Note on A. Barbour's paper on Stein's method for diffusion approximations

In (Barbour, 1990) foundations for diffusion approximation via Stein's method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein's method. A semigroup argument is used therein to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in (Barbour, 1990), the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0,1] growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of (Barbour, 1990) hold true.


Introduction
In [2] a claim is made that the semigroup defined by (2.4) thereof is strongly continuous on space L defined on page 299 thereof. We prove that this is not the case.
Nevertheless, we show that the only assertion of the paper following from the aforementioned assumption of strong continuity, namely the claim that (2.20) solves the Stein equation (2.1), remains true. This may be proved by adapting the proof of [5, Proposition 9, p. 9] and noting that in the case of interest in [2], the point-wise continuity of the semigroup is sufficient. It then follows that all the other results of [2] hold true.
In Section 2 we recall the relevant definitions and notation from [2]. In Section 3 we give a counterexample to the strong continuity of the semigroup. In Section 4 we provide a proof of the fact that the function (2.20) of [2] does actually solve the Stein equation. We do this by following the steps of the proof of [5, Proposition 9, p. 9] and proving each of the assertions therein for the semigroup of interest by hand.
sup {h: h =1} |B[h, ..., h]|. We will also often write D 2 f (w)[h (2) Let: We define: for any f ∈ L for which the expressions exist and where, for 2 n ≤ k < 2 n+1 : We also define a semigroup acting on L: For any g ∈ M with Eg(Z) = 0, the Stein equation is given by: The idea of Stein's method applied in [2] is to find a bound on EAf (X), where f is a solution to this equation, in order bound |Eg(X) − Eg(Z)|, for some stochastic process X on [0, 1].

Counterexample to strong continuity
It is well known that the Ornstein-Uhlenbeck semigroup is not strongly continuous on the space C b (R), see [3]. More generally, given a separable Hilbert space H, in [8] it is noted that this semigroup is also not strongly continuous on the space C b,k of all continuous functions ψ : H → R such that x → ψ(x)/(1 + |x| k ) is uniformly continuous and sup x∈H 1+|x| k < ∞. Following these two results, in this section we shall show that the semigroup T u defined by (2.1) is not strongly continuous on the Banach space L by constructing an explicit counterexample. Proof. Consider f ∈ L defined by: Note that: Furthermore, given > 0, consider R > 0 such that P( Z > R) < . Fix δ > 0, such that for any a, b ∈ R: |a − b| < δ ⇒ | sin(a) − sin(b)| < . Now, for any u such that σ(u)R < δ and for every w ∈ D, we have: Therefore: Therefore:

Solution to the Stein equation
We first show that the function, which in Lemma 4.3 is shown to solve the Stein equation, exists and belongs to the domain of A.
Proof. Note that: 1]. This follows from the fact that: Now, we note that, as a consequence of (4.1), we have: for some constant C. Since L is complete, this guarantees the existence of φ(g).
Note on A. Barbour's paper on Stein's method for diffusion approximations As noted in (2.23) and (2.24) of [2], dominated convergence may be used, because of (4.2) to obtain that: and, as a consequence, that φ(g) ∈ M . This is enough to conclude that φ(g) belongs to the domain of A by the observation directly above the formulation of A labelled as (2.9) in [2].

Remark 4.2.
The argument of (2.23) and (2.24) in [2] also readily gives that for any g ∈ M and t > 0: We now prove that observation (2.19) of [2] is true for all g ∈ M :  Proof. We will follow the steps of the proof of Proposition 1.5 on p. 9 of [5]. Observe that for all w ∈ D[0, 1] and h > 0:  Similarly: Therefore, as h → 0, the right-hand side of (4.5) converges to T t g − g, which finishes the proof.
Proof. We note that for any h > 0 and for any f ∈ M : We also note that for any w ∈ D[0, 1], g ∈ M and some constant K 1 depending only on f : as noted on page 300 of [2]. Therefore, we can apply dominated convergence to obtain: = 0 again, by dominated convergence. It can be applied because of (4.7) and the observation that for any z ∈ D[0, 1] and h ∈ [0, 1]: and so for any h ∈ [0, 1], Af (we −s+h + σ(s − h)Z) − Af (we −s + σ(s)Z) is bounded by a random variable with finite expectation.
Thus, for all w ∈ D[0, 1] and s > 0: and so, by the Fundamental Theorem of Calculus: Now, we take t → ∞. Let Z be an independent copy of Z. We apply dominated convergence, which is allowed because of (4.2) and the following bound for ϕ t (w) = t 0 T u g(w)du: where the second inequality follows again by dominated convergence applied because of (4.2) in order to exchange integration and differentiation in a way similar to (4.3). Then, we obtain: Now, by Lemma 4.1, we can divide both sides by r and take r → 0 to obtain: