semigroup (T (t)) t≥0 generated by (A, D M (A)), for every t ≥ 0, for every sequence (k(n)) n≥1 of positive integers such that k(n)/n → t (n → ∞) and for every f ∈ C([0, 1