Countable Systems of Degenerate Stochastic Diﬁerential Equations with Applications to Super-Markov Chains

We prove well-posedness of the martingale problem for an inﬂnite-dimensional degenerate elliptic operator under appropriate H˜older continuity conditions on the coe–cients. These martingale problems include large population limits of branching particle systems on a countable state space in which the particle dynamics and branching rates may depend on the entire population in a H˜older fashion. This extends an approach originally used by the authors in ﬂnite dimensions.


Introduction.
We prove existence and uniqueness of the martingale problem for the infinite-dimensional degenerate operator under suitable Hölder continuity assumptions on the coefficients γ i and b i . Here S is a countably infinite discrete set, we write x = (x i ) i∈S with x i ≥ 0 for each i, L operates on the class of finite-dimensional cylindrical functions, and f i and f ii denote the first and second partials of f in the direction x i .
In the last ten years there has been considerable interest in infinite-dimensional operators whose coefficients are only Hölder continuous rather than Lipschitz continuous. See [CD96], [D96], [L96], [Z00], and [DZ02], for example, which consider operators that are perturbations of either the infinite-dimensional Laplacian or of the infinite-dimensional Ornstein-Uhlenbeck operator. The operator L given above is not only infinite-dimensional, but also degenerate, due to the x i factor in the second order term. This degeneracy also means that the diffusion coefficient will not have a Lipschitz square root even for smooth γ i , invalidating the standard fixed point approaches.
The principal motivation for this work is the question of uniqueness for measurevalued diffusions which behave locally like a superprocess. In general assume S is a Polish space and let M F (S) denote the space of finite measures on S with the weak topology.
Write m(f ) = f dm for m ∈ M F (S) and an appropriate R-valued f on S. Assume {A x : x ∈ M F (S)} is a collection of generators, all defined on an appropriate domain D 0 of bounded continuous functions on S, and γ : S × M F (S) → R + . Let Ω be C(R + , M F (S)), equipped with the topology of uniform convergence on bounded intervals, its Borel σ-field F, canonical right-continuous filtration F t , and coordinate maps X t (ω) = ω t .
For each law µ on M F (S), a probability P on (Ω, F) is a solution of the martingale problem associated with A, γ and initial law µ, written M P (A, γ, µ), if for each f ∈ D 0 , where M f is a continuous F t -martingale such that Under appropriate continuity conditions on (A x , γ) one can usually construct solutions to M P (A, γ, µ) as the weak limit points of large population (N ), small mass (N −1 ) systems of branching particle systems. In these approximating systems a particle at x in population X t branches into a mean 1 number of offspring with rate N γ(x, X t ), and between branch times particles evolve like a Markov process with generator A X t (see e.g. [MR92]). The main difficulty lies in questions of uniqueness of solutions to M P (A, γ, µ). A case of particular interest is S = R d and in which particles evolve according to a state dependent Itô equation between branch times. For γ = γ 0 constant, uniqueness is proved in [DK98] under appropriate Lipschitz conditions on a, b, using methods in [P95]. The latter also effectively handles the case γ(y, X) = γ(y) (and some other special cases of X-dependence) by proving uniqueness for an associated strong equation and historical martingale problem.
Even in the case where S is finite, the problem of handling general γ was only recently solved in [ABBP02] and [BP03]. If S = {1, . . . , d}, then M F (S) = R d + and A x f (i) = d j=1 q ij (x)f (j), where for each x ∈ R d + , (q ij (x)) is a Q-matrix of a Markov chain on S, that is, q ij ≥ 0 for i = j and q ii = − j =i q ij . Then X solves M P (A, γ, µ) if and only if X t ∈ R d + solves the degenerate stochastic differential equation (1.1) Here γ i : R d + → [0, ∞), i = 1, . . . , d, B 1 , . . . , B d are independent one-dimensional Brownian motions, and X 0 has law µ for a given probability measure µ on R d + . More generally, consider the generator , the space of bounded continuous functions on R d + whose first and second partials are also bounded and continuous; b i (x) = j x j q ji (x) would correspond to (1.1). If µ is a law on R d + , a probability P on C(R + , R d + ) solves M P (L, µ) if and only if for all , Lf (X s ) ds is an F t -martingale under P and X 0 has law µ.

1.3)
Then there is a unique solution to M P (L, µ) for each law µ on R d + . A similar existence and uniqueness theorem was proved in [ABBP02] (see Theorem A of [BP03]) assuming only continuity of γ i and b i but with (1.2) strengthened to (1.4) A simple one-dimensional example shows these results are sharp in the sense that uniqueness fails if only continuity and (1.2) are assumed (see Section 8 of [ABBP03]). Clearly (1.2) is needed to ensure solutions remain in the positive orthant.
In this work we extend the method of [BP03] to the case where S is a countably infinite discrete set and hence take a step towards resolving the general uniqueness problem described above. Both [ABBP02] and [BP03] adapt the perturbation approach of [SV79] to this setting by considering L as a perturbation of If R λ is the resolvent associated with L 0 , the key step is to show that on a suitable Banach space (B, · ), one has (1.5) the space B is a weighted Hölder space ((4.9) below gives the precise norm). In both cases the constant C in (1.5) is independent of d. The L 2 setting in [ABBP02], however, does not appear to extend readily to infinite dimensions. There is the question of an appropriate measure on R ∞ + , there are problems extending the Krylov-Safonov type theorems on regularity of the resolvents which are required to handle all starting points as opposed to almost all starting points, and, as in the finite-dimensional setting, (1.4) will not hold for the most natural Q-matrices such as nearest random walk on the discrete circle. We therefore will extend the weighted Hölder approach in [BP03]. This approach has also been effective in other (non-singular) infinite-dimensional settings ( [ABP04]).
Our main result (Theorem 2.7) states that the natural infinite dimensional analogue of M P (L, ν) has a unique solution when S is a discrete countably infinite set and X takes on values in an appropriate space of measures. To understand the nature of the assumptions made on the coefficients b i and γ i , consider the Corollaries 2.10-2.12 when S = Z d . Basically, we require b i and γ i to be Hölder continuous in the j th variable, where the Hölder constant approaches 0 at a certain polynomial rate as |i − j| approaches ∞. The state space of X will be measures x(·) satisfying i |i| q x(i) < ∞ where q > 0 may approach zero for α close to 1 but becomes large as α gets small. There are cases where infinite measures are allowed but they require stronger Hölder conditions on the coefficients as the mass gets large (see Remark 2.13).
The main result and a number of corollaries are stated in Section 2. In Section 3 we prove a more general existence theorem (Theorem 2.4) by truncating to a finite-dimensional system and taking weak limits. Although these type of arguments are well-known (see [SS80]), we could not find the particular result we needed in the literature and have included the proof for completeness: in addition, there is an unexpected mild condition needed; see Theorem 2.4 and Remark 2.5. The weighted Hölder spaces are introduced in Section 4 where the infinite dimensional analogues of (1.5) are derived. Since the constants in [BP03] are independent of dimension this should be easy, but some complications arise in infinite dimensions since boundedness of the weighted Hölder norms does not imply continuity, in contrast to the case of finite dimensions. We must establish uniform convergence of the appropriate derivatives of the resolvent by the corresponding quantities for a sequence of approximating finite-dimensional functions to carry over the finite-dimensional estimates from [BP03] and obtain continuity (Proposition 4.8). The key bounds on the weighted Hölder norm then follow from the finite-dimensional result in [BP] (Corollary 4.10). This approximation is also used to derive the perturbation equation for the resolvent of strong Markov solutions of M P (L, µ) in terms of R λ = (λ − L 0 ) −1 (Proposition 5.4).
In Section 5 local uniqueness is established (Theorem 5.5), i.e., if γ i and b i are sufficiently close to constant functions, uniqueness is shown. In this setting our state space may include counting measure, but as the coefficients become asymptotically constant this is not surprising. In Section 6 we use the local uniqueness and a localization argument to prove Theorem 2.7. Localization in infinite dimensions still seems to be an awkward process and our arguments here are surely not optimal-we believe some of the additional continuity conditions in Assumption 2.6 may be weakened. Still it is important to note that the weighted Hölder spaces at least allow for localization. In their ground-breaking paper [DM95], Dawson and March establish a quite general uniqueness result in the Fleming-Viot setting but were unable to carry out the localization step. Nonetheless [DM95] still represents the best available uniqueness result in general infinite dimensional settings albeit in the Fleming-Viot setting and for close to constant coefficients. Finally in Section 7 we prove the various corollaries to Theorem 2.7.

Notation and statement of results.
We will use the letter c with or without subscripts to denote positive finite constants whose exact value does not matter and which may change from line to line. We use κ with subscripts to denote positive finite constants whose value does matter.
Let S be a countable set equipped with a map | · | : S → [0, ∞) such that S n = {i ∈ S : |i| < n} is finite for all n ∈ Z + . (S 0 = ∅). Our prototype is of course S = Z d with | · | equal to the usual Euclidean length of i. Let ν : S → (0, ∞) and for x ∈ R S let ν will be called a weight functon. We will use both ν i and ν(i) for the i th coordinate of ν and similarly for other maps defined on S.
We let and consider elements of M ν (S) as measures on S with It is easy to see that M ν (S) is a Polish space when equipped with the distance |x − x | ν . If ν i ≡ 1, it is easy to check that M ν (S) = M F (S) is the usual space of finite measures on S equipped with the topology of weak convergence for the discrete topology on S.
is the set of bounded continuous functions f : R S n + → R whose first and second partial derivatives f i , f ij are bounded and continuous. If x i = 0, then the partials f i , f ij are interpreted as right-hand derivatives.
Define the projection operator π n : R S + → R S n + by and define the lift operator Π n : These are the functions which only depend (in a C 2 way) on the coordinates provided these partial derivatives exist and the above series is absolutely convergent. Note that this is the case if f ∈ C 2 b,F (M ν (S)). Let Ω ν equal C(R + , M ν (S)), equipped with the topology of uniform convergence on bounded intervals. Let X t (ω) = ω(t) for ω ∈ Ω ν , let F 0 u be the universal completion of σ(X s : s ≤ u), and set Definition 2.1. Let µ be a probability on M ν (S). A probability P on (Ω ν , F) solves MP(L, µ), the martingale problem for L in M ν (S) started at µ, if P(X 0 ∈ ·) = µ(·) and for any f ∈ C 2 b,F (M ν (S)) Note that s → Lf (X s ) and t → f (X t ) and hence t → M f t are all necessarily continuous functions.
Remark 2.2 (a) As in Remark 1.1(d) of [BP03] one could also consider test functions f (x) = f n (π n x) for some f n ∈ C 2 b (R S n ), (i.e., those which extend in a C 2 manner to all of R S n instead of R S n + ).
(b) Changing the class of test functions changes the martingale problem. The smaller the class of test functions for which one establish uniqueness, the stronger the theorem. C 2 b,F (M ν (S)) is a reasonably small class. (c) Let {B i : i ∈ S} be a sequence of independent one-dimensional adapted Brownian motions on a filtered probability space (Ω, F, F t , P) and consider the stochastic differential equation Here Y (0) is an F 0 -measurable random vector in M ν (S). If Y is a continuous M ν (S)valued solution to (2.4), a simple application of Itô's formula shows that the law P Y of Y is a solution to MP(L, µ) where µ is the law of Y (0). Conversely, given a solution P of MP(L, µ), a standard construction allows one to build Brownian motions {B i : i ∈ S} and a solution Y to (2.4) on some Ω such that P is the law of Y .
We introduce conditions on b i , the first of which we assume throughout this work: (2.6) Assumption 2.3(a) will avoid explosions in finite time while a condition such as Assumption 2.3(b) is needed to ensure that our solutions have non-negative components (although a weaker condition b i (x) ≥ 0 if x i = 0 sufficed in finite dimensions -see [BP03].) Note that no analogue of (2.3) is needed for the diffusion coefficients.
Our focus is on uniqueness in law of solutions to MP(L, δ x 0 ), but as our setting is slightly different from that considered in the literature (e.g., Shiga and Shimizu [SS80]), we state a general existence result. The proof is given in Section 3.
Then for any x 0 ∈ M ν (S) there is a solution to MP(L, δ x 0 ).
Remark 2.5. Note the above continuity condition is trivially satisfied if γ i , b i are given as continuous functions on R S + with the product topology (as in Shiga and Shimizu [SS80]). This condition is only needed to obtain a compact containment condition in the usual tightness proof and will be easy to verify in the examples of interest. If γ i , b i on M ν (S) are uniformly continuous with respect to | · | βν , the above extensions exist.
Then the hypotheses of Theorem 2.7 are valid and so MP(L) is well-posed in M ν (S).
Corollary 2.9. Let (q ij ) be a Q-matrix satisfying  Consider now the case when S = Z d and |i| is the usual Euclidean norm.
Corollary 2.11. Let p : where q is as in Corollary 2.10. Let q ij = λp(j −i) (i = j) be the Q-matrix of a random walk which takes steps distributed as p with rate λ > 0. Let ν, ( b i ), and (γ i ) satisfy the hypotheses of Corollary 2.10, and Corollary 2.8 is a version of Theorem 2.7 where the hypotheses are given in terms of the Hölder constants of the γ i and b i ; Corollary 2.9 applies Corollary 2.8 to the case of super-Markov chains. Corollaries 2.10-2.12 are the application of Corollaries 2.8 and 2.9 to the case where S = Z d and an explicit bound is given for the Hölder constants of the γ i and b i .
Remark 2.13. If we assume Assumptions 2.6 (a),(b) and (2.9) we may take ν i → 0 so that M ν (S) will contain some infinite measures, that is, points x such that i∈S x i = ∞. In this case the Hölder condition Assumption 2.6 (b) becomes rather strong if x j gets large.

Existence.
If ε = {ε n } is a sequence in (0, ∞) decreasing to 0 and S = {S n } is a sequence of finite subsets of S which increases to S with S 0 = ∅, let Write K ε for the above in the case when S n = S n for all n ∈ Z + .
Lemma 3.1. (a) For any ε, S as above, K ε,S is a compact subset of M ν (S).
Proof. The standard proof is left for the reader.
, then for any sequence ε n decreasing to 0, it is easy to check that σ N (K ε ) ⊂ K ε .
Proof of Theorem 2.4. First, let X n t be the solution to Here (X n s ) + = ((X n,i s ) + : i ∈ S) and {B i } is a sequence of independent one-dimensional Brownian motions on some filtered probability space. Note that is a continuous function on R S n with linear growth (by (2.5) and the continuity assumptions on b i ). The same is true of , one can use Tanaka's formula to see that X n,i t ≥ 0 for all t ≥ 0 and for all i almost surely, and one may therefore is a martingale by (3.3) and so (2.5) implies The left hand side is clearly finite by the definition of T n k , Gronwall's lemma implies and so Fatou's lemma gives Therefore, using (2.6) in (3.4), we see that X n t is a submartingale. The weak L 1 inequality and (3.5) imply and define Then K is compact in M νβ (S) by Lemma 3.1. By (3.8) with ε 0 = β m k 1/2 and m = m k and (3.6) with k = K 0 we get that for each n P(X n t ∈ K for all t ≤ T ) (3.10) by (3.9). This will give us the compact containment required for the tightness of {P(X n · ∈ ·) : n ∈ N}. We claim next that if i ∈ S is fixed, then Let M n,i t denote the stochastic integral on the right hand side of (3.2). Then for s ≤ t, In addition by (2.5), This, (3.13), (3.2), and standard arguments now imply (3.12).
By Skorokhod's theorem we may first extract a weakly convergent subsequence {X n m } and then assume X n m → X a.s. in C([0, ∞), M νβ (S)). It is easy to use the continuity of b i , γ i on M νβ (S) to let n = n m → ∞ in (3.2) and conclude for all t ≥ 0 and all i ∈ S, a.s. Fatou's lemma implies for any t > 0 and so an elementary argument implies the last by (3.6). This proves and so X · has M ν (S) valued paths a.s. To show that X · has continuous M ν (S)-valued paths a.s. we use the following lemma, whose elementary proof is left to the reader. (3.16) (3.15) and (2.5) show that i∈S n which is bounded uniformly on compact time intervals a.s. as n → ∞, and so as n → ∞ by dominated convergence. By (3.16) and (2.5), By (3.15) and the Dubins-Schwartz theorem, this means { M n T } remains bounded in probability as n → ∞. Therefore A standard square function inequality now implies in probability as m, n → 0 for all T > 0, and so we may take a subsequence such that M n k converges uniformly on compact time intervals a.s. Let n = n k → ∞ in (3.16). The above and (3.17) show that the right hand side of (3.16) converges uniformly on compact time intervals a.s. to a necessarily continuous process. As the left hand side converges to |X t | ν for all t ≥ 0, a.s., it follows that t → |X t | ν is continuous. Lemma 3.3 therefore shows t → X t is a continuous M ν (S)-valued process. By (3.14) and Remark 2.2(c), the law of X is a solution of the martingale problem for L starting at x 0 .

Estimates.
We first obtain some key analytic estimates for the special case when for functions f for which the partial derivatives exist and the sum is absolutely convergent. By Theorem 2.4 there is a solution P 0 In fact it is easy to see that under P 0 x 0 , the processes {X i : i ∈ S} are independent diffusions and each X i is a suitably scaled squared Bessel process whose law is that of the pathwise unique solution to where the B i s are independent one dimensional Brownian motions. (Theorem 2.4 is only needed here to ensure X has paths in Ω ν .) An explicit formula for the transition kernel of p i t (x i , dy i ) of X i is given in (2.2) and (2.4) of [BP03]. Let (P t ) t≥0 and (R λ ) λ≥0 be the semigroup and resolvent, respectively, of the M ν (S)-valued diffusion X t = (X i t ) i∈S .
Lemma 4.1. For any compact set K ∈ M ν (S), T > 0, and ε > 0, there is a sequence η = {η n } decreasing to zero such that Proof. By Lemma 3.1(b) we may assume K = K δ for some sequence δ n decreasing to zero. Set for n > N ≥ 0 and where δ(N ), B(N ) ↓ 0 by (4.2). Since Z n,N (t) is a submartingale, the weak L 1 inequality implies that for any and then η > 1 sufficiently large so that The latter is possible by (4.4) with N = 0 since Z 0 (t) = |X t | ν . Now define Then for any by (4.5) and (4.6).
Remark 4.2. One difference between our infinite dimensional setting and the finite dimensional setting in [BP03] is that sup i |f | α,i < ∞ does not imply that f is uniformly continuous on I = {x ∈ M ν (S) : x(i) > 0 for all i ∈ S} and hence has a continuous extension to M ν (S). This is true on R d + (see Lemma 2.2 of [BP03]). Suppose we define f (x) to be 1 if infinitely many of the x(i) = 2 −i /ν(i) and 0 otherwise. Then |f | α,i = 0 but f is discontinuous on I. This complicates things a bit when checking whether various operators preserve C α .

Remark 4.3.
A key fact in our argument is that the estimates on (R λ f ) i and x i (R λ f ) ii from [BP03] are independent of the dimension of the space. Recall that the way we obtained the estimates in [BP03] was to first consider the one-dimensional case. If P i t denotes the semigroup corresponding to the operator xγ 0 , we then derived bounds on |(P t f ) i (x)| and on |(P t f ) i (x + ∆e j ) − (P t f ) i (x)| with the same constants (Propositions 5.1 and 5.2 of [BP03]); hence the constants did not depend on the dimension d of the underlying space. We then deduced estimates on (R λ f ) i . The same reasoning was applied for x i (R λ f ) ii .
Lemma 4.4. Let f ∈ C α , λ > 0, and i ∈ S. There is a κ 4.4 = κ 4.4 (α) independent of f, i, λ such that the following hold: (a) The partial derivative (R λ f ) i (x) exists for every x ∈ M ν (S) and satisfies (4.10) Proof. We only prove (b) as (a) is similar but easier. Let t > 0. Then argue as in the finite-dimensional argument (Proposition 5.1 of [BP03]), noting the constants there are independent of dimension, to see that (P t f ) ii exists on M ν (S) (in fact on R S + ) and satisfies If x i > 0, this allows one to differentiate through the time integral (by the dominated convergence and the mean value theorem) and conclude for x i > 0 that (R λ f ) ii exists and satisfies A simple calculation using (4.12) leads to (4.10) for x i > 0. The fact that x i (R λ f ) ii approaches 0 uniformly as x i ↓ 0 is then immediate, as is (4.11). Proof. By Lemma 3.1(b) we may assume K = K η for some sequence η = {η n } decreasing to 0. Then sup Since π n (K η ) ⊂ K η (recall Remark 3.2) and f is uniformly continuous on K η , the result follows.
Corollary 4.6. If f ∈ C b (M ν (S)) and f n = f • π n , then for any λ > 0, R λ f n → R λ f uniformly on compact subsets of M ν (S).
Proof. Let K be a compact subset of M ν (S) and ε > 0. Lemma 4.1 shows there is a compact K η such that sup Lemma 4.5 implies that R λ f 1 K η − R λ f n 1 K η ∞ → 0 as n → ∞ and so lim sup Let R n λ denote the resolvent of the finite-dimensional diffusion (X i ) i∈S n under {P x 0 }. Then R n λ is a Feller resolvent (i.e., it maps C b (R S n + ) to itself) and so if f n ((x i ) i∈S n ) = f (Π n (x)) for f ∈ C b (M ν (S)), then f n ∈ C b (R S n + ) and R λ f n (x) = R n λ f n ((x i ) i∈S n ) is continuous on M ν (S). (Π n is defined in (2.2).) The convergence in Corollary 4.6 therefore shows Our immediate goal is to extend the continuity on M ν (S) to (R λ f ) i and x i (R λ f ) ii for f ∈ C α . As explained earlier, this is more delicate in our infinite-dimensional setting.
Proof. (a) In view of (b) we may assume R ≥ 1. The integral in (a) is bounded by which is bounded by cR −1 r −1 (r + 1); see, e.g., Lemma 3.2(a) of [BP03]. This gives the required bound if r ≥ 1 (recall R ≥ 1). Assume now that 0 < r < 1. The integral in (a) is at most (b) If r ≥ 1 this is immediate from Lemma 3.2(a) of [BP03]. If 0 < r < 1, then the required integral is at most Proposition 4.8. Let f ∈ C α and f n = f • π n . If i ∈ S and λ > 0, then for any compact Proof. Note first that f ∈ C α implies f n ∈ C α and so the existence of the above partial derivatives follows from Lemma 4.4. We focus on the convergence of the second order derivatives as the first order derivatives are handled in a similar and slightly simpler way, while the resolvents themselves were handled in Corollary 4.6. Fix f ∈ C α . If y ∈ M ν (S), write is the point which has the same coordinates as y except that the i th coordinate is equal to v instead of y i . We may then define d n (v; If |d n (·; y i )| α denotes the | · | α,i norm of d n (·; y i ) with S = {i}, then |d n (·; y i )| α ≤ 2|f | α,i , and so the above derivative exists and satisfies (see Lemmas 4.1, 4.3, 4.5, and 4.6 of [BP03]) (4.14) Note that where differentiation through the integrals is justified by the above bound and dominated convergence. If g : R + → R, let g R = sup{|g(y)| : y ≤ R}. Assume first b 0 i > 0 and use (4.14) in [BP03] Use this and Lemma 4.7(b) in (4.16) to see that (4.17) A simple calculation (see Lemma 3.3(b) of [BP03]) shows that If N is a Poisson random variable with mean w, the series in I 2 is E |N − w|(r i + N + 1) −1/2 1 (N +r i >R/2γ 0 i t) . (4.19) (4.20) If r i < R/(4γ 0 i t), recalling x i < R/8 by (4.20), the expectation in (4.19) is at most If r i ≥ R/(4γ 0 i t), then (4.19) is bounded by the last by (4.20). Therefore under (4.20) Now use the above bounds in (4.17) to see that for R ≥ 8 max(1, γ 0 i t), (4.21) If b 0 i = 0, a slightly simpler argument starting with (4.6) in [BP03] will lead to the same bound.
Now choose a compact set K in M ν (S), T > 1 and ε > 0. Assume R is large enough and f ∞ R −1/4 < ε. Let η n be such that K η is a compact set satisfying the conclusion of Lemma 4.1. Let π i (y) = (y(j), j ∈ S − {i}) be the projection of y ∈ M ν (S) onto Then it is easy to use Lemma 3.1 to check that K η is compact, and so by Lemma 4.5 lim n→∞ sup y∈ K η |f n (y) − f (y)| = 0.

This implies lim
Use this with (4.14), (4.21), and (4.22) in (4.15) and conclude that for n ≥ N and t ∈ [0, T ] Use the above for t ∈ [T −1 , T ] and (4.14) for t ∈ [T −1 , T ] c to see that for x ∈ K and n ∈ N As T > 1 and ε > 0 are arbitrary, this gives The differentiation through the time integral in (4.24) does require x i > 0 as in the proof of Lemma 4.4, but that result shows the left-hand side is 0 if x i = 0.
Corollary 4.9. If f ∈ C α , then for any i ∈ S and λ > 0, Proof. Fix i and consider n large enough so that i ∈ S n . Recall R n λ is the resolvent of (X i ) i∈S n and f n (x) = f •Π n (x) on R S n + . Then R λ f n (x) = R n λ f n (π n x) and so by Proposition 5.3 of [BP03] (R λ f n ) i (x) = (R n λ ) i f n (π n x) and  There is a κ 4.10 > 0 depending only on α such that for all f ∈ C α , λ > 0, and i, j ∈ S, Proof. The proof is almost the same as that of Proposition 5.3 of [BP03] for the finitedimensional case -again the constants given there are independent of dimension; cf. Remark 4.3. The only change is that once the bounds on the increments of x i (R λ f ) ii are established for x i > 0, they follow for x i = 0 by the continuity established in Corollary 4.9; this is in place of the use of Lemma 2.2 in [BP03].

Local uniqueness.
We make the following assumption. 3) The following result uses only Assumption 5.1(a)-(b).
Note that L − L 0 can be well-defined for a larger class of functions than the domain of L and L 0 thanks to Assumption 5.1 and the results of Section 4. In particular, this lemma shows that (L − L 0 )R λ f is well-defined for f ∈ C α .
Proof. Lemma 4.4 shows that for f ∈ C α i∈S  Assumptions 5.1(a),(b) imply that if then {f n } is uniformly integrable with respect to counting measure on S and hence by (5.4) so is This allows us to take the limit as n → ∞ through the summation and conclude by the continuity of This proves (L − L 0 )R λ f is continuous.
Proof. Use Lemma 4.4 and Corollary 4.10 to see that for f ∈ C α , j ∈ S, and h > 0, The first summation is bounded by (use Assumption 5.1(c)) The second summation is bounded by (use Assumption 5.1(a)) We may therefore conclude Combine this with Lemma 5.2 to see that This, together with Lemma 5.2, shows B λ : C α → C α is a bounded operator with B λ ≤ 1/2 for ρ ≤ ρ 0 (α), λ ≥ λ 0 (α).
Let P µ be a solution of MP(L, µ) for some law µ on M ν (S) and for λ > 0, let The following result uses only (2.5), (2.7), and Assumption 5.1(a) (it only requires the bound in Lemma 5.2 and so does not require Assumption 5.2(b)). Recall the constant κ 2.3a in (2.5). We use bp −→ to denote bounded pointwise convergence.
Here P n t is the semigroup of {X i : i ∈ S n } under P 0 x . The finite dimensional analysis in the proof of Lemma 6.1 of [BP03] shows that g δ ∈ C 2 b (R S n + ). Use (2.4), (5.5), (2.5), a stopping time argument, and a Gronwall argument (cf. the proof of Theorem 2.4) to see that This and (2.7) shows that the stochastic integrals in (2.4) are square integrable martingales and by Itô's formula, the same is true of M g δ t , the martingale entering in MP(L, µ). Take expectations in MP(L, µ) to see that Let λ > κ 2.3a , multiply the above by λe −λt , and integrate over t ∈ [0, ∞) to conclude (5.7) Note here that (5.6) and λ > κ 2.3a are needed to apply Fubini's theorem, since ≤ c n,δ e κ 2.3a s |x| ν dµ + κ 2.3a s .
Now let δ ↓ 0 in (5.7). As δ → 0, g δ bp −→R λ f , and so λS λ g δ → λS λ R λ f and g δ dµ → R λ f dµ by dominated convergence. The finite-dimensional arguments in Lemma 6.1 of [BP03] show that as δ ↓ 0, The latter implies that S λ ((L−L 0 )g δ ) bp −→S λ B λ f and the former gives S λ (L 0 g δ ) bp −→λS λ R λ f − S λ f . Therefore we may let δ → 0 in (5.7) to derive the required equality. Now derive the result for a general f ∈ C α by approximation. Recall f n (x) = f • π n (x), and so |f n | α.i ≤ |f | α,i implies f n ∈ C α . By the above (5.9) Now Proposition 4.8 shows that each of the summands approaches 0 as n → ∞, while Lemma 4.4 and |f n | α,i ≤ |f | α,i show that the i th summand is at most This is summable by Assumption 5.1(a) and we may use dominated convergence in (5.10) to see that |B λ f n (x) − B λ f (x)| → 0 as n → ∞. The bound in Lemma 5.2 shows that the convergence is also bounded and so S λ B λ f n → S λ B λ f . Since f n bp −→f (Lemma 4.5), we also have S λ f n → S λ f and R λ f n dµ → R λ f dµ. Therefore we may let n → ∞ in (5.9) to complete the proof under (5.5).
To remove (5.5), let P N be the restriction of P µ to {ω ∈ Ω ν : Note B λ f and hence H B λ f,λ are bounded by the upper bound in Lemma 5.2. Now let N → ∞ in the above to finish the proof.
Proof. Let λ 0 be as in Proposition 5.3 and assume λ > λ 1 ≡ max(λ 0 , κ 2.3a ). Let P µ satisfy MP(L, µ). If f ∈ C α , then B λ f ∈ C α by Proposition 5.3, and so iterating Proposition 5.4 gives This shows the last term in (5.11) converges to 0 as n → ∞ and ∞ k=0 R λ B k λ f converges uniformly on M ν (S) to a bounded continuous function (recall (4.13)). Therefore letting n → ∞ in (5.11) we arrive at Inverting the Laplace transform (t → E µ (f (X t )) is continuous) one sees that for any t ≥ 0, E µ (f (X t )) is uniquely defined for all f ∈ C α . This shows P µ (X t ∈ ·) is unique (C α contains C 1 functions of finitely many coordinates with compact support). A standard result (see, e.g., Theorem 4.4.2 of Ethier-Kurtz [EK86]) now implies P µ is unique. Strictly speaking, the latter requires that Lf be bounded for our test functions f and M f t should be a martingale. However the only test functions we actually used were the functions g δ = ∞ δ e −λt P t f dt with f a function in C α depending on finitely many coordinates. In the proof of Lemma 5.4, the boundedness of Lg δ was made clear (see (5.8)), as was the fact that M g δ t is then a martingale (which is also then immediate as it is bounded on bounded time intervals).

Uniqueness.
Proof of Theorem 2.7. A standard argument shows that it suffices to show that for each z ∈ M ν (S) there is a unique solution to MP(L, δ z ) (see p. 136 of [Ba97].) Indeed, once this is established, Ex. 6.7.4 in [SV79] shows the laws of P z are Borel measurable in z and then it is easy to see P µ (·) = P z (·) µ(dz) is the unique solution to MP(L, µ).
Assumption 2.6 implies the continuity of b i and γ i on M βν (S). It is therefore easy to check that all the hypotheses of Theorem 2.4 are in force and hence existence holds.
Turning to uniqueness in MP(L, δ z ), let C be a compact set in M ν (S) containing z. Assume the following: , has a unique solution) for all y ∈ M ν (S). (6.1) We first show that the theorem would then follow by a minor modification of the localization argument in [SV79] (Theorem 6.6.1). Let P be a solution of MP(L, δ z ) and let T C = inf{t : X t / ∈ C}.
The tightness of P on Ω ν shows there are compact sets C n in M ν (S) increasing in n such that T C n ↑ ∞ P-a.s. It therefore suffices to show If δ(x 0 ) is as in (6.1) we may choose a finite subcover Let λ > 0 be a Lebesgue number for this cover, that is, a number λ such that for each Note T i ↑ T C a.s. as i → ∞ by the continuity of X in M ν (S). Let { P x 0 x : x ∈ M ν (S)} be the unique solutions to MP( L x 0 , δ x ) in (6.1). As noted above, x → P x 0 x is Borel measurable. If B(X T i , λ) ⊂ B j (where we choose the minimal such j = j(X T i )) and τ = inf{t : |X t − X 0 | > λ or X t ∈ C c }, then the uniqueness of P x j(X T i ) shows that conditional on F T i , X((· + T i ) ∧ T i+1 ) has law P x j(X T i ) X(T i ) (X(· ∧ τ ) ∈ ·). As in the proof of Theorem 6.6.1 of [SV79], this easily gives (6.2).
Next we claim Assume this for the moment. Then we may use this and the trivial bound δ j ≤ 1 in (6.8) to derive To prove (6.9) use the bound on β in Assumption 2.6 and x j < θ(j) ≤ c 5 ν(j) −1 to see As sup j γ 0 j < ∞, (6.9) follows and hence so does (6.10). Next we show If Assumption 2.6(b) holds this is immediate because h j ≤ h and R 2 = 0 if x j ≥ θ j . Assume Assumption 2.6(c). Then as h j ≤ θ j ≤ θ ∞ < ∞, we may apply Assumption 2.6(c) with h = h j and assume x j ≤ θ j ≤ θ ∞ to conclude Finally (6.10) and (6.11) show Assumption 5.1(c) holds for (γ j ,b j ). Next consider Assumption 5.1(b). Let x ∈ M ν (S) and η > 0. If as well. Then γ i (x) = γ i (x ) = γ 0 i and b i (x) = b i (x ) = b 0 i and so Assumption 5.1(b) holds as both sides are zero. (Here we are taking limits in the weaker norm | · | βν , as will be the case below.) Assume therefore that |(x ∧ θ) − x 0 | βν ≤ 2 3 δ 0 = 2δ. By Assumption 2.6(a) we may choose δ 1 < η so that if |x − x| βν < δ 1 , then Use (6.12) and (6.13) to see that Use Assumption 2.6(a) and our choice of δ 0 to see that (recall This completes the proof of (6.1) and hence the theorem if b i ≥ 0.
Hence (2.5) holds with b replaced by b; it follows that Assumption 2.3(a) holds for b. By (2.14), (2.10) with b replaced by b, and (2.11), Hence (2.10) holds with b. Next consider Assumption 2.6 with β = ν −α/2 . We may assume without loss of generality that b i satisfies Assumption 2.6(a) and (c) with β = ν −α/2 . Note that if x, x 0 ∈ M ν (S), then Assumption 2.6(a) follows for (b i ) i∈S as does the fact that b i , and hence b i , has a continuous extension to M ν 1−α/2 (S). If h ∈ (0, 1] and j ∈ S, then In view of (2.7) and (2.11) this shows b i , and hence b i , satisfies Assumption 2.6(c). This establishes the hypotheses of Theorem 2.7.
Proof of Corollary 2.10. Our choice of p implies sup j C(j) < ∞. The choice of q then easily gives (2.13). The required result now follows from Corollary 2.8.
This gives (2.15) and completes the proof.