Invariance principle for non-homogeneous random walks

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in $\mathbb{R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X}$ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq2$. To characterise $\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in $\mathbb{R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X}$ and thus develop the excursion theory of $\mathcal{X}$ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X}$ in $\mathbb{R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X}$ is time-reversible. If so, the excursions of $\mathcal{X}$ in $\mathbb{R}^d$ generalise the classical Pitman-Yor splitting-at-the-maximum property of Bessel excursions.

subsequent evolution. The latter is given in terms of a Bessel process and a time-changed angular process solving (1.3) as above: (r t φ ρs(t) , t ≥ s) with φ 0 :=X s . The random vector X s is forced to be independent of r s and distributed according to the stationary law µ of φ, due to the rapid spinning of the process X as it leaves 0: ρ s (t) → ∞ as s ↓ 0 for fixed t > 0 (see Lemma 3.12 below). As ρ s (t) = ρ s (1) + ρ 1 (t) for any s, t > 0, the processes (r t ψ ρ1(t) , t > 0) and (X t , t > 0) are equal in law, where ψ (the stationary solution to (1.3)) and r are independent. The analogy with the classical case of the skew product of BM on R d in both cases X 0 = 0 and X 0 = 0 (see [25,§IV.35,p. 73] and [13, p. 276]) is clear.

Point-recurrent case: skew-product decomposition of excursions of X
Assume V /U ∈ (1, 2) and X 0 = 0. The process X returns to 0 infinitely often since X / √ U is Bessel of dimension V /U . As the excursions of X turn out to exhibit the rapid spinning behaviour at each end, its excursion measure may be constructed as follows.
Mark each Bessel excursion by an independent draw ψ from the law P Ψ on C(R, S d−1 ), the stationary solution to (1.3) given in Prop. 3.7 below. Since, due to rapid spinning at the beginning of each excursion of X , the angular component of the excursion is distributed according to the stationary measure µ of SDE (1.3) at all times, we need to map the marked Bessel excursion by time-changing the mark ψ via an additive functional of the Bessel excursion, see Section 3.6.1 below for details. Note that the mapping has to be defined for Bessel excursions lasting longer than a (for any fixed a > 0), since the time-change can only be "anchored" at a pre-specified time during the life time of the excursion. Although this causes some technical difficulties, the mapped Poisson point processes can be interpreted consistently (for all a > 0). Its excursion measure turns out to be that of X . We stress that this construction of the excursion measure depends only on σ 2 , which specifies the dimension of the Bessel process and hence its excursion measure and determines the marks via SDE (1.3) (the mapping uses only the information contained in the Bessel excursion). Moreover, the local time at 0 of X can be defined as that of X at 0, without a reference to the strong Markov property of X . Hence, once the excursion measure has been constructed (Section 3.6.1 below), the key step in the proof of Theorem 1.1 consists of establishing that (without the strong Markov property) the point process of excursions of X is the Poisson point process with the excursion measure described above. The details are in Section 3.6.2 below.
In the case X 0 = 0, up to the first hitting time of 0, the skew product of excursions coincides with the generalized Lamperti representation for self-similar Markov processes on R d \ {0} [1], where the Lévy process is a scalar BM with drift and the angular component equals the diffusion on S d−1 in (1.3) started atX 0 . Note also that there is a literature (see e.g. [28] and the reference therein) on the extensions of strong Markov processes on R d \ {0} with skew-product decomposition beyond the first hitting time of the origin, of which X is an example.

Splitting excursions at the maximum: a generalized Pitman-Yor representation
If the vector field V 0 in (1.4) has a potential, the excursions of X provide a multidimensional generalization of the famous Pitman-Yor [23] representation of the Bessel excursions with dimension δ = V /U ∈ (1, 2). Let U = 1 and recall from [23] that the unique maximum M of the Bessel excursion e r is drawn from the σ-finite density m → m δ−3 on the interval (0, ∞). Then, conditional on M , the excursion e r is obtained by joining back to back two independent Bessel processes β and β of dimension 4 − δ, both started at 0 and run until the first times (T M and T M respectively) they hit M : e r (t) = In the limit as U ↑ V , which is excluded from our results, the angular motion degenerates to a constant as the trace of σ 2 equals the radial eigenvalue. The radial part becomes the modulus of the scalar BM, while rapid spinning and (1.5) suggest that the singular diffusion in the limit changes the ray it lives on every time it hits the origin according to a law on S d−1 , which is the limit of the stationary measures of SDE (1.3) as V /U ↓ 1. It hence appears that the limiting singular diffusion is a generalization of the Walsh BM (or Brownian spider) [2] to R d .

Smooth square roots and pathwise uniqueness: the Stroock-Yor phenomenon
SDE (1.1) need not (but clearly could) possess pathwise uniqueness even if σ 2 is the identity (consider σ(u) = diag (sgn(u 1 ), . . . , sgn(u d )) and recall the scalar Tanaka SDE [24, §IX.1, Ex. (1.19)]). This behaviour persists even for smooth square roots σ. Below we give a generalization of the SDE for complex Brownian motion in [27,Thm 3.12], with the property that the failure of pathwise uniqueness occurs precisely when the solution starts from (or visits) 0.
Note first that a simple application of the occupation times formula and the fact that X t = 0 if and only if X t = 0 imply that if X solves SDE (1.1) for a given choice of0, then it also solves the SDE for any other choice0 ∈ S d−1 . If a square root σ satisfies (I) P σ(u) = σ(P u) for all u ∈ S d−1 , where P ∈ SO(d) \ {I} 1 , then Itô's formula and the remark above imply that for any solution (X , W ) of (1.1) started from 0, the process (Y, W ), where Y := P X , is also a solution. By Theorem 1.1, X and Y have the same law but are clearly not equal. If, in addition, σ satisfies (II) u = σ(u)c for all u ∈ S d−1 and some c ∈ S d−1 , the BM driving the process X equals c W (Lemma 3.2 below), making X adapted to W . Moreover, assuming X never visits 0, the BM driving the angular component via SDE (1.3) is a time-change of · 0 X s −1 dW s (see (3.15) and Prop. 3.11 below). Hence the skew product X t φ ρ0(t) , t ∈ R + , where ρ 0 (t) = t 0 X u −2 du, makes X a strong solution of (1.1).
It remains to exhibit a smooth σ satisfying (I) and (II) above. Note first that (I) may only hold in even dimensions. We rely on the Lie group structure of the spheres in dimensions d ∈ {2, 4} for our examples. Pick a positive-definite A ∈ R d ⊗ R d and let σ(u) = R(u)A, where R : S d−1 → SO(d) is smooth. For d = 4, view S 3 as unit quaternions and define R by R(u)v := u • v, where u • v denotes the multiplication of quaternions v ∈ R 4 and u (see e.g. [25, p. 229]). It is easy to check that R(u) ∈ SO(4) and R(u)e 1 = u for all u ∈ S 3 , where e 1 is the first standard basis element of R 4 , i.e. the real unit quaternion. If Ae 1 = e 1 (as is the case if (A6) holds), then (II) holds. Moreover, σ(u) is a smooth square root of σ 2 (u) = R(u)A 2 R(u) −1 . Pick a unit quaternion p ∈ S 3 \ {e 1 } 1 SO(d) is the group of orientation-preserving orthogonal matrices in R d ⊗ R d and I is the identity matrix.

Angular convergence and the first exit from large balls of the random walk
We now describe the behaviour of the angular component of the random walk X and its asymptotic law at τ n a := inf{m ∈ Z + : X m ≥ a √ n} its first exit out of the ball centred at 0 with radius a √ n (for some a > 0). Both statements are easy consequences of Theorem 1.2.

Corollary 1.3.
Let the random walk X satisfy the assumptions of Theorem 1.2 with U = 1 and define δ := V . Let the random vector θ with the law µ on S d−1 , whose density satisfies (1.4), be independent of r. Then, as n → ∞, the following weak limits hold: n −1/2 X n ⇒ r 1 θ (and henceX n ⇒ θ) and (τ n a /n, n −1/2 X τ n a ) ⇒ (τ a , aθ).

Assumptions and examples
Let {e 1 , . . . , e d } be the standard orthonormal basis in R d (d ≥ 2) with respect to the Euclidean inner product ·, · on R d , and S d−1 := {u ∈ R d : u = 1} the unit sphere in R d , where · is the Euclidean norm. For x ∈ R d \ {0} and the origin 0, letx := x/ x and0 := e 1 , respectively. Let X = (X n , n ∈ Z + ) be a discrete-time, time-homogeneous Markov process on an unbounded Borel subset X of R d . Suppose X 0 is a non-random point in X. Denote the increments of X by ∆ n := X n+1 − X n . Since the law of ∆ n depends only on X n , we often take n = 0 and write ∆ for denote the probabilities and expectations when the walk is started from x ∈ X. We make the following assumptions. (A1) Suppose that µ(x) = 0 for all x ∈ X.
The next assumption ensures that ∆ is uniformly non-degenerate.
For a matrix M ∈ R d ⊗ R d define the norm M := sup u∈S d−1 M u . Throughout the paper, let σ 2 (u) be a positive-definite matrix for all u ∈ S d−1 .
(A3) Suppose that, as r → ∞, we have ε(r) := sup x∈X: Under assumptions (A0)-(A4), it was proved in [9] that the walk is transient if and only if 2U < V , while [10] gives an invariance principle for the radial component X . The full invariance principle of the present paper requires additional structure on the limiting covariance matrix σ 2 to ensure that the angular part is a suitably well-behaved process on the sphere.
Controlling the dependence between the radial and angular components requires the following.
(A6) Suppose that u is an eigenvector of σ 2 (u) for all u ∈ S d−1 .
Following [9, §3], we describe a family of examples satisfying (A0)-(A6) in which the increment distribution is supported on an ellipsoid having one distinguished axis aligned in the radial direction. The model is specified by positive constants a, b. Let Qx be an orthogonal matrix representing a transformation of R d that maps e 1 tox, and write D = √ d diag (a, b, . . . , b). Given X 0 , the law of X 1 is generated by taking ζ uniform on S d−1 ; if X 0 = 0 set X 1 = ζ, otherwise set X 1 − X 0 = QxDζ. In words, from X 0 = 0 the position X 1 is generated by taking a uniform point on the unit sphere centred at X 0 , stretched differentially in the radial and transverse directions to give a point on an ellipsoid. The special case a = b is a Pearson-Rayleigh random walk. A calculation [9, p. 104] shows that In particular, tr σ 2 (u) = a 2 + (d − 1)b 2 , σ 2 (u)u = a 2 u, and u, σ 2 (u)u = a 2 , while M (x) = σ 2 (x) for x = 0. Thus (A0)-(A6) hold. Without loss of generality, we may take U = a = 1. Then V = 1 + (d − 1)b 2 , and σ sy (u) = uu + b(I − uu ), so that the spherical part of X is driven by the SDE (3.8), which reduces in this case Xt dt, which corresponds to a BM on S d−1 sped up by a factor of b. The diffusion limits generated by this family of random walks thus include the classical skew-product description of BM as a special case, but also include examples where 0 is recurrent.

Overview
Let σ sy : S d−1 → R d ⊗ R d be the unique positive-definite matrix-valued function satisfying σ sy σ sy = σ 2 , i.e. σ sy is the unique symmetric square root of σ 2 . Pick any measurable square root σ : S d−1 → R d ⊗R d of σ 2 and note that, since σ 2 and σ sy commute, the matrix σ −1 sy (u)σ(u) is orthogonal for all u ∈ S d−1 . By Lévy's characterisation of Brownian motion, it is hence sufficient to prove Theorem 1.1 for the SDE The next step is to establish weak existence for SDE (3.1). We start with a simple lemma.
Proof. Since σ 2 is positive-definite, by (A5) and the compactness of S d−1 there exists ε > 0 such that det(σ 2 ) > ε on S d−1 . By (A4) we have tr σ 2 (u) = V . Hence the smallest eigenvalue λ min (u) of σ 2 (u) satisfies ε < λ min (u)V d−1 for all u ∈ S d−1 . Since σ sy is symmetric and non-degenerate, its eigenvalues are positive and the smallest one is equal to λ min (u). Hence the inequality in the lemma holds for the constant Since the function x → σ sy (x) is bounded and uniformly elliptic by Lemma  The proof of uniqueness in law proceeds as follows. Throughout this section, assume U = 1 in (A4). In Section 3.2 we prove that the radial component of any solution of (3.1) is Bessel of dimension V > 1. Section 3.3 introduces the Riemannian structure on the sphere, needed in Section 3.4 to characterize the law of a stationary diffusion on S d−1 indexed by R. This process is a key ingredient in the description of the projection of the path of the solution X of SDE (3.1) (away from 0) onto S d−1 . In Section 3.5 we analyse the case when 0 is polar for the radial process (V ≥ 2). We prove that any solution has a skew-product decomposition constructed using the components from Sections 3.2 and 3.4 that are unique in law. In Section 3.6 we consider the recurrent case (1 < V < 2). We develop the excursion theory (away from 0) of the solution X of (3.1) without reference to the strong Markov property of X . We characterize the excursion measure in terms of the excursion measure of the radial part, given in [23], and the law of the diffusion on S d−1 from Section 3.4. This implies the uniqueness in law for SDE (3.1).

The radial process
Let r := X be the radial part of a solution X of SDE (3.1).

Lemma 3.2.
Let (A4) hold and σ 2 : For any solution (X , W ) of SDE (3.1), adapted to a filtration (F t , t ≥ 0), the process y = (y t , t ≥ 0), y t := X t 2 , is the unique strong solution of SDE where (Z t , t ≥ 0) is an (F t ) Brownian motion given by Z t := t 0X s σ sy (X s )dW s . In particular, the law of r = √ y is BES V X 0 .
Proof of Lemma 3.2. For any solution (X , W ) of (3.1), the processes y and Z defined in the lemma are (F t )-adapted. Itô's formula and the assumption (A4) imply that equation (3.2) holds. The process Z is a Brownian motion by Lévy's characterisation, (A4) and assumption U = 1. Since SDE (3.2) has weak existence and pathwise uniqueness, the law of y is BESQ V X 0 2 .

A Riemannian structure on S d−1
This section introduces a Riemannian metric g on S d−1 , gives an explicit description of its inverse tensor in local coordinates and relates it to the Laplace-Beltrami operator corresponding to g (see [14] as reference on Riemannian geometry).
for any x ∈ S d−1 and v 1 , v 2 ∈ T x S d−1 .
(3.4) By (A5), g is a symmetric positive-definite (0, 2)-tensor field, i.e., a Riemmanian metric on the smooth manifold S d−1 . The metric g provides a canonical way of identifying tangent and cotangent vectors: the mapg : . Hence grad f is the unique vector field satisfying the identity g(grad f, X) = df X for all X ∈ Γ(T S d−1 ). Moreover, the operator grad : , which is metric and torsion-free [14,Thm 4.3.1]. In short, the connection ∇ allows us to compare tangent vectors in nearby tangent spaces in a way that is compatible with the geometry induced by the metric g, cf. [14, § §4.1 & 4.2]. In particular, a vector field X ∈ Γ(T S d−1 ) gives rise to a linear endomorphism (∇X) x : T x S d−1 → T x S d−1 for any x ∈ S d−1 [14,Def. 4.1.1]. Put differently, ∇ v X is the derivative of the vector field X at x in the direction v ∈ T x S d−1 . Define the divergence of the vector field X to be the trace of this linear endomorphism, (div X)(x) := tr(∇X) x . This yields a coordinate-free definition of the divergence operator div : Γ(T S d−1 ) → C ∞ (S d−1 , R). The Laplace-Beltrami operator ∆ g : C ∞ (S d−1 , R) → C ∞ (S d−1 , R) on the Riemannian manifold (S d−1 , g) can now also be defined in a coordinate-free way as ∆ g f := div(grad f ) for any f ∈ C ∞ (S d−1 , R).
, where δ ij is the Kronecker delta. We interpret the tangent vector E i as a linear map E i : , where ∂ i is the partial derivative in the i-th component [12, p. 247].
For any point z ∈ B d−1 and tangent vector u ∈ R d−1 we have dz −1 q (z)u = u − e q z, u / z −1 q (z), e q . Since g ij (x) = g x (dz −1 q (z q (x))e i , dz −1 q (z q (x))e j ) for any i, j ∈ [q], the formula for g ij (x) follows by (3.4). We now prove that (g ij (x)) i,j∈ [q] , defined in the lemma, is the inverse of (g ij (x)) i,j∈ [q] . Define (d − 1)-dimensional square matrices S − and S as follows: S − ij := σ −2 ij (x) and S ij := σ 2 ij (x) for any i, j ∈ [q]. Define (d − 1)-dimensional vectors S − q , S q by S − q,i := σ −2 qi (x) and S q,i := σ 2 qi (x) for i ∈ [q]. Let s := σ 2 qq (x) and s − := σ −2 qq (x). Since σ −2 (x)σ 2 (x) is the identity on R d , we have where I denotes the identity matrix on R d−1 . Denote z := z q (x), and D := ± 1 − z 2 . Since x = z + De q ∈ S d−1 , the assumption in (A6) implies σ −2 (x)(z + De q ) = z + De q (recall U = 1). Hence the following identities hold, where z q denotes the (d − 1)-tuple of coordinates of z expressed in the basis {e i ; i ∈ [q]} of R d−1 . Define (d − 1)-dimensional square matrices G, G − as follows: A direct calculation, using identities in (3.5)-(3.6) and the fact that S = S and S − = S − , yields GG − = I. It remains to note that G − ij = g ij (x) and G ij = g ij (x) for all i, j ∈ [q]. The expression for the Laplace-Beltrami operator ∆ g = div grad = tr Hess g in local coordinates in terms of the Christoffel symbols Γ k ij is well-known, cf. [ , and note that it is an extension of σ sy : exists since, by Lemma 3.1, σ sy can be expressed as an absolutely convergent power series in σ 2 , which is smooth by (A5).   (c) Let P x denote the law of the solution of (3.7) on C(R + , S d−1 ). Then {P x , x ∈ S d−1 } is a strongly Markovian system [12, p. 204], determined uniquely by its generator G, where the vector fields S i , i ∈ {0, . . . , d}, are viewed as linear (over R) maps C ∞ (S d−1 , R) → C ∞ (S d−1 , R) satisfying the Leibniz rule.
(d) V 0 := G − 1 2 ∆ g is a vector field in Γ(T S d−1 ), making the solution of (3.7) a Brownian motion with drift on the Riemannian manifold (S d−1 , g) with generator 1 2 ∆ g + V 0 .
(e) Any solution (X, W ) of the Itô SDE Proof. The vector fields S j , j ∈ {0, . . . , d}, are tangential to S d−1 by (A6) and smooth by (A5 where the second equality holds by Dz q = z q , and where S i , and all f ∈ C ∞ (H ± q , R). The definition of S j above, (A4), (A6) and Lemma 3.5 imply . Hence, by the definition of G in the lemma and the expression for ∆ g in the local coordinates . Since such an equality holds for every q ∈ {1, . . . , d} and choice of ± (i.e. for every chart in our atlas), V 0 satisfies the Leibniz rule and is hence an Extend the vector fields S 0 , S 1 , . . . , , DA i (y) e j = tr(DA j (y)A(y)). Together with (A4), this implies tr A(y) = V + 2 A 0 (y), y and (3.10) follows.
Let (X, W ) be a solution of (3.8). A simple application of Itô's formula yields d X t 2 = 0, implying the first statement in (e). By (3.10) it follows that X in fact satisfies the . . , d}, are defined above (3.10). By the definition of the Stratonovich integral on R d [12, Ch III, §1, . . , d}, on S d−1 and X stays on the sphere for all time, SDE (3.7) holds for X (see [12, Ch V, Rem 1.1]).
. In particular, the law P of the solution of (3.7), started according to a probability measure ν on S d−1 , equals P Proposition 3.7. Let (A4)-(A6) hold. There exists a unique probability measure µ on S d−1 with full support, such that µ(·) = S d−1 µ(dx)P t (x, ·) for all t ∈ R + and the transition function P t (x, ·) converges to its stationary measure µ in the following sense: 2 Furthermore, there exists a unique law P Ψ [ · ] on the Borel sets of C(R, S d−1 ) with compact-open topology, satisfying P Ψ [ψ s ∈ · ] = µ( · ) and P Ψ [ψ s+t ∈ · | ψ s ] = P t (ψ s , · ) for all (s, t) ∈ R × R + , where (ψ u , u ∈ R) denotes the coordinate process on C(R, S d−1 ).
The unique stationary measure µ exists and has full support essentially because the vector fields S 1 , . . . , S d in Lemma 3.6(a) span T x S d−1 at every x ∈ S d−1 . The proof uses the representation in Lemma 3.6(d) of the process as a Brownian motion with drift and applies the well-known results for the stability of elliptic diffusions on compact Riemannian manifolds [22]. (b) The geometry introduced in Section 3.3 allows us to characterize the time-reversibility of the diffusion X satisfying SDE (3.7). This leads to an explicit description, given in (1.5) of Section 1.1 above, of the excursions of the process X appearing in Theorem 1.1. imply that P Ψ [ · ] exists and is unique: Proof of Proposition 3.7. By Lemma 3.6(d), the generator of the strong Markov process [12, p. 291] and Lemma 3.5 above). Let G be the adjoint of G with respect to the measure d g x. Assumptions of [22, Ch 4, Thm 11.1] are satisfied for the generator G since its second order term is the Laplace-Beltrami operator and the vector field V 0 is smooth by (A5). Hence by [22,Ch 4,Thm 11.1], all harmonic functions for G are constant and there exists a unique positive

Proof of Theorem 1.1 when 0 is polar for the radial process
Assume throughout this section that V ≥ 2 (and U = 1) and let (X , W ) be any solution to (3.1), adapted to (F t , t ≥ 0), on a probability space that supports a one-dimensional (F t ) Brownian motion, independent of (X , W ). By Lemma 3.2, 0 is polar for r = X .
We now show B is independent of Z. By the Markov property, B t depends on . It remains to prove that B is independent of (Z t − Z s , t ≥ s). Note that by (3.15) and Lemma 3.9 it holds that Z cs(t) − Z s = cs(t) s the process ϕ satisfies SDE (3.8) driven by (B t , t ≥ 0) as required.
Proof of Theorem 1.1 in the transient case with X 0 = 0. By Prop. 3.11 (enlarge the probability space if needed), the law of any solution X of SDE (3.1), (3.12) and ϕ is the unique solution of (3.8) with ϕ 0 =X 0 , independent of r.
In order to characterize the law of X in the case V ≥ 2 with X 0 = 0, we need to understand the law of theX s (for any fixed s > 0) and its dependence on the path of the radial process r.
Recall that by Prop. 3.7, the process ϕ defined in Prop. 3.11 has a unique stationary measure µ. Lemma 3.12. Suppose that (A4), (A5) and (A6) hold. Then for any t > 0,X t has the law µ and is independent of F r ∞ . Put differently, the conditional law takes the form Proof. Fix t > 0 and let s ∈ (0, t). By Prop. 3.11 and Lemma 3.9 we haveX t = ϕ ρs(t) , where ϕ satisfies SDE (3.8). By (e), (b) and (c) of Lemma 3.6 and Prop. 3.7, ϕ is strong Markov with the transition function P u (x, · ) that does not depend on s. Hence, for as ϕ ρs(t) depends on F r ∞ only through ρ s (t) and ϕ 0 =X s . Crucially, (3.17) holds for any fixed time s ∈ (0, t), and also for any random time s By Lemma 3.10 we have lim s↓0 ρ s (t) = ∞. Hence, for sufficiently small s, an arbitrarily large time interval separates ϕ 0 =X s and ϕ ρs(t) , and so stationarity must be attained at the latter, regardless ofX s . Formally, we apply the uniform ergodicity of ϕ in (3.11). Lemmas 3.9 and 3.10 imply that for any u > 0, there is an F r ). Therefore the finite-dimensional distributions of (X t , t > 0) are uniquely determined by P Ψ [ · ] and the law of r. Moreover, by Lemma 3.2, the law of ( X ,X ), and hence of X , is uniquely determined by BES V (0) and P Ψ [ · ]. The uniqueness in law of (3.1) implies that X is strong Markov and Theorem 1.1 follows in the transient case. In this section we assume V ∈ (1, 2) and U = 1. Hence, by Lemma 3.2, r = X is BES V (0) where X is a solution of SDE (3.1). We recall briefly the necessary elements of excursion theory (see [23,Ch XII], [3, Ch IV] as a general reference). Since 0 is regular and instantaneous for r, there exists Markov local time L = (L t , t ≥ 0) at 0. By [24, Prop. XI.1.11], up to a constant factor, L is a time-change of the local time at 0 of a Brownian motion, where the time-change is a constant multiple of ( a Lévy process with non-decreasing paths). Furthermore, as L tends to infinity, L −1 is not killed: , is a Poisson point process (PPP) with excursion measure µ r on E 1 .

Marked Bessel excursions
Pick a ∈ (0, ∞) and let t ∧ a := min(t, a), t ∨ a := max(t, a) for any t ∈ R. For any w ∈ E 1 Lemma 3.13. The following statements hold for any fixed a ∈ (0, ∞).
Due to the product structure of the image, the w , is measurable, which is equivalent to g s : E Hence, g s (w)−g s (u) ≤ w(c a w (s))−w(c a u (s)) + w(c a u (s))−u(c a u (s)) ≤ ε/2+ε/2 = ε and the inclusion N δ (K 1 ) ⊂ g −1 s (B), implying the continuity of g s , follows. Since δ 0 could be arbitrarily small, the bound in (3.19) also implies the continuity of w → c a w (s).
The equality in part (iv) follows from (ii) and (iii). What remains to be proved is that , the uniform continuity of θ 0 on any compact, together with the proximity of (b 0 , w 0 ) and (b, w), yields a uniform control on compacts of the first two terms. The third term is controlled by the proximity of θ 0 and θ in C(R, S d−1 ). The estimates, analogous to the ones in the proof of (iii), are omitted.   Proof of Proposition 3. 16. In order to establish µ r (E 1 \ E + 1 ) = 0, note that by [23], the excursion measure µ r has the following representation: any excursion e r λ has a finite maximum and this maximum is attained at a unique time. Furthermore, conditional on the maximum being at some level M > 0, the excursion has the same law as the path formed by taking two independent BES 4−δ (0) processes, both run up until their first hitting time of the level M , and placing them end-to-end. Since 2 < 4 − δ < 3, by Lemma 3.10, any excursion in the support of µ r is in E + and Mapping theorems of [17] (the latter applies since Φ a is measurable and bijective by Lemma 3.13(iii)), the point process e r,Ψ,a = (e r,Ψ,a λ , λ ≥ 0), defined by e r,Ψ,a λ := δ d , if τ r λ ≤ a, and e r,Ψ,a is atomless. Hence any measure ν satisfying the identity in the proposition for all a ∈ (0, ∞) is also atomless, σ-finite and unique.
The proposition now follows from the claim that µ r ⊗ P It remains to establish this claim. Consider Q : , implying the claim.

Proof of Theorem 1.1 in the recurrent case
Let (X , W ) be a solution of SDE (3.1) with X 0 = 0, adapted to (F t , t ≥ 0). Since we are only interested in the law of the solution, we may assume that we are in the canonical setting, i.e. the probability space is Ω = C(R + , R n ) (for some n ∈ N) and the filtration satisfies the usual conditions with respect to the probability measure P on Ω. Define the point process e X = (e X , ≥ 0) of excursions of X away from 0 by e X := δ d if ∈ R + \ Λ r , and e X : R + → R d , where e X (u) := X L −1 − +u u ∈ (0, τ r ), 0 u ∈ R + \ (0, τ r ), (3.20) if ∈ Λ r (the notation introduced earlier in Section 3.6 will be used throughout Section 3.6.2). The point process e X = ( e X , ≥ 0) with excursions e X (u) = r L −1 − +u 1{u ≤ τ r }, u ∈ R + , for any ∈ Λ r , is clearly equal to the PPP e r defined above. Since X t = 0 if and only if r t = 0, e X takes values in E + d ∪ {δ d }. The key step in the proof of Theorem 1.1 is to characterize e X : this will establish uniqueness in law of X (see Corollary 3.23), and, at the same time, show that e X is a PPP with excursion measure a.s.
Here the law P Ψ on C(R, S d−1 ) is defined in Prop. 3.7 and µ r is the excursion measure of the PPP e r . In particular, the excursion e X Lτ +µ n a is independent of F L −1 Lτ and its law on 1 ), depends neither on n ∈ N nor on the stopping time τ . Remark 3.19. Theorem 3.18 would follow trivially if we knew that X was strong Markov.
However, this cannot be assumed a priori. Once the uniqueness in law of SDE (3.1) has been established, the strong Markov property of X follows.
As e X Lτ +µ n a ∈ E Proof. Since C(R, S d−1 ) is Polish, the regular conditional distribution P[θ a,n ∈ · | F L −1 Lτ ∨ F r ∞ ] exists. Moreover, as every trajectory of θ a,n is continuous, it is sufficient to prove that P-a.s. the finite-dimensional distributions at rational times coincide with those of P Ψ . Since the set of all finite subsets of the rationals is countable and the Borel σ-algebra on S d−1 is generated by a countable family of open balls, by a diagonalization argument it suffices to prove that the finite-dimensional distributions at a given set of (rational) times (evaluated on the products of the finite intersections of generating sets) coincide P-a.s. We establish this in two steps. First, we show that the process (θ a,n t , t ≥ 0) solves SDE (3. We next show that B is independent of Z. Recall that η a (0) and L −1 Lτ +µ n a are (F t )stopping times. Since B 0 = 0, B is independent of G 0 = F ηa(0) and hence of (Z s , 0 ≤ s ≤ η a (0)). B is measurable with respect to s∈R+ G s ⊆ F L −1 driven by the Brownian motion A defined above. It is easy to see from the definition of the Brownian motion B above that t 0 (σ sy (θ a,n u ) − θ a,n u (θ a,n u ) )dB u = t 0 (σ sy (θ a,n u ) − θ a,n u (θ a,n u ) )dA u for all t ≥ 0. Hence (θ a,n u , u ≥ 0) satisfies SDE (3.8) driven by B independent of F r ∞ .
The second step in the proof of the lemma analyses the conditional law of θ a,n . The number of excursions longer than b started before the start of the n-th excursion of r of length at least a, i.e. N b (L −1 µ n a − ), is F r ∞ measurable. Fix t ∈ R and note that by For all b ∈ (0, a) such that I a b (e r Lτ +µ n a ) > −t, the first step of the proof implies is a probability measure on S d−1 , P is the transition function from Prop. 3.7 and µ denotes its stationary measure. By (3.11) in Prop. 3.7, Lemma 3.13(ii) and (3.21), for any ε > 0 there exists b ∈ (0, a) such that |P[θ a,n ]. An analogous argument shows that finite-dimensional distributions of P Ψ [ · ] and P[θ a,n The remaining step in our characterization of e X is provided by the following result, which will enable us to describe finite-dimensional distributions. Proposition 3.21. Pick k ∈ N and indices 0 =: i 0 < i 1 < i 2 < · · · < i k−1 < i k . Define n := i k and choose measurable sets B 1 , . . . , B n ⊆ R d and times 0 < u 1 < u 2 < · · · < u n .
for any j ∈ {1, . . . , n} (recall thatL depends on τ ). Then, on the event E k : Remark 3.22. In (3.22), for any p ∈ {i l + 1, . . . , i l+1 }, it holds thatL up =L ui l +1 and hence erL up refers to a single excursion. Note also that E k depends on the sequence i 1 < · · · < i k and not just on the index k. This information is suppressed from the notation for brevity.
the number of excursions orr that started prior toL −1 u1 with length of at least a 1 . Clearly, Since this identity is independent of b and E b 1 E 1 as b ↓ 0, the proposition holds for k = 1 and any i 1 = n ∈ N.
We proceed by induction: assume that (3.22) holds for some k ∈ N and any increasing sequence of indices of length at most k. Pick an event E k+1 . Put differently, choose a sequence of indices 0 = i 0 < i 1 < · · · < i k < i k+1 = n. The (F t )-stopping time ρ : Lρ is an (F t )-stopping time, the σ-algebra F L −1 Lρ is well-defined and contains F L −1

Lτ
. For the sequence 0 < i 1 < · · · < i k , define the event E k as in the statement of the proposition. Note that which equals the left-hand side in (3.22). The proposition follows by the induction hypothesis.
Corollary 3.23. Let X be a solution of SDE (3.1) started at 0 and adapted to (F t , t ≥ 0).
(a) Let τ be a finite (F t )-stopping time. Then the processX = (X t , t ≥ 0), defined bỹ X t := X L −1 Lτ +t , is independent of F L −1 Lτ and has the same law as X .
(b) Let Y be a solution of SDE (3.1) started at 0. Then the laws on C d of X and Y coincide.
Proof. (a) If we prove that for any 0 < u 1 < u 2 < · · · < u n and measurable sets B 1 , . . . , B n ⊆ R d , the equality P[X u1 ∈ B 1 , . . . ,X un ∈ B n |F L −1 Lτ ] = P[X u1 ∈ B 1 , . . . , X un ∈ B n ] holds P-a.s., part (a) follows by a diagonalization argument (cf. first paragraph in the proof of Lemma 3.20), sinceX 0 = X 0 and all the trajectories ofX are continuous. Recall that L L −1 Lτ +u = L τ +L u . Hence, for all u ≥ 0,X u = e X Lτ +Lu (u −L −1 Lu− ) and in particular (take τ ≡ 0) X u = e X Lu (u − L −1 Lu− ). Note that the set E k in Prop. 3.21 is determined by k ∈ {1, . . . , n} and the indices i 1 < . . . < i k−1 (with i 0 = 0 and i k = n) and should be denoted by E Note that F is defined P-a.s. on Ω and is measurable. Furthermore, F is a function only of the radial componentr = X ofX . By Prop. 3.21, we get P[X u1 ∈ B 1 , . . . ,X un ∈ B n |F L −1 Lτ ∨ F r ∞ ] = F (X ). An identical argument applied to X (with τ ≡ 0) yields P[X u1 ∈ B 1 , . . . , X un ∈ B n |F 0 ∨ F r ∞ ] = F (X ). By the strong Markov property of r, the processr, and therefore F (X ), is independent of F L −1 Page 25/38 http://www.imstat.org/ejp/ (b) As before it is sufficient to show P[X u1 ∈ B 1 , . . . , X un ∈ B n ] = P [Y u1 ∈ B 1 , . . . , Y un ∈ B n ] for any 0 < u 1 < u 2 < · · · < u n and measurable sets B 1 , . . . , B n ⊆ R d , where P [ · ] is the probability measure on the space where Y is defined. Prop. 3.21 implies this statement, using the same argument as in part (a) as the processes X and Y have the same law. Proof. Let X be adapted to (F t , t ≥ 0). Pick λ ∈ R + and recall that L −1 λ is an (F t )stopping time. DefineX = (X t , t ≥ 0) byX t := X L −1 λ +t .    a ij ∂ i ∂ j f (for a smooth f : R d → R with compact support) and v is a distribution R d . For n ∈ N, let Z n be a process with sample paths in D d and let A n = (A ij n ) be a symmetric R d ⊗ R d -valued process started at zero, such that A ij n has sample paths in D 1 and A n (t) − A n (s) is non-negative definite for all t > s ≥ 0. Set F n t := σ(Z n (s), A n (s), s ≤ t). Suppose that Z i n and Z i n Z j n − A ij n are F n t -adapted local martingales for each i, j ∈ {1, . . . , d}. Let τ r n := inf{t ≥ 0 : Z n (t) ≥ r or Z n (t−) ≥ r} (with convention inf ∅ := ∞) and suppose that for every r > 0, T > 0, and i, j ∈ {1, . . . , d}, and, as n → ∞, where P −→ denotes convergence in probability and s ∧ t := min{s, t} for s, t ∈ [0, ∞]. Assume sup n∈N E Z n (0) 2 < ∞. Suppose that Z n (0) and Z n converge weakly to a probability law v on R d and the law of a Bessel process of dimension greater than one, respectively. Then Z n converges weakly to the solution of the martingale problem for (G, v).
The underlying idea for the proof of Theorem 4.1 is standard: show that every subsequence of (Z n ) n∈N has a further subsequence converging weakly to the law given by the solution of the martingale problem (G, v) (cf. proof of [8, Thm 7.4.1, p. 354]). Since a in Theorem 4.1 is bounded, a i := sup x∈R d a ii (x) is finite for each i ∈ {1, . . . , d}. Since A ii n (t) ≥ A ii n (t−) for all t ≥ 0 and i ∈ {1, . . . , d},  Define for given r > 0, n ∈ N and i, j ∈ {1, . . . , d} the processesZ r n andÃ ij n bỹ Z r n (t) := Z n (t ∧ η n ∧ τ r n ) andÃ ij n (t) := A ij n (t ∧ η n ∧ τ r n ), (4.5) respectively (Ã ij n depends on r but this is suppressed from the notation as it is clear from the context). Observe that for any T >  http://www.imstat.org/ejp/ n (0) = Z n (0) is integrable by assumption, the local martingaleZ r n is of class (DL) and therefore a martingale [24,Prop. IV.1.7]. An analogous argument, relying on (4.1)-(4.2), the inequality |Z r,i nZ r,j n | ≤ (Z r,i n ) 2 + (Z r,j n ) 2 and the square integrability of Z n (0) , shows thatZ r,i nZ r,j n −Ã ij n is also a martingale. Furthermore, since A ii n (0) = 0 for all indices i ∈ {1, . . . , d}, for any t ≥ 0 we havẽ Proof. We prove the lemma by establishing the sufficient condition for the relative compactness of the sequence (Z r n ) n∈N given in [8,Thm 3.8.6,. Fix an arbitrary T > 0 and let B K denote a closed ball of radius K > 2r + 1 in R d . Note that the bound in (4.6) and the Markov inequality imply sup n∈N E Z n (0) 2 , which are finite by assumption. As K is independent of n and can be arbitrarily large, the compact containment condition [8,Eq (7.9) for any t, h ≥ 0. With this in mind, define for any δ > 0. In order to compare γ n (δ) with the corresponding quantity for the limiting process, let t+δ t a ii (Z r n (s))ds. Page 28/38 http://www.imstat.org/ejp/ both tend to zero in probability, implying that Γ n (δ) also tends to zero in probability: Since the upper bound in (4.7) is non-decreasing in t, we get By (4.2) the right-hand side of this inequality converges in L 1 as n → ∞. Thus the sequence (Γ n (δ)) n∈N must be uniformly integrable and hence by (4.8) converges to zero in L 1 . By adding and subtracting the relevant term we find Hence it clearly holds that lim δ→0 lim sup n→∞ E γ n (δ) = 0 and the relative compactness ofZ r n now follows from [8, Thm 3.8.6, p. 137-138] (see also [8,Rem 8.7(b), p. 138]).
For any path x ∈ D d , we define the time τ r (x) of its first contact with the complement of the open ball of radius r in R d (centred at the origin) by τ r (x) := inf{t ≥ 0 : x(t) ≥ r or x(t−) ≥ r}, (4.9) where inf ∅ = ∞. If it is clear from the context which path x we are considering, to simplify the notation we sometimes write τ r for τ r (x). Note that if x is continuous, then τ r (x) = inf{t ≥ 0 : x(t) ≥ r}. The following lemma is important in the proof of  To prove Lemma 4.3 we first need to establish properties of the function r → τ r . Lemma 4.4. Fix x ∈ D d . The function r → τ r (x), mapping R + into [0, ∞], is nondecreasing, has right limits and is left continuous. Put differently, for any r ∈ R + the limit lim s↓r τ s =: τ r+ exists in [0, ∞] and, for r > 0, it holds that lim s↑r τ s = τ r . Furthermore, for any r ∈ R + the following hold: (i) if τ r = ∞ then lim s→r τ s = τ r ; (ii) if τ r < ∞ then for any ε > 0 there are at most finitely many s ∈ [0, r] such that τ s+ > τ s + ε. Proof of Lemma 4.4. Observe that τ r (x) = inf{t ≥ 0 : sup 0≤s≤t x(s) ≥ r} is the generalized inverse [7] of the non-decreasing right-continuous function t → sup 0≤s≤t x(s) . Thus [7,Prop 2.3] r → τ r (x) is non-decreasing, has right limits and is left-continuous. It follows from the left continuity and monotonicity that τ r = ∞ implies the limit in (i). Assume τ r < ∞ and pick ε > 0. The  : P[τ s+ > τ s + ε] ≥ δ} for arbitrary ε, δ > 0, r ∈ R + . Claim. A r ε,δ is at most countable. Note first that the claim implies the lemma. By Lemma 4.4, the following equivalence holds for any r ∈ R + : lim s→r τ s = τ r ⇐⇒ τ r+ = τ r . Hence it suffices to show the set is at most countable, which clearly holds by the claim, where (ε k ) k∈N , (δ i ) i∈N and (s n ) n∈N are monotone sequences satisfying ε k ↓ 0, δ i ↓ 0 and s n ↑ ∞.
Proof of Claim. Assume that A r ε,δ is uncountable and let I be the set of its isolated points (i.e. x ∈ I if and only if x ∈ A r ε,δ and there exists a neighbourhood U of x in R + such that {x} = U ∩ A r ε,δ ). Then I is at most countable. To see this, note that for each x ∈ I there exists a rational number q x ≤ x, such that [q x , x) ∩ A r ε,δ = ∅ (for x ∈ I ∩ Q we may take q x := x). For any distinct points x, y ∈ I, it clearly holds q x = q y . Hence the cardinality of I is at most that of Q and the uncountable set A r ε,δ \ I has no isolated points. Consider r 1 := sup{y ∈ A r ε,δ \ I} ≤ r. There exists a strictly increasing sequence (p 1 i ) i∈N in A r ε,δ \ I with limit p 1 i ↑ r 1 . It is also clear that any ] ≥ δ and, for each path x ∈ B r1 , the function s → τ s (x) has infinitely many jumps of size at least ε on the interval [0, r 1 ]. Furthermore, since these jumps occur along a subsequence of (p 1 i ) i∈N , Lemma 4.4 implies for any x ∈ B r1 that τ s (x) < ∞ for all s ∈ [0, r 1 ) and τ r1 (x) = ∞.
We can now repeat the construction above, with A r ε,δ substituted by A r ε,δ , to define the event B r2 (for some r 2 ∈ (0, r ]) with properties analogous to those of B r1 . In particular P[B r2 ] ≥ δ and, since each x ∈ B r2 satisfies τ r2 (x) = ∞, it must hold B r1 ∩ B r2 = ∅. As before, there exists r < r 2 such that A r ε,δ is uncountable. By the same construction there exists r 3 ∈ (0, r ] and an event B r3 satisfying P[B r3 ] ≥ δ and B r3 ∩ (B r1 ∪ B r2 ) = ∅, since x ∈ B r3 satisfies τ r3 (x) = ∞ while for any x ∈ B r1 ∪ B r2 we have τ r3 (x) < ∞. We can thus inductively construct a sequence of pairwise disjoint events (B rn ) n∈N in D d each of which has probability at least δ > 0. This contradicts the fact that the total mass of P is equal to one. Lemma 4.7. Pick r > 0. Assume that x ∈ D d satisfies lim s→r τ s (x) = τ r (x) (see (4.9) for definition of τ r (x)). Then the function D d → [0, ∞], given by y → τ r (y), is continuous at x. If in addition it holds that either x(τ r (x)−) < r or x(τ r (x)) ≤ r, then the map D d → D d , given by y → y( · ∧ τ r (y)), is continuous at x.
The next task in the proof of Theorem 4.1 is to construct a limiting process. Lemma 4.9. Fix r 0 > 0. There exists a process Z r0 with paths a.s. in C d , such that for all but countably many r ∈ (0, r 0 ) it holds that (Z n k ( · ∧ τ r n k ), τ r n k ) ⇒ (Z r0 ( · ∧ τ r ), τ r ), (4.11) where τ r n = τ r (Z n ) is given in Theorem 4.1, τ r = τ r (Z r0 ) is defined in (4.9) and ⇒ denotes the weak convergence of probability measures on D d × [0, ∞]. Furthermore, the law of Z r0 ( · ∧ τ r ) equals that of a Bessel process (of dimension greater than one) stopped at level r. In particular it holds that (Z r0 ( · ∧ τ r ), τ r ) ∈ D d × R + a.s.
Proof. Lemma 4.2 implies the existence of a convergent subsequence (Z r0 n k ) k∈N of the sequence (Z r0 n ) n∈N defined in (4.5). Denote its limit by Z r0 . By (4.4) and the definition of the metric d : It hence follows that the sequence (Z n k ( · ∧ τ r0 n k )) k∈N also converges weakly to Z r0 . Furthermore, by [8, Thm 3.10.2, p. 148] and assumption (4.1), the process Z r0 is continuous, i.e. the support of its law is contained in C d .
Pick r ∈ (0, r 0 ). It follows from Lemmas 4.3 and 4.7 and the mapping theorem (see [4, p. 20]) that the joint convergence in (4.11) holds for all but countably many r < r 0 . Furthermore, from (4.11) we have that Z n k ( · ∧ τ r n k ) ⇒ Z r0 ( · ∧ τ r ) for all but countably many r < r 0 . By assumption in Theorem 4.1, the weak limit of Z n k is a Bessel process. Hence, again by Lemmas 4.3 and 4.7, the fact that a Bessel process EJP 24 (2019), paper 48.
Page 32/38 http://www.imstat.org/ejp/ has continuous trajectories and the mapping theorem [4, p. 20], the law of Z r0 ( · ∧ τ r ) equals that of a Bessel process stopped at level r for all but countably many r < r 0 . The final statement in the lemma is equivalent to saying that a Bessel process of dimension greater than one reaches every positive level with probability one. This is immediate in the transient case. In the recurrent case it follows from the fact that the height of excursions away from zero is not bounded.
Define the function F i,j : D d × R + → R by the formula F i,j (y, T ) := T 0 a ij (y(s))ds for any i, j ∈ {1, . . . , d}, where a ij is a coefficient in the generator G in Theorem 4.1.
Lemma 4.10. Fix r 0 > 0. Then for all but countably many r ∈ (0, r 0 ), the sequence of processes F i,j (Z n k , · ∧ τ r n k ) = (F i,j (Z n k , t ∧ τ r n k ); t ≥ 0) converges weakly to the process F i,j (Z r0 , · ∧ τ r ) = (F i,j (Z r0 , t ∧ τ r ); t ≥ 0) as k → ∞ for any i, j ∈ {1, . . . , d}, implicitly and follows directly from the continuity assumption on a ij in [8, Thm 7.4.1, p. 355] (which implies that F i,j is itself continuous at any continuous path) and the analogue of the the weak limit in (4.11). In our case the coefficient a ij is discontinuous at the origin and the process Z r0 may visit zero infinitely many times. Hence we must rely on the more detailed information about the limit law Z r0 ( · ∧ τ r ) . In particular, we use the fact that the Bessel process of dimension greater than one is a continuous semimartingale and apply the occupation times formula to quantify the amount of time it spends around zero.
Proof of Lemma 4.10. Let ε > 0 and take smooth functions φ ε 1 , φ ε 2 : Then since a ij is continuous on R d \ {0} and φ ε 1 is continuous and vanishes in a neighbourhood of 0, we have that F 1,ε i,j : D d × R + → R is continuous at any point (x, T ) ∈ C d × R + . Hence (4.11) in Lemma 4.9 implies the convergence F 1,ε i,j (Z n k , · ∧ τ r n k ) ⇒ F 1,ε i,j (Z r0 , · ∧ τ r ) for all but countably many r < r 0 .
Consider now F 2,ε i,j : D d × R + → R. Since a ij is globally bounded, there exists a constant C > 0 such that (4.12) By Lemma 4.9, we may assume that Z r0 ( · ∧ τ r ) is a Bessel process (of dimension greater than one) stopped at level r. The random field (L t (a)) t,a∈R+ of Bessel local times exists by [ (4.13) since the quadratic variation of Z r0 ( ·∧τ r ) is dominated by that of the Brownian motion and the support of φ ε 2 is contained in [0, ε]. Since (x, t) → t 0 φ ε 2 ( x(s) )ds is continuous on D d × R + , Lemma 4.9 and the mapping theorem [4, p. 20] imply (4.14) EJP 24 (2019), paper 48.
Lemma 4.10 is key in proving that the processes in (4.19) are true martingales, which will in turn imply that the limit Z r0 is a solution of the stopped martingale problem. We establish the martingale property in the next lemma.
Page 34/38 http://www.imstat.org/ejp/ Proof. Recall that the sequence (Z r0 n ) n∈N , defined in (4.5), is relatively compact by Lemma 4.2. Furthermore, the process Z r0 was defined as a weak limit of a convergent subsequence (Z r0 n k ) k∈N . For any i, j ∈ {1, . . . , d} the processesZ r0,i n k andÃ ij n k (see (4.5) for definition) give rise to martingalesZ r0,i n kZ r0,j n k −Ã ij n k (see the argument following the display in (4.6)). Hence, for any index i ∈ {1, . . . , d} and k ∈ N, we have that E[(Z r0,i n k (t)) 2 ] = E[Z i n k (0) 2 ] + E Ã ii n k (t) for all t ≥ 0.
Thus by (4.2), (4.7) and the assumption on the square integrability of Z n k (0) in Theorem 4.1, we have that sup k∈N E[ Z r0 n k (t) 2 ] < ∞ and hence the family ( Z r0 n k (t) ) k∈N is uniformly integrable for every t ≥ 0.
To prove that the components of Z r0 are martingales with respect to the natural filtration (σ(Z r0 u : u ∈ [0, s]), s ∈ R + ), note first that each σ-algebra σ(Z r0 u : u ∈ [0, s]) is generated by the π-system of events of the form {Z r0 (s 1 ) ∈ A 1 , . . . , Z r0 (s p ) ∈ A p } for any p ∈ N and s 1 , . . . , s p ∈ [0, s], where A 1 , . . . , A p are rectangular boxes in R d . Hence it is sufficient to show that for any 0 ≤ s 1 < . . . s p ≤ s < t and a non-negative, bounded, continuous f : R d ⊗ R p → R it holds that E[ Z r0,i (t) − Z r0,i (s) f (Z r0 (s 1 ), . . . , Z r0 (s p ))] = 0. By the Skorokhod representation theorem [8, Thm 3.1.8, p. 102] we may assume that the zero mean random variables Z r0,i n k (t) −Z r0,i n k (s) f (Z r0 n k (s 1 ), . . . ,Z r0 n k (s p )) converge almost surely as k → ∞ to the random variable in (4.20). Furthermore, since f is bounded, this sequence is uniformly integrable by the argument in the first paragraph of this proof. This implies the convergence in L 1 and hence the identity in (4.20). Since Z r0 is a martingale, so is Z r0 ( · ∧ τ r ) for any r ∈ (0, r 0 ).
Consider now the process in (4.19). We start by establishing the following fact.
To prove the claim it therefore suffices to show thatÃ ij n k ( ·∧τ r n k ) ⇒ ·∧τ r 0 a ij (Z r0 (s))ds.
SinceZ r0,i n kZ r0,j n k −Ã ij n k is a martingale by the argument following (4.6), the stopped process M k :=Z r0,i n k ( · ∧ τ r n k )Z r0,j n k ( · ∧ τ r n k ) −Ã ij n k ( · ∧ τ r n k ) is also a martingale for every EJP 24 (2019), paper 48.
Page 35/38 http://www.imstat.org/ejp/ k ∈ N. Hence the process in (4.19) will be a martingale by the analogous argument to the one that established the martingale property of Z r0,i above, if we prove that for any t ≥ 0 the family of random variables {M k (t) : k ∈ N} is uniformly integrable. With this in mind, note that 2|Ã ij n k | ≤Ã ii n k +Ã jj n k since the matrixÃ n k is non-negative definite. The elementary inequality 2|Z r0,i n kZ r0,j n k | ≤ (Z r0,i n k ) 2 + (Z r0,j n k ) 2 implies |M k (t)| ≤Z r0,i n k (t ∧ τ r n k ) 2 +Z r0,j n k (t ∧ τ r n k ) 2 +Ã ii n k (t ∧ τ r n k ) +Ã jj n k (t ∧ τ r n k ).
The right-hand side converges in L 1 by (4.1). Hence {Z r0,i n k (t ∧ τ r n k ) 2 : k ∈ N} is uniformly integrable and the lemma follows for all but countably many r ∈ (0, r 0 ). Note however that there exist r n ↑ r 0 such that the martingale properties in the lemma hold for all r n . Since a stopped martingale is a martingale, the lemma follows for all r ∈ (0, r 0 ).

Proof of Theorem 1.2
Recall the definition of the scaled process X n = ( X n (t), t ≥ 0) in (1.2) in terms of the chain X = (X m , m ∈ Z + ), X n (t) = n −1/2 X nt for t ∈ R + . Theorem 1.2 will follow from Theorem 4.1 and the main result of [10]: Lemma 4.13. Suppose that (A0)-(A4) hold. Without loss of generality assume that U = 1. Then X n converges weakly to the V -dimensional Bessel process started at 0.
Proof of Theorem 1.2. Define A n (t) = 1 n nt −1 m=0 M (X m ) , where M (x) is the covariance matrix of the increment at x ∈ X and, as usual, an empty sum is 0. Define Z n := X n and note that Z i n Z j n − A ij n is a local martingale for all i, j ∈ {1, . . . , d}. By Lemma 4.13 we have Z n ⇒ BES V (0) as n → ∞. Let a(x) := σ 2 (x) be a non-negative definite matrixvalued function on R d , where σ 2 satisfies (A3)-(A6). Let the generator G be defined as in Theorem 4.1 for this coefficient a. Then the C d martingale problem for (G, δ 0 ) is well-posed by Theorem 1.1, where δ 0 denotes the delta measure on R d concentrated at the origin. In order to apply Theorem 4.1, it remains to establish the assumptions (4.1), (4.2) and (4.3) for Z n and A n . Condition (4.1) follows from [10, Lem 2]. Since by assumption |M ij (y)| ≤ sup x∈X: x ≥r M (x) < ∞ for a sufficiently large r > 0 and any y ∈ X with y ≥ r, condition (4.2) follows from lim n→∞