Williams decomposition for superprocesses

We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and H\'{e}nard [Electron. J. Probab.,18,1-43,2013] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time. This generalizes a result of Tribe [Ann. Probab.,20,286-311,1992] in the sense that our branching mechanism is more general.


Introduction
Let X be a superprocess with a spatially dependent branching mechanism. We assume that the extinction time H of X is finite. In this paper we study the genealogical structure of X. More precisely, we give a spinal decomposition of X involving the ancestral lineage of the last individual alive, conditioned on H = h with h > 0 being a constant. This decomposition is called a Williams decomposition, in analogy with the terminology of Delmas and Hénard [4]. For a superprocess with spatially independent branching mechanism, the spatial motion is independent of the genealogical structure. As a consequence, the law of the ancestral lineage of the last individual alive does not depend on the original motion. Therefore, in this setting, the description of X conditioned on H = h may be deduced from Abraham and Delmas [1] where no spatial motion is taken into account. On the contrary, for a superprocess with nonhomogeneous branching mechanism, the law of the ancestral lineage of the last individual alive should depend on the spatial motion and the extinction time h. Delmas and Hénard [4] gave a Williams decomposition for superprocesses with a spatially dependent quadratic branching mechanism given by Ψ(x, z) = β(x)z + α(x)z 2 , under some conditions on β(x) and α(x) (see (H2) and (H3) in [4]). In [4], the Williams decomposition was established for superprocesses with spatially dependent quadratic branching mechanism by using two transformations to change the branching mechanism Ψ(x, z) to a spatially independent one, say ψ 0 , and then using the genealogy of superprocesses with branching mechanism ψ 0 given by the Brownian snake. As mentioned in [4], the drawback of the approach in [4] is that one has to restrict to quadratic branching mechanisms with bounded and smooth parameters.
The goal of this paper is to establish a Williams decomposition for more general superprocesses. Our superprocesses are more general in two aspects: first the spatial motion can be a general Markov process and secondly the branching mechanism is general and spatially dependent (see (2.1) below). We will give conditions that guarantee our general superprocesses admit a Williams decomposition. The conditions should be satisfied by a lot of superprocesses. We obtain a Williams decomposition by direct construction. For any fixed constant h > 0, we first describe the motion of a spine up to time h and then construct three kinds of immigrations (continuous immigration, jump immigration and immigration at time 0) alone the spine. We prove that, conditioned on H = h, the sum of the contributions of the three types of immigrations has the same distribution as X before time h, see Theorem 3.5 below. Note that for quadratic branching mechanisms, there is no jump immigration.
As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time, see Theorem 3.7 below. This generalizes a result of Tribe [15] in the sense that our branching mechanism is more general.

Superprocesses and assumptions
In this subsection, we describe the superprocesses we are going to work with and formulate our assumptions.
Suppose that E is a locally compact separable metric space. Let E ∂ := E ∪ {∂} be the onepoint compactification of E. ∂ will be interpreted as the cemetery point. Any function f on E is automatically extended to E ∂ by setting f (∂) = 0.
Let D E be the set of all the càdlàg functions from [0, ∞) into E ∂ having ∂ as a trap. The filtration is defined by F t = F 0 t+ , where F 0 t is the natural canonical filtration, and F = t≥0 F t . Consider the canonical process ξ t on (D E , {F t } t≥0 ). We will assume that ξ = {ξ t , Π x } is a Hunt process on E and ζ := inf{t > 0 : ξ t = ∂} is the lifetime of ξ. We will use {P t : t ≥ 0} to denote the semigroup of ξ. We will use B b (E) (B + b (E)) to denote the set of (non-negative) bounded Borel functions on E. We will use M F (E) to denote the family of finite measures on E and M F (E) 0 to denote the family of non-trivial finite measures on E.
Let M F (E) be the space of finite measures on E, equipped with the topology of weak convergence. As usual, f, µ := E f (x)µ(dx) and µ := 1, µ . According to [13,Theorem 5.12], there is a Hunt process X = {Ω, G, G t , X t , P µ } taking values in M F (E), such that, for every where u f (t, x) is the unique positive solution to the equation is called a superprocess with spatial motion ξ = {ξ t , Π x } and branching mechanism Ψ, or sometimes a (Ψ, ξ)-superprocess. In this paper, the superprocess we deal with is always this Hunt realization. For the existence of X, see also [3] and [5].
Define v(t, x) := − log P δx ( X t = 0), and H := inf{t ≥ 0 : X t = 0}. It is obvious that v(0, x) = ∞. In this paper, we will consider the critical and subcritical case. More precisely, throughout this paper, we assume that X satisfy the following uniform global extinction property.
We also assume that (H2) For any x ∈ E and t > 0, exists. Moreover, for any 0 < r < t, Note that, since t → v(t, x) is decreasing, we have w(t, x) ≥ 0. We also use v t and w t to denote the function x → v(t, x) and x → w(t, x) respectively.
Example 1 Assume that the spatial motion ξ is conservative, that is P t (1) ≡ 1, and the branching mechanism is spatially independent, that is where a ≥ 0, b ≥ 0 and ∞ 0 (y ∧ y 2 )n(dy) < ∞. We also assume that Ψ satisfies the Grey condition: Then { X t , t ≥ 0} is a continuous state branching process with branching mechanism Ψ(z). So v(t, x) = v(t) < ∞ does not depend on x, and lim t→∞ v(t) = 0, thus Assumption (H1) holds immediately. Moreover, for t > 0, we have that Thus Assumption (H2) is satisfied. See [10, Theorem 10.1] for more details.
In Section 5, we will give more examples, including some class of superdiffusions, that satisfy Assumptions (H1)-(H2).

Excursion law of {X t , t ≥ 0}
We use D to denote the space of M F (E)-valued càdlàg functions t → ω t on (0, ∞) having zero as a trap. We use (A, A t ) to denote the natural σ-algebras on D generated by the coordinate process.
Let {Q t (µ, ·) := P µ (X t ∈ ·) : t ≥ 0, µ ∈ M F (E)} be the transition semigroup of X. Then by (2.3), we have This implies that Q t (µ 1 + µ 2 , ·) = Q t (µ 1 , ·) * Q t (µ 2 , ·) for any µ 1 , µ 2 ∈ M F (E), and hence Q t (µ, ·) is an infinitely divisible probability measure on M F (E). By the semigroup property of Q t , V t satisfies that Moreover, by the infinite divisibility of Q t , each operator V t has the representation For λ > 0, we use V t λ to denote V t f when the function f ≡ λ. It then follows from (2.10) that for every x ∈ E and t > 0, The left hand side tends to − log P δx (X t = 0) as λ → +∞. Therefore, Assumption (H1) implies that λ t (x, E) = 0 for all t > 0 and hence x ∈ E 0 , which says that E = E 0 .
For x ∈ E, we get from (2.10) that It then follows from [13, Proposition 2.8 and Theorem A.40] that for every x ∈ E, the family of measures {L t (x, ·) : t > 0} on M F (E) 0 constitutes an entrance law for the restricted semigroup {Q 0 t : t ≥ 0}. It is known (see [13,Section 8.4]) that one can associate with (2.11) and, for every 0 < t 1 < · · · < t n < ∞, and nonzero µ 1 , · · · , µ n ∈ M F (E), (2. 12) This measure N x is called the Kuznetsov measure corresponding to the entrance law {L t (x, ·) : t > 0} or the excursion law for superprocess X. For earlier work on excursion law of superprocesses, see [6,8,12]. It follows from (2.11) that for any t > 0,

Main results
In this and the next section we will always assume that Assumptions (H1)-(H2) hold.
Recall that H := inf{t ≥ 0 : X t = 0}. Note that By the continuity of v(t, x) with respect to t ∈ (0, ∞), we get that for any t > 0, For h > 0, define Then, under P µ , {M h t , 0 ≤ t < h} is a nonnegative martingale with mean one (see Lemma 4.2 below).
We define, for each h > 0, Then, by Theorem 3.1, {X t , t < h; P µ (·|H = h)} has the same law as {X t , t < h; P h µ }, where P h µ is a new measure defined via the martingale M h t : Corollary 3.2 For any A ∈ G t , we have Proof: It follows from Fubini's theorem that where in the fifth equality we use the fact that ∂z . Then we have the following result whose proof will be given in Section 4.
. Given the trajectory of ξ h , we define three processes as follows: Immigration at time 0 Let {X 0,h t , 0 ≤ t < h} be a process distributed according to the law P µ (X ∈ ·|H < h).
We assume that the three processes X 0,h , X 1,h,N and X 2,h,P are independent given the trajectory of ξ h . Define We write the law of Λ h as P (h) µ .

Theorem 3.5 Under P
(h) µ , the process {Λ h t , t < h} has the same law as {X t , t < h} conditioned on H = h.
If we define Λ h t = 0, for any t ≥ h, then we get the following result.
Corollary 3.6 {X t ; P µ } has the same finite dimensional distribution as Proof: Let f k ∈ B + b (E), k = 1, 2, · · · , n and 0 = t 0 < t 1 < t 2 < · · · < t n . We put t n+1 = ∞ and define (t n , t n+1 ] := (t n , ∞). We will show that where the second equality follows from Theorem 3.5, and the third equality follows from Corollary 3.2. The proof is now complete. ✷ The decomposition (3.6) is called a Williams decomposition or spinal decomposition of the supperprocess {X t , t < h} conditioned on H = h, and ξ h = {(ξ t ) 0≤t<h , Π h ν } is called the spine of the decomposition. It gives us a tool to study the behavior of the superprocesses X near extinction, see Theorem 3.7 below. To state Theorem 3.7, we need the following assumption: (H3) For any bounded open set B ⊂ E and any t > 0, the function is finite for x ∈ B and locally bounded.
Theorem 3.7 Assume that (H1)-(H3) hold and that for any µ ∈ M F (E), where the limit above is in the sense of weak convergence. Moreover, conditioned on {H = h}, Z has the same law as , Note that, if the martingale {Y h t , 0 ≤ t < h} is uniformly integrable, then condition (3.7) holds. Now we give an example that satisfies Assumption (H3).
which satisfies the following two conditions: (A) (Uniform ellipticity) There exists a constant γ > 0 such that (B) a ij and b j are bounded Hölder continuous functions.
Suppose that the branching mechanism Ψ(x, z) satisfies that, for some α ∈ (1, 2] and c > 0, where R is the range of X, which is the minimal closed subset of R d which supports all the measures Therefore the superprocess X satisfies Assumption (H3).
Compared with [15], the example above assumes that the spatial motion ξ is a diffusion, while in [15], the spatial motion is a Feller process. However, in [15], the branching mechanism is binary (Ψ(z) = z 2 ), while in the example above, the branching mechanisms is more general.

Proofs of Main Results
We will use P r,δx to denote the law of X starting from the unit mass δ x at time r > 0. Similarly, we will use Π r,x to denote the law of ξ starting from x at time r > 0. First, we give an useful lemma.
For any 0 < t 1 ≤ t 2 ≤ · · · ≤ t n and 0 ≤ r ≤ t n , we have In particular, for any Proof: By [13, Propersition 5.14], we have that, for 0 ≤ r ≤ t n , Differentiating both sides of (4.3) with respect to θ and then letting θ → 0, we get that Proof: For any h > 0 and 0 ≤ t < h, by Assumption (H2) and the dominated convergence theorem, we get that where in the second equality, we used the Markov property of X. Thus, it follows that P µ (M h t ) = 1.
By the Markov property of X, we obtain that, for s < t < h, which implies that, under P µ , {M h t , t < h} is a nonnegative martingale. The proof is complete. ✷ Proof of Theorem 3.1: For any A ∈ G t , by the Markov property of X, By Assumption (H2), we get that and lim Note that, for 0 < ǫ < 1, Thus, it follows from the dominated convergence theorem that Thus, by (4.5) and (4.7), we have that The proof is now complete. ✷ Proof of Lemma 3.3: By the Markov property of X, we get that, . By (4.4) with h = t + s and µ = δ x , we get that where in the last equality we used Lemma 4.1 and the fact that u vs (t, x) = v(t + s, x). Thus, it follows immediately that For 0 < s < t, by the Markov property of ξ, we have that where the last equality above follows from (4.9). The proof is now complete. ✷

Williams decomposition
Proof of Theorem 3.5: By the definition of Λ h t , we have By the construction of Λ h t , we have We first deal with part (I). By (4.12), we have (4.14) Next we deal with part (II). By the definition of X 1,h,N and Fubini's theorem, we have Therefore, By the dominated convergence theorem, we obtain that, for s = t j , j = 1, 2, · · · , n, Hence, Now we deal with (III). Using arguments similar to those leading to (4.15), we get that Thus, Combining (4.16) and (4.17), we get that By (4.11), (4.14) and (4.18), we get that, for h > t, So, by (4.10), we obtain that Now we calculate J s (h, x) defined in (4.12). For 0 ≤ s < t < h, by the Markov property of X, we have that (4.20) Using Lemma 4.1 with r = 0, we have that Thus, by (4.19), we get that Now, the proof is complete. ✷

The behavior of X t near extinction
Recall that, for any Proof: By the decomposition (3.6), we have Note that E ∂ is a compact separable metric space. According to [14, Exercise 9.1.16 (iii)], C b (E ∂ ; R), the space of bounded continuous R-valued functions f on E ∂ , is separable. Therefore, C + b (E), the space of nonnegative bounded continuous R-valued functions f on E, is also a separable space. It suffices to prove that, for any f ∈ C + b (E), Therefore, it suffices to prove that, for any f ∈ C + b (E), Step 1 We first prove that given ξ h , Note that given ξ h , Let I 1 be the support of the measure N 1,h . Note that I 1 is a random subset of [0, h) × D.
In the remainder of this proof, we always assume that ξ h is given.
Let I 2 be the support of the measure N 2,h . Note that I 2 is a random countable subset of [0, h) × D.
Using arguments similar to those leading to (4.25), we get that where S 1 and S 2 are the set defined in (4.27) and (4.30). It follows that where in the second inequality we used the fact that Thus, N 2,h (S 1 ) < ∞, a.s., which implies that (4.36).
To prove (4.37) we only need to show that, given ξ h , yn(ξ s , dy)P yδ ξs (X ∈ dω)ds < ∞. In fact, where in the second inequality, we used the fact that P yδ ξs The proof is now complete. ✷ Proof of Theorem 3.7: Since {X t , t ≥ 0} is a Hunt process, t → X t is right continuous, which implies that where Q is the set of all rational numbers in [0, ∞). And, note that Thus, by Corollary 3.6 and Lemma 4.3, we get that Let V := lim t↑H Xt Xt . Then, for any f ∈ B + b (E), by Lemma 4.3, Thus, V is a Dirac measure of the form V = δ Z and the law of Z satisfies (3.8). The proof is now complete. ✷

Examples
In this section, we will list some examples that satisfy Assumptions (H1) and (H2). The purpose of these examples is to show that Assumptions (H1) and (H2) are satisfied in a lot of cases. We will not try to give the most general examples possible.
Example 3 Suppose that P t is conservative and preserves C b (E). Let A be the infinitesimal generator of P t in C b (E) and D(A) be the domain of A. Also assume that where sup x∈E α(x) ≤ 0 and inf x∈E b(x) > 0 and 1/b ∈ D(A). Then by Remark 2.2, we know that Auumption (H1) is satisfied. One can check that is a positive martingale under Π x . Thus we define another probability measure Π 1/b x by [4, (3.10) and Lemma 4.9] that w(t, x) exists and satisfies where c, β 0 are positive constants. Using this, one can check that Assumption (H2) is satisfied. This example shows that our result covers Delmas and Hénard [4,Corollary 4.14].
Now we give some examples of superprocesses, with general branching mechanisms, satisfying Assumptions (H1) and (H2).
Recall that the general form of branching mechanism is given by By (2.2), there exists K > 0, such that In the next two examples, we always assume that E = R d and that Ψ satisfies (2.7) and the following condition: for any M > 0, there exist c > 0 and γ 0 ∈ (0, 1] such that By Remark 2.2, condition (2.7) implies that Assumption (H1) is satisfied. Therefore, in the following examples, we only need to check that Assumption (H2) is satisfied.
Example 4 Assume that the spatial motion ξ is a diffusion on R d satisfying the conditions in Example 2. The branching mechanism Ψ is of the form in (2.1) and satisfies (2.7) and (5.2). Then the (ξ, Ψ)-superprocess X satisfies Assumptions (H1) and (H2).
We now proceed to prove the second assertion of the example above.
Furthermore, there exists a constant c such that for any t ∈ (0, 1], x ∈ R d and f ∈ B b (R d ), Proof: For t ∈ (n, n + 1], P t f (x) = P t−n (P n f )(x). Thus, we only need to prove the differentiability for t ∈ (0, 1]. It follows from [11,IV.(13.1) Thus by the dominated convergence theorem we have that for all t ∈ (0, 1] and x ∈ R d , and that for all t ∈ (0, 1], x ∈ R d and bounded Borel function f on R d , The proof is now complete. ✷ Then there is a constant c such that for any t ∈ (0, 1] and x, Proof: It follows from [11,IV.(13.1)] that there exist constants c 1 , c 2 > 0 such that for all t ∈ (0, 1] and x, x ′ ∈ R d , Hence for any t ∈ (0, 1] and x,  (iii) There exist constants s 0 ∈ (0, 1), C > 0 and γ ∈ (0, 1] such that for all s ∈ [0, s 0 ] and First, we will show that for any x ∈ R d , Thus, it suffices to prove that as t → 0. Thus, (5.10) is valid. For any 0 < t < t + r < s 0 , by the definition of G(t, x), Now we deal with part (I). For 0 < t < t + r < s 0 , using (5.27), we obtain that Thus, using the dominated convergence theorem, we get that, for any 0 ≤ t < t + r < s 0 , Combining (5.12) and (5.14), we get that Using similar arguments, we can also show that Thus, (5.9) follows immediately. The proof is now complete.

Recall that v(s, ·) is a bounded function and
Lemma 5.4 For any s > 0, there is a constant c(s) such that for t ∈ [0, 1/2) and x, y ∈ R d , Moreover, c(s) is decreasing in s > 0.
Since t → v(t, x) is continuous, the function s → Ψ t+s (x) = Ψ(x, v(t + s, x)) is continuous and, by (5.1), sup By (5.13) and Lemma 5.5 (2), we get that, for any s > t 0 , Combining (5.22) -(5.24), we get that, for t 0 > 0, which implies that, for any s 0 > 0, sup s>s 0 sup x∈R d w(s, x) < ∞. ✷ Now we give an example of a superprocess with discontinuous spatial motion and general branching mechanism such that Assumptions (H1) and (H2) are satisfied.
Example 5 Suppose that B = {B t } is a Brownian motion in R d and S = {S t } is an independent subordinator with Laplace exponent ϕ, that is The process ξ t = B St is called a subordinate Brownian motion in R d . Subordinate Brownian motions form a large class of Lévy processes. When S is an (α/2)-stable subordinator, that is, ϕ(λ) = λ α/2 , ξ is a symmetric α-stable process in R d . Suppose that Ψ is of the form in (2.1) satisfying (2.7) and (5.2). Suppose further that ϕ satisfies the following conditions: r dr < ∞. 2. There exist constants δ ∈ (0, 2] and a 1 ∈ (0, 1) such that a 1 λ δ/2 ϕ(r) ≤ ϕ(λr), λ ≥ 1, r ≥ 1. Now we proceed to prove the second assertion of the example above. The arguments are similar to that for the second assertion of Example 4. Without loss of generality, we will assume that ϕ(1) = 1. First we introduce some notation. Put Φ(r) = ϕ(r 2 ) and let Φ −1 be the inverse function of Φ. For t > 0 and x ∈ R d , we define For t > 0, x ∈ R d and β, γ ∈ R, we define Let p(t, x, y) = p(t, x − y) be the transition density of ξ and let {P t : t ≥ 0} be the transition semigroup of ξ. It is well known that {P t : t ≥ 0} satisfies the strong Feller property, that is, for any t > 0, P t maps bounded Borel functions on R d to bounded continuous functions on R d . Now we list some other properties of the semigroup {P t : t ≥ 0} which will be used later.
Furthermore, there exists a constant c such that for any t ∈ (0, 1], Proof: For t ∈ (n, n + 1], P t f (x) = P t−n (P n f )(x). Thus, we only need to prove the differentiability for t ∈ (0, 1]. It follows from [ and that for all t ∈ (0, 1], x ∈ R d and bounded Borel function f on R d , The proof is now complete. ✷ Lemma 5.8 Assume that f s (x) is uniformly bounded in (s, x) ∈ [0, 1] × R d , that is, there is a constant L > 0 so that, for all s ∈ [0, 1] and x ∈ R d , |f s (x)| ≤ L. Then there is a constant c such that for any t ∈ (0, 1] and x, Proof: It follows from [9, Proposition 3.3] that there exists a constant c 1 > 0 such that for all t ∈ (0, 1] and x, x ′ ∈ R d , Thus using (5.28) we get that for any t ∈ (0, 1] and x, It is well known that ϕ, the Laplace exponent of a subordinator, satisfies ϕ(λr) ≤ λϕ(r), λ ≥ 1, r > 0.
Proof: The proof of this lemma is similar as that of Lemma 5.4. We use Lemma 5.8 instead of Lemma 5.2. Here we omit the details. ✷
Proof: The proof of part (1) is exactly the same as that of part (1) of Lemma 5.5.
Using arguments similar to that in the proof of part (2) of Lemma 5.5 and using Lemma 5.10 instead of Lemma 5.4, we can get the result in part (2). Here we omit the details. ✷ Lemma 5.12 The functiom t → v t (x) is differentiable in (0, ∞), and for any s > 0 and t ∈ [0, 1/2), w(t + s, x) = − ∂ ∂t v t+s (x) satisfies that Moreover, t → w(t, x) is continuous and for any s 0 > 0, sup s>s 0 sup x∈R d w(s, x) < ∞.
Proof: Combining Lemmas 5.7, 5.9 and 5.11, and using arguments similar to that in the proof of Lemma 5.6, Lemma 5.12 follows immediately. ✷ Remark 5.13 Actually, by the same arguments and the results from [9], one check that in the example above, we could have replaced the subordinate Brownian motion by the non-symmetric jump process considered there, which contains the non-symmetric stable-like process discussed in [2].
Using these instead of (5.4) and (5.5), and repeating the arguments for Example 4, we can get the following example.

Example 6
Assume that E be is bounded smooth domain in R d and that the spatial motion is ξ E , which is the diffusion ξ of Example 2 killed upon exiting E. The branching mechanism Ψ is of the form in (2.1) and satisfies (2.7) and (5.2) on E. Then the (ξ E , Ψ)-superprocess X satisfies Assumptions (H1) and (H2).