On the Relation of One-Dimensional Diffusions on Natural Scale and Their Speed Measures

It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.

1. On the Relation of Diffusions and their Speed Measures 1.1.Introduction.It is well-known (see e.g.[3,11]) that the law of a one-dimensional regular continuous strong Markov process on natural scale (called diffusion in this short section) is fully characterized by its speed measure.Among other things, Stone [22] proved that diffusions depend continuously on their speed measures and Brooks and Chacon [4] established the converse direction for real-valued diffusions, i.e. they proved a continuous dependence of the speed measures on the diffusions.
The main contribution of this paper is a converse to Stone's theorem for general diffusions.The real-valued and the general case distinguish in two important points: For real-valued diffusions there is no issue with the boundary behavior and the corresponding speed measures are locally finite, which in particular means they can be endowed with the vague topology.To treat the general case we use a new method of proof, which is quite different to those of Brooks and Chacon.Below we comment in more detail on the methods and compare them to each other.
As a second contribution, we consider the relation of diffusions and their speed measures from a topological perspective.Namely, for suitable topologies, we deduce a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.As an application of the homeomorphic relation we discuss properties of certain subsets of the set of diffusions without absorbing boundaries, namely those with the Feller-Dynkin property and Itô diffusions with open state space.More precisely, we show that both of these subsets are dense Borel sets which are neither closed nor open.
The remainder of this paper is structured as follows.In Section 1.2 we introduce our setting and recall the most important terminologies.Thereafter, in Section 1.3 we present our results and we comment on related literature.Finally, in Section 2 we present the proof of our main theorem, i.e. the converse to Stone's theorem.1.2.Setting.We work with the canonical setting for diffusions as introduced in [19, Section V.25].A quite complete treatment of the theory is given in the monograph of Itô and McKean [11].Shorter introductions can also be found in the monographs [3,18].Let J ⊂ R be a finite or infinite, closed, open or half-open interval and define Ω to be the space of continuous functions R + → J endowed with the local uniform topology.The coordinate process on Ω is denoted by X, i.e.X t (ω) = ω(t) for t ∈ R + and ω ∈ Ω.The Borel σ-field on Ω is given by F σ(X s , s ≥ 0).For any time t ∈ R + we also set F t σ(X s , s ≤ t) and we define the shift operator θ t : Ω → Ω by (θ t ω)(s) = ω(t + s) for s, t ∈ R + .Let M 1 be the set of probability measures on (Ω, F ) endowed with the weak topology.
We call (J ∋ x → P x ∈ M 1 ) a (canonical) diffusion, if x → P x (A) is measurable for all A ∈ F ,1 P x (X 0 = x) = 1 for all x ∈ J, and for any (F t+ ) t≥0 -stopping time τ and any x ∈ J, P Xτ is the regular conditional P x -distribution of θ τ X on {τ < ∞}.The final part is the strong Markov property.A diffusion (x → P x ) is called regular if for all x ∈ J • and y ∈ J where γ y inf(s ≥ 0 : X s = y), and it is called completely regular2 if (1.1) holds for all x, y ∈ J. Clearly, regularity and complete regularity are equivalent for open J.In case J is closed or half-open, complete regularity means that closed boundaries are reflecting (i.e.neither exit nor absorbing in the language from [3,Section 16.7]).We say that a regular diffusion (x → P x ) is on natural scale if for all a, b, x ∈ J with a < x < b we have Any regular diffusion can be brought to natural scale via a homeomorphic space transformation ([3, Proposition 16.34]).Let (x → P x ) be a regular diffusion on natural scale.According to [3,Theorem 16.36], there exists a unique locally finite measure m on (J • , B(J • )) such that for any a < b with [a, b] ⊂ J • we have where G (a,b) is the Green function as defined in [3,Eq. 16.35].Furthermore, by [3,Theorem 16.47], if the left boundary point l is in J, then m({l}) can be defined such that for any b ∈ J where G [l,b) is the symmetrized Green function as defined in [3,Eq. 16.46].A similar statement holds for right boundary points which are in J.In this manner we get a measure m on (J, B(J)) which is called the speed measure associated to the regular diffusion (x → P x ).The speed measure is locally finite if and only if the corresponding diffusion is completely regular.Within the class of regular diffusions the speed measure determines a diffusion uniquely ([3, Corollary 16.73]).
1.3.Main results.In the following let l be the left boundary point of J and let r be the right boundary point.Take a sequence m 0 , m 1 , m 2 , . . . of speed measures on J.
Definition 1.1.We say that the sequence m 1 , m 2 , . . .converges in the speed measure sense to m 0 , which we denote by m n ⇒ m 0 , if the following hold: . .are locally finite, then m n ⇒ m 0 if and only if m n → m 0 vaguely.
For each n ∈ Z + let (x → P n x ) be the regular diffusion on natural scale with speed measure m n .Moreover, take a sequence x 0 , x 1 , x 2 , • • • ∈ J.Among other things, Stone [22] proved the following theorem.

Theorem 1.3 ([22]
).If m n ⇒ m 0 and x n → x 0 , then P n x n → P 0 x 0 weakly.Corollary 1.4.If (x → P x ) is a regular diffusion on natural scale, then x → P x is a continuous function from J into M 1 , i.e. (x → P x ) ∈ C(J, M 1 ).
To explain the main idea behind Theorem 1.3, let us prove it for the case J = [0, ∞) and x n ≡ x 0 .Of course, the main steps of the proof are borrowed from [22].
Proof for Theorem 1.3 in case J = [0, ∞) and x n ≡ x 0 .The key idea is to use the Itô-McKean construction of a regular diffusion on natural scale as a time change of Brownian motion.Let B be a Brownian motion starting in x 0 which is reflected at the origin, denote its local time process by {L(t, y) : t, y ∈ R + } and set The following discussion should be read up to a null set which depends on standard properties of the local time process (see e.g.[9, Section 2.8]).As explained in the discussion below [3,Definition 16.55], t → S n t is finite, continuous and increasing.If t < γ 0 (B), then x → L(t, x) is continuous and supported on a compact subsets of J • such that part (a) of Definition 1.1 yields that T n t → T 0 t .By the Corollary on p. 640 in [22] and the representation of the local time process for reflected Brownian motion as discussed above [3,Definition 16.55], if t > γ 0 (B) then the map x → L(t, x) has the same properties as f in part (b) of Definition 1.1, which then yields that T n t → T 0 t .In summary, m n ⇒ m 0 implies that T n t → T 0 t for all t = γ 0 (B).We conclude that S n t → S 0 t for all t ∈ R + , see [10,Lemma 1.1.1].As S 0 , S 1 , S 2 , . . .are increasing and continuous, [12,Theorem VI.2.15] further yields that S n → S 0 uniformly on compact time intervals.Hence, we also have that a.s.B S n → B S 0 uniformly on compact time intervals.Since B S n has law P n x 0 by [3,Theorem 16.56], we get that P n x 0 → P 0 x 0 weakly.
The following theorem is our main result.It can be viewed as a converse to Theorem 1.3.Its proof is given in Section 2 below.Theorem 1.6.If P n x → P 0 x weakly for all x ∈ J, then m n ⇒ m 0 .
Combining Corollary 1.5 and Theorem 1.6 gives us the following: Corollary 1.7.The following are equivalent: x → P 0 x weakly for all x ∈ J.It is interesting to note that on the set of regular diffusions on natural scale the sequential topologies of pointwise and local uniform convergence coincide.
Comments on related literature.For the real-valued case, i.e.J = R, Theorem 1.6 was proved in [4].Theorem 1.6 seems to be new in its generality.The result provides the complete picture for general state spaces and arbitrary boundary behavior.Further, our method of proof is new and quite different to those from [4].As the method from [4] heavily relies on the concept of vague convergence, it seems not to work for Theorem 1.6, so that new ideas are necessary.We now outline the main steps of the argument from [4] and compare it to ours.In the following we take J = R.Note that m 0 , m 1 , m 2 , . . .are locally finite and that m n ⇒ m 0 if and only if m n → m 0 vaguely.The first step is to prove that the sequence m 1 , m 2 , . . . is uniformly bounded on compact subsets of the reals, which shows that {m n : n ∈ N} is vaguely relatively compact ([17, Proposition 3.16]).In [4] this is done by a contradiction argument.To conclude m n → m 0 vaguely it suffices to show that any vague accumulation point q of m 1 , m 2 , . . .coincides with m 0 .Assume that m n → q vaguely.First, q is shown to be positive on any compact subset of the reals, which means it is a speed measure.In [4] this is again done by a contradiction argument.Second, let B be a Brownian motion started at x 0 , denote its local time process by {L(t, y) : t ∈ R + , y ∈ R} and let S be the inverse of t → L(t, y)q(dy).By the argument outlined in the proof of Theorem 1. 3, P n x 0 converges weakly to the law of B S and consequently, by the uniqueness of the limit, B S has law P 0 x 0 .Now, q = m 0 follows from the definition of the speed measure via the canonical form (i.e. as time change of Brownian motion, see [14,Theorem 33.9]).It follows that m n → m 0 vaguely.
From a technical point of view this proof heavily relies on the Itô-McKean construction of a diffusion as a time change of Brownian motion, properties of the Brownian local time and the characterization of the speed measure via the canonical form.Our proof for Theorem 1.6 uses non of these tools.Instead, we use the definition of the speed measure from the monographs [3,18], a uniform second moment bound for exit times and the continuous mapping theorem.
A topological point of view.In the remainder of this section we look at the relation of diffusions and their speed measures from a topological point of view.Let S be the set of all locally finite speed measures and let D be the set of all completely regular diffusions on natural scale.Our goal is to establish a homeomorphic relation between S and D. We endow S with the vague topology, which turns it into a metrizible space.Thanks to Corollary 1.4, we can treat D as a subspace of C(J, M 1 ) endowed with the local uniform topology, which renders it into a metrizible space.
Remark 1.8.It would also be natural to consider regular diffusions as elements of the product space M J 1 .However, M J 1 is not first countable ([21, Theorem 7.1.7])and hence we cannot a priori3 check continuity via sequential continuity.The space C(J, M 1 ) on the other hand is metrizible and therefore also sequential.
Finally, let Φ : D → S be the function which maps a completely regular diffusion on natural scale to its speed measure.Corollary 1.7 gives us the following result, which we call a theorem rather than a corollary, since we think it deserves this name.
Related to Remark 1.8, Corollary 1.7 also shows the following: Corollary 1.10.In case D is seen as a subspace of M J 1 endowed with the product weak topology, then Φ is a sequential homeomorphism, i.e. a sequentially continuous bijection with sequentially continuous inverse. 4n the remainder of this section we apply Theorem 1.9 to study properties of certain subsets of D.More precisely, we consider the set of completely regular diffusions with the Feller-Dynkin property and the set of Itô diffusions with open state space.
On the set of diffusions with the Feller-Dynkin property.Let C 0 (J) be the set of all continuous functions J → R which are vanishing at infinity.We say that a diffusion (x → P x ) has the Feller-Dynkin property if (x → E x [f (X t )]) ∈ C 0 (J) for all f ∈ C 0 (J) and t > 0. Let O be the set of all completely regular diffusions with the Feller-Dynkin property.Proof.According to [7, Theorem 1.1], a regular diffusion with speed measure m has the Feller-Dynkin property if and only if any infinite boundary point of J is natural, i.e. 5 Hence, by Theorem 1.9, the same is true for O. Finally, the claim that O is dense in D follows from Theorem 1.9 and the fact that any locally finite measure can be approximated in the vague topology by a sequence of discrete measures ([2, Theorem 30.4]).To be more precise, let m 0 ∈ S, m ∈ O and let n 1 , n 2 , . . .be a sequence of discrete measures such that n n → m 0 vaguely.Then, m n (dx) n n (dx) + 1 n m(dx) is the speed measure of a diffusion from O and m n → m 0 vaguely.This shows that Φ(O) is dense in S and hence, by Theorem 1.9, O is dense in D. The proof is complete.
On the set of Itô diffusions.Let us assume that J = (l, r) is open.We call a completely regular diffusion an Itô diffusion if its speed measure m is absolutely continuous w.r.t. the Lebesgue measure, i.e. m(dx) = f (x)dx for some f ∈ L 1 loc (J).Denote the set of Itô diffusions by I.The non-closedness of the set of real-valued Itô diffusions with drift was already observed in [20].Corollary 1.12 provides a refined picture for the set of Itô diffusions without drift.
Proof of Corollary 1.12.Let A be the set of speed measures which are absolutely continuous w.r.t. the Lebesgue measure and let R be the Polish space of locally finite measures on (J, B(J)) with the vague topology.Furthermore, let S + (J) be the set of all f ∈ L 1 loc (J) such that b a f (x)dx > 0 for all a, b ∈ J with a < b and We endow L 1 loc (J) with the local L 1 topology which renders it into a Polish space.It is not hard to see that S + (J) ∈ B(L 1 loc (J)).Now, consider the map ψ : As ψ is a continuous injection from a Borel subset of a Polish space into a Polish space, [6, Theorem 8.2.7] yields that ψ(S + (J)) ∈ B(R).As ψ(S + (J)) = A by [3, Proposition 16.43, Theorem 16.56],A is a Borel subset of S. It is not hard to see that A is neither closed nor open. 6We conclude from Theorem 1.9 that I has the same properties, i.e. it is a Borel set but neither closed nor open.Finally, let us explain that I is dense in D. By Theorem 1.9, it suffices to show that A is dense in S. It is clear that any discrete measure can be approximated in the vague topology by a sequence of absolutely continuous measures. 7Thus, as the set of discrete measures is dense in R, it follows that A is dense in S. The proof is complete.
The remainder of this paper is devoted to the proof of Theorem 1.6.

Proof of Theorem 1.6
In this section we assume that P n x → P 0 x for all x ∈ J. Our goal is to show that m n ⇒ m 0 .The proof for this is split into two parts.In the first we establish property (a) from Definition 1.1 and in the second we deal with the properties (b) and (c).

2.1.
Proof for convergence on the interior.In this section we prove property (a) from Definition 1.1.Recall [18, Corollary VII.3.8]:For each n ∈ Z + , all a < b such that [a, b] ⊂ J • and every f ∈ C c (J) the following holds: where G (a,b) denotes the Green function as given in [3,Eq. 16.35].Recall our notation J • = (l, r).Lemma 2.1.Suppose that P n x 0 → P 0 x 0 weakly for some x 0 ∈ J • .Then, there exist two sets A ⊂ (l, x 0 ) and B ⊂ (x 0 , r) with countable complements (in (l, x 0 ) and (x 0 , r), respectively) such that for all a ∈ A, b ∈ B and f as n → ∞.
The test functions G (a,b) (x, •)f are sufficient to characterize vague convergence of locally finite measures on J • .Thus, by virtue of the r.h.s. in (2.1), Lemma 2.1 implies Lemma 2.2.For any y ∈ J • the functions σ ± y are upper semi-continuous and the functions τ ± y are lower semi-continuous.Proof.The claim is implied by [ For every s ∈ Q ∩ [0, t] and ω, ω ′ ∈ Ω we also have inf r∈Q,r≤t Taking the infimum over s and using symmetry yields that inf r∈Q,r≤t Consequently, ω → inf s∈Q,s≤t d y (ω(s)) is continuous, {τ + y ≤ t} is closed and τ + y is lower semi-continuous.
Let A be the set of all a ∈ (l, x 0 ) such that τ − a is P 0 x 0 -a.s.continuous and let B be the set of all b ∈ (x 0 , r) such that τ + b is P 0 x 0 -a.s.continuous.Lemma 2.3.The complements of A and B are both at most countable.
Proof.We restrict our attention to the set B. The claim for A follows the same way.Note that σ + b = τ + b+ for all b ∈ J • .It is well-known ([8, Lemma 7.7 on p. 131]) that the set {b > x 0 : P 0 x 0 (τ + b = τ + b+ ) > 0} is a most countable.Consequently, by virtue of Lemma 2.2, the complement of B is at most countable.
It remains to prove (2.2).The key step is the following lemma.
To prove this we first establish two preliminary results.
Lemma 2.5.For all n ∈ Z + , t > 0 and a ≤ x, x 0 ≤ b we have Let us first take a ≤ x ≤ x 0 .The strong Markov property yields that This yields the claimed inequality for a ≤ x ≤ x 0 .For x 0 ≤ x ≤ b we get from the same computation that The proof is complete.Lemma 2.6.For all t > 0 and a < b we have Proof.By Lemma 2.2, σ ± a are upper semi-continuous.Hence, by [1,Theorem 15.5], the maps M 1 ∋ P → P (σ ± a ≥ t) ∈ [0, 1] are also upper semi-continuous.As we assume that P n x 0 → P 0 x 0 weakly, the set {P n x 0 : n ∈ Z + } is compact in M 1 .Now, since upper semicontinuous functions attain a maximum value on a compact set ([1, Theorem 2.43]), there exists an N ± a ∈ Z + such that sup As, thanks to [5, Theorem 1.1], regular diffusions hit points arbitrarily fast with positive probability, we have P N ± a x 0 (σ ± a ≥ t) < 1 and (2.3) follows from Lemma 2.5.
Proof of Lemma 2.4.We fix t > 0. Using the Markov property, for every m ∈ Z + we get Thus, by induction we obtain for every m ∈ Z + that As the final term is finite by Lemma 2.6 and independent of n, the proof is complete.
We are in the position to finish the proof of Lemma 2.1.Take f ∈ C c (J • ), a ∈ A and b ∈ B. First of all, as a < x 0 < b, for every n ∈ Z + we have P n x 0 -a.s.
Then, by Lemma 2.4, we have Thus, the family f (ω(s))ds is P 0 x 0 -a.s.continuous.Consequently, the continuous mapping theorem yields (2.2).The proof of Lemma 2.1 is complete.

2.2.
Proof of convergence up to the boundaries.We now prove property (b) from Definition 1.1.Assume that l ∈ J.In the following we distinguish between the cases where (x → P 0 x ) is absorbing or reflecting8 at the boundary point l.
The absorbing case.Assume that l is an absorbing boundary point of (x → P 0 x ).This case is captured by the speed measure via the property m 0 ({l}) = ∞.
where G [l,z) is the symmetrized Green function as given in [3,Eq. 16.46].Now, we obtain This completes the proof for the absorbing case.
The reflecting case.Next, we assume that l is a reflecting boundary point of (x → P 0 x ).Take z ∈ J • and t > 0. Using Lemma 2.2, the Portmanteau theorem and [5, Theorem 1.1], we get lim inf n→∞ P n l (σ + z < t) ≥ P 0 l (σ + z < t) > 0. Recall the Urysohn property: A sequence in a metrizible space converge to a limit L if and only if any of its subsequences contains a further subsequence which converges to L. Thus, to prove f (x)m n (dx) → f (x)m 0 (dx) we have to prove that any subsequence of f (x)m 1 (dx), f (x)m 2 (dx), . . .contains a further subsequence which converges to f (x)m 0 (dx).Let k(1), k(2), . . .be an arbitrary subsequence of 1, 2, . ...Then, lim inf Thus, there exists a subsequence m(1), m(2), . . . of k(1), k(2), . . .such that P m(n) l (σ + z < t) > 0, ∀n ∈ N. Let us take this subsequence and prove part (b) of the definition of m m(n) ⇒ m 0 .Once we have done this, we can conclude part (b) of m n ⇒ m 0 .To simplify our notation, we assume that P n l (σ + z < t) > 0 for every n ∈ N. Consequently, for each n ∈ Z + the point l is a reflecting boundary of (x → P n x ) and, recalling again [5, Theorem 1.1], we have P n l (σ + c < t) > 0 for all n ∈ Z + and c ∈ J • .(2.5)

Corollary 1 . 11 .
If J is bounded, then O = D and, in particular, O is clopen in D. Conversely, if J is unbounded, then O is a dense Borel subset of D and it is neither closed nor open in D.

Corollary 1 . 12 .
I is a dense Borel subset of D and it is neither closed nor open in D.