Conditional Excursion Representation for a Class of Interacting

lwx3 (2004/05/15)

Conditional Excursion Representation for a Class of Interacting Superprocesses Zenghu Li 1 Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China E-mail: [email protected] Hao Wang 2 Department of Mathematics, University of Oregon, Eugene OR 97403-1222, U.S.A. E-mail: [email protected] Jie Xiong 3 Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, U.S.A. and Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, P.R. China E-mail: [email protected] Abstract A class of interacting superprocesses, called superprocesses with dependent spatial motion (SDSMs), has been introduced and characterized in Wang [22] and Dawson et al. [7]. In this paper, we give a construction or an excursion representation of the non-degenerate SDSM with immigration by making use of a Poisson system associated with the conditional excursion laws of the SDSM. As pointed out in Wang [22], the multiplicative property or summable property is lost for SDSMs and immigration SDSMs. However, summable property is the foundation of excursion representation. This raises a sequence of technical difficulties. The main tool we used is the conditional log-Laplace functional technique that gives the conditional summability, the conditional excursion law, and the Poisson point process for the construction of the immigration SDSMs. AMS Subject Classifications: Primary 60J80, 60G57; Secondary 60J35 Key words and phrases: superprocess, interaction, immigration, non-linear SPDE, conditional log-Laplace functional, excursion law

1

Supported by the NSFC (No. 10121101 and No. 10131040). Corresponding author. Research partially supported by the research grant of UO. 3 Supported partially by NSA and by Alexander von Humboldt Foundation. 2

1

1

Introduction

Wang ([21] [22]) and Dawson et al [7] have considered a model of interacting branching particle systems in which each individual is governed by a stochastic differential equation (SDE): Z dxi (t) = c(xi (t))dBi (t) + h(y − xi (t))W (dt, dy), t ≥ 0, i = 1, 2, · · · (1.1)

R

where B1 , B2 , · · · are independent Brownian motions and W is a Brownian sheet or a spacetime white noise which is independent of Bi ’s, c ∈ Cb2 (R), h ∈ Cb2 (R) is square-integrable with square-integrable derivative. Above Cbk (R) denotes the set of functions that, together with their derivatives up to order k inclusive, are bounded continuous on R. The subset of non-negative elements of Cbk (R) is denoted by Cbk (R)+ . Suppose that σ ∈ Cb2 (R)+ is the branching rate for all particles in the system. Let M (R) be the space of all finite measures on R with weak topology and let W E := C([0, ∞), M (R)). Then the corresponding limit superprocesses, which are called superprocesses with dependent spatial motion (SDSMs), of such interacting branching systems can be constructed and characterized as the unique solution on W E of the martingale problem associated with the operator L = A + B defined as follows: Z 1 d2 δF (µ) AF (µ) = µ(dx) a(x) 2 2 R dx δµ(x) Z δ 2 F (µ) 1 d2 µ(dx)µ(dy) (1.2) + ρ(x − y) 2 R2 dxdy δµ(x)δµ(y) 1 BF (µ) = 2

Z σ(x)

R

δ 2 F (µ) µ(dx), δµ(x)2

(1.3)

where a(x) = c2 (x) + ρ(0) and Z ρ(x) :=

R

h(y − x)h(y)dy.

(1.4)

For more details, see Wang [22] and Dawson et al. [7]. The SDSM can also be uniquely characterized by the following simplified martingale problem: ∀ φ ∈ S(R), the space of the infinitely differentiable functions that, together with all their derivatives, are rapidly decreasing at infinity, Z ® 1 t ­ 00 Mt (φ) ≡ hφ, Xt i − hφ, µi − aφ , Xu du 2 0 is a martingale with quadratic variation process Z t Z t Z ­ 2 ® hM (φ)it = σφ , Xu du + du 0

R

0

d2 dx2

2

ρ(y − z)φ0 (y)φ0 (z)Xu (dy)Xu (dz),

d where φ0 (x) = dx φ(x) and φ00 (x) = φ(x). Regarding the investigation of this model in different assumptions, it was proved in Wang [21] that Xt is absolutely continuous and Dawson et al [9] derived a stochastic partial differential equation (SPDE) for the density process for the

2

case that c(·) ≡ ² > 0. The absolute continuity was proved in Dawson et al [7] under the weaker assumption |c(·)| ≥ ² > 0. Wang [24] proved the convergence of SDSM to a super-Brownian motion in the case that c(·) ≡ ² > 0 and h converges to a singular function. When c(·) ≡ 0, Wang ([21], [23]) proved that Xt is purely atomic and Dawson et al [8] derived a degenerated SPDE and proved the existence of a unique strong solution. Li et al. [16] has constructed a singular, degenerate SPDE that connected to coalescing Brownian motions. For a general term of the excursion representation of a stochastic process, we mean a construction or representation of the stochastic process in terms of excursions and Poisson point processes. The first breakthrough in the excursion representation of Brownian motion was the seminal paper of Itˆo [12], although some ideas were already, at an intuitive level, in the work of L´evy, it was Itˆo who put the subject on a firm mathematical basis, thus supplying another cornerstone to Probability Theory. Since then, there are many important progresses in this field (see Dawson and Li [6] and references therein). In the present paper the following problems are tackled: Recently Dawson and Li [6] gives an excursion representation of the immigration SDSM with c ≡ 0 (degenerate SDSM) and Fu and Li [10] gives an excursion representation of immigration Dawson-Watanabe process. This naturally raises a question: Given c ∈ Cb2 (R) satisfing c(·) ≥ ² > 0, can we give an excursion representation of the immigration SDSM with this non-degenerate coefficient c? In the case that c ≡ 0, the SDSM is a purely-atomic measure-valued process. The corresponding location processes of the purely-atomic SDSM are (smooth) coalescing stochastic flows which never meet each other if their initial locations are separated. Over each stochastic flow or location process, it is just one dimensional Feller branching diffusion. [6] has well used this purely-atomic property of the degenerate SDSM and transformed the question of the excursion representation into the one dimensional case. Then, the problem of the excursion representation of the degenerate SDSM is solved based on the results of the one dimensional continuous state branching processes (for more details, See Dawson and Li [6] and Pitman and Yor [18]). In the case of immigration Dawson-Watanabe process, the motions of particles are independent and infinite divisible property holds. Then, the cluster representation of the Dawson-Watanabe process gives the entrance law (See Dawson [5], Fu and Li [10]). For our case, c(·) is a strictly elliptic function (i.e. there exists an ² > 0 such that c(·) ≥ ²). This corresponds to the non-degenerate case. In this case, the SDSM has density at each t > 0. On the other hand, the motions of particles of SDSM are always dependent based on the assumption of coefficient h 6≡ 0 and ρ is twice continuously differentiable with ρ0 and ρ00 bounded. Thus, to give an excursion representation for the non-degenerate SDSM with immigration, our first difficulty is how to construct the entrance law in this case. We already mentioned that in this case we don’t have cluster representation any more due to the dependency of particles’ motions. An even more challenging difficulty for us is the loss of summable property of the SDSM due to interaction. The summable property is just the multiplicative property which is the fact that a measure-valued Markov process has summable property if two such processes start at µ1 and µ2 , respectively, then their sum is equal in law to the same process starting at µ1 +µ2 . This is an obvious consequence of the independent behavior of the particles in the population. The summable property is the key to the successes in the study of the independent measure-valued Markov processes whatever the approaches to different problems be. In particular for the immigration and the excursion representation of the measure-valued Markov processes, even the definition of immigration process is directly given 3

based on the summable property (See Li-Shiga [14] Definition 1.1). Nevertheless, it is obvious that an interacting particle system has lost such a property and is only allowed instantaneous perturbation of motion, branching, and immigration due to loss of independent behavior of individuals. For more information, see the counter example given in Wang [22] for the summable property of SDSM. From above analysis, this seems to us that to find the excursion representation of the non-degenerate SDSM would be hopeless. However, the following intuitive idea brought sunshine into our dull struggling with this problem. The idea is looking the common space-time white noise or the Brownian sheet as a shared random environment or a common living ground. If we freeze the shared random environment (conditioned on the space-time white noise), the individual particles are still independent and summable. This is called conditional summability. We will see that the successful realization of this plan is based on the construction of the conditional or stochastic log-Laplace functional which is a unique strong solution of a nonlinear, backward stochastic partial differential equation. This stochastic log-Laplace functional plays the role for SDSM same as the log-Laplace functional does for the super-Brownian motion. The basic idea, thus, can be intuitively explained as follows: When freezing the random environment, the SDSM becomes a generalized, inhomogeneous super-Brownian motion if condition c(·) ≥ ² > 0 holds. Thus, all the results are intuitively natural generalization of the classical super-Brownian motion under conditional argument. However, this is by no means that everything is straight forward copy from the case of super-Brownian motion after freezing the random environment. Actually, although the stochastic log-Laplace functional technique can provide conditional summable property, it raises lot of new challenging problems related to the nonlinear SPDE and other issues. We will see that the stochastic log-Laplace functional will serve as a basic tool for a sequence of works. (See Xiong [26], [25] and Li et al [15] for motivations and more details of conditional log-Laplace functionals). For fixed integers m ≥ 1 and 0 ≤ k ≤ ∞, let C k (Rm ) be the set of functions on Rm having continuous derivatives up to order k and C∂k (Rm ) be the set of functions in C k (Rm ), which, ¯ m := Rm ∪ {∂}, together with their derivatives up to order k, can be extended continuously to R the one point compactification of Rm . Let C0k (Rm ) denote the set of functions in C k (Rm ), which, together with their derivatives up to order k, vanish at infinity. Cck (Rm ) stands for the set of functions in C0k (Rm ), which, together with their derivatives up to order k, have compact support. Let B(R) denote all the Borel functions on R. L2 (R) denotes the Hilbert space of square integrable function classes with inner product h·, ·i0 and norm k · k0 . If h is a function d h or ∂x h. Let Hm (R) denote the Sobolev space of classes of of x, we use h0 to denote either dx functions that, together with their derivatives in the sense of distribution up to mth order, are square integrable on R with norm defined by v um uX kφkm := t k∂xi φk20 , φ ∈ Hm (R), i=0

where k · k0 is the norm of L2 (R). {Hm (R) : m ≥ 0} are Hilbert spaces. In particular, we have T H0 (R) = L2 (R). Cbk (R) Hk (R) denotes the set of functions that, together with their bounded continuous R derivatives up to order k, are square integrable. For f ∈ B(R) and µ ∈ M (R), set hf, µi = R f dµ. Above notations will keep same meaning throughout the paper. To simply the statement of each theorem, here we give a statement that put several required 4

conditions together.

Basic Condition (A): In our model, we assume that the coefficients σ ∈ Cb2 (R), h ∈

T Cb2 (R) H2 (R), c ∈ Cb2 (R), and there exist constants ² > 0 and σb > σa > 0 such that c2 (x) ≥ ² and σa ≤ σ(x) ≤ σb . This article is organized as follows: In section 2, we discuss the stochastic log-Laplace equation, the existence of its unique strong solution, as well as the regularity of the solution. We generalize the results discussed in Li et al. [15] based on the results of Kurtz-Xiong [13] and Rozovskii [19]. We derive the ψ-semigroup property. In section 3, the conditional or stochastic log-Laplace functionals for SDSM will be discussed. In section 4, we will investigate the conditional generalized super-Brownian motion and give some basic tools for the conditional excursion representation of the SDSM. Section 5 will discuss the construction of the conditional entrance law for SDSM. The conditional excursion law of the SDSM, and the conditional excursion representation of the immigration SDSM (SDSMI) is discussed in section 6.

2

Stochastic log-Laplace equations and ψ-semigroups

Recall that we have assumed the basic condition (A) for the model coefficients. Let W (ds, dx) be a space-time white noise. For any given initial data φ ∈ L2 (R), we consider the following forward non-linear SPDE: ¸ Z t· 1 1 2 2 a(x)∂xx ψs (x) − σ(x)ψs (x) ds ψt (x) = φ(x) + 2 2 0 Z tZ + h(y − x)∂x ψs (x)W (ds, dy), t ≥ 0, (2.1) 0

R

and the following backward nonlinear SPDE: ¸ Z t· 1 1 2 2 ψr,t (x) = φ(x) + a(x)∂xx ψs,t (x) − σ(x)ψs,t (x) ds 2 2 r Z tZ + h(y − x)∂x ψs,t (x) · W (ds, dy), t ≥ r ≥ 0, r

R

(2.2)

where “·” denotes the backward Itˆo stochastic integral. In the following, first we discuss SPDEs (2.1) and (2.2) with initial data φ ∈ L2 (R). Although φ≡1∈ / L2 (R), we will need the solutions of SPDEs (2.1) and (2.2) with initial data φ ≡ 1. We will handle this case by a monotone convergence method. Let L2 ([0, T ], P, L2 (R)) stand for the space of all classes of predictable random mappings from [0, T ] to L2 (R), which are square integrable with respect to measure ` × P, where ` is Lebesgue measure on R. Definition 2.1 For a given initial data φ ∈ L2 (R), a stochastic process u ∈ L2 ([0, T ], P, L2 (R)) is called a generalized solution of equation (2.1) if, for every f ∈ Cc∞ (R), the space of infinitely differentiable with compact support, it satisfies the following equation : ¶ Z t µ 1 hut , f i0 = hφ, f i0 + − hau0s , f 0 i0 − ha0 u0s + σu2s , f i0 ds 0 2 Z tZ + hh(y − ·)u0s , f i0 dW (ds, dy), for any t ≥ 0, P-a.s.. (2.3) 0

R

5

For the case that φ ≡ 1, a continuous stochastic process u is called a generalized solution of equation (2.1) if, for every f ∈ Cc∞ (R), it satisfies the following equation : Z Z Z t µZ 1 ut (x)f (x)dx = φ(x)f (x)dx + us (x)(a(x)f 0 (x))0 dx 2 R RZ 0 RZ ¶ 0 0 2 + us (x)(a (x)f (x)) dx − σ(x)us (x)f (x)dx ds R R ¸ Z tZ ·Z ∂ − us (x) (h(y − x)f (x))dx dW (ds, dy), t ≥ 0, P-a.s.(2.4) ∂x 0 R R For any r ∈ [0, T ], a stochastic process vr,· ∈ L2 ([r, T ], P, L2 (R)) is called a generalized solution of equation (2.2) if, for every f ∈ Cc∞ (R), it satisfies the following equation : ¶ Z t µ 1 0 0 0 0 2 hvr,t , f i0 = hφ, f i0 + − havr,s , f i0 − ha vr,s + σvr,s , f i0 ds r 2 Z tZ 0 + hh(y − ·)vr,s , f i0 · dW (ds, dy), for any t ≥ r, P-a.s.. (2.5) r

R

For the case that φ ≡ 1, a continuous stochastic process u is called a generalized solution of equation (2.2) if, for every f ∈ Cc∞ (R), it satisfies the following equation : Z Z Z t µZ 1 vr,s (x)(a(x)f 0 (x))0 dx vr,t (x)f (x)dx = φ(x)f (x)dx + 2 RZ R r ¶ ZR 0 0 2 + vr,s (x)(a (x)f (x)) dx − σ(x)vr,s (x)f (x)dx ds R R ¸ Z tZ ·Z ∂ vr,s (x) (h(y − x)f (x))dx · dW (ds, dy), t ≥ r,P-a.s.(2.6) − ∂x r R R Remark: In order to use Rozovskii’s results (see [19]), above definition, we have used the divergent form of the principle term since a ∈ Cb2 (R). For the existence, the uniqueness, and the regularity of solution of the SPDEs (2.1) and (2.2), we generalize two theorems from Li et al. [15] in the following. T Theorem 2.1 Suppose that the basic condition (A) holds. Then, for any φ ∈ {Cb (R)+ T H1 (R)}, equation (2.1) has a unique Cb (R) H1 (R)-valued, non-negative, strong solution {ψt : t ≥ 0}. Furthermore, if φ ≡ 1, equation (2.1) has a unique Cb (R)-valued, non-negative, strong T solution {ψt : t ≥ 0}. For any φ ∈ {Cb (R)+ H1 (R)} ∪ {1}, kψt ka ≤ kφka holds P − a.s. for all t ≥ 0, where kφka is the supremum of φ. T Proof: For the case that φ ∈ {Cb (R)+ H1 (R)}, see the proof of Theorem 4.1 of Li et al. [15]. T For φ ≡ 1, let {φn : n ≥ 1} ⊂ {Cb (R)+ H1 (R)} be a monotone increasing sequence such that the following conditions are satisfied: 0 ≤ φn ≤ 1 and limn→∞ φn (x) = 1 for all x ∈ R. Let ψtn be the unique solution of (2.1) with initial function φn . Let ut (x) = ψsn+1 (x) − ψsn (x). Define ds (x) := σ(x)(ψsn+1 (x) + ψsn (x)),

6

(2.7)

which is nonnegative. Then ut is a solution to the following SPDE: ¸ Z t· 1 1 2 ut (x) = φn+1 (x) − φn (x) + a(x)∂xx us (x) − ds (x)(us (x)) ds 2 2 0 Z tZ + h(y − x)∂x us (x)W (ds, dy), t ≥ 0. 0

R

(2.8)

Note that ds (x) is not a Lipschitz function, the results of Kurtz-Xiong [13] is not directly applicable. However, the existence of a nonnegative solution given by the particle representation is still true. To prove ut (x) ≥ 0, we only need to prove the uniqueness for the solution of (2.8). Let vt (x) be the difference of any two solutions of (2.8). Then ¶ Z tµ 1 1 2 vt (x) = a(x)∂x vs (x) − ds (x)vs (x) ds 2 2 0 Z tZ + h(y − x)∂x vs (x)W (dsdy). 0

R

By an argument similar to the proof of Lemma 4.2 in [15], we can show that there exists a finite nonnegative K such that Z t

Ekvt k20 ≤ K

0

Ekvs k20 ds.

Then, Gronwall’s inequality implies v = 0. According to Kurtz-Xiong [13], equation (2.8) has a unique, non-negative solution. By direct calculation, we will see that (ψtn+1 − ψtn ) is a solution of (2.8). Thus {ψtn (x)} is a bounded increasing sequence for each x ∈ R. By definition 2.1, the limit is a unique solution of (2.1). T Theorem 2.2 Suppose that the basic condition (A) holds. Then, for any φ ∈ {Cb (R)+ T H1 (R)}, equation (2.2) has a unique Cb (R) H1 (R)-valued, non-negative, strong solution {ψr,t : t ≥ r ≥ 0}. Furthermore, if φ ≡ 1, equation (2.2) has a unique Cb (R)-valued, non-negative, T strong solution {ψr,t : t ≥ r ≥ 0}. For any φ ∈ {Cb (R)+ H1 (R)} ∪ {1}, kψr,t ka ≤ kφka holds P − a.s. for all t ≥ r ≥ 0, where kφka is the supremum of φ. T Proof: For any φ ∈ {Cb (R)+ H1 (R)}, see the proof of Theorem 4.2 of Li et al. [15]. For φ ≡ 1, the existence and the uniqueness can be proved by an argument similar to the proof of Theorem 2.1. Since the solution of (2.1) depends on the initial function φ(·), we can rewrite the solution of (2.1) as ψt (x) = ψt (x, φ). Based on this new notation, we say that ψt (x, φ), the solution of (2.1), defines a ψ-semigroup if there exists a set N ⊂ Ω such that P(N ) = 0 and for any T φ ∈ C(R)+ H1 (R) and 0 ≤ s ≤ t, ψs+t (x, φ) = ψt (x, ψs (·, φ)) holds for all ω ∈ / N . Based on this definition, we have following theorem.

7

(2.9)

T Theorem 2.3 Suppose that the basic condition (A) holds. Then, for any φ ∈ {Cb (R)+ T H1 (R)}, equation (2.1) has a unique {Cb (R) H1 (R)}-valued, non-negative, strong solution {ψt : t ≥ 0}. Moreover, the solution defines a ψ-semigroup. Proof: Based on the Theorem 2.1, we only need to prove the semigroup property. Let ψt be the unique strong solution of (2.1). Then, for any nonnegative s, t, we have ¸ Z s· 1 1 2 ψs (x) = φ(x) + a(x)∂xx ψu (x) − σ(x)ψu (x)2 du 2 2 Z sZ 0 + h(y − x)∂x ψu (x)W (du, dy), s ≥ 0. (2.10)

R

0

Z

s+t ·

¸ 1 1 2 2 a(x)∂xx ψu (x) − σ(x)ψu (x) du 2 2

ψs+t (x) = φ(x) + 0 Z s+t Z + h(y − x)∂x ψu (x)W (du, dy),

R

0

s≥0

(2.11) minus (2.10). Then, we get ¸ Z s+t · 1 1 2 2 a(x)∂xx ψu (x) − σ(x)ψu (x) du ψs+t (x) − ψs (x) = 2 2 s Z s+t Z + h(y − x)∂x ψu (x)W (du, dy), s≥0 s

R

t ≥ 0.

(2.11)

t ≥ 0. (2.12)

For any one-dimensional Borel set A, any 0 ≤ u ≤ v, and any fixed s ≥ 0 , define W s ([u, v], A) = W ([u + s, v + s], A) as the s-shifted space-time white noise of W . Since a(x), σ(x), h are time homogeneous, we can reform (2.12) to get ¸ Z t· 1 1 2 2 a(x)∂xx ψs+u (x) − σ(x)ψs+u (x) du ψs+t (x) = ψs (x) + 2 2 0 Z tZ + h(y − x)∂x ψs+u (x)W s (du, dy), s≥0 t ≥ 0. (2.13) 0

R

By the uniqueness of the strong solution of (2.1), for any fixed s ≥ 0, ψs+t (x, φ) is the unique strong solution of (2.12). On the other hand, for the same fixed s ≥ 0, just following the same idea to prove Theorem 2.1 we can prove that (2.13) has a unique strong solution which is just ψt (x, ψs (·, φ)) since the initial value is ψs (·, φ). This obviously gives that for any fixed s ≥ 0, ψs+t (x, φ) = ψt (x, ψs (·, φ)),

t ≥ 0,

(2.14)

holds for all ω ∈ / N with P(N ) = 0. The existence of the set N comes from the continuity of the unique strong solution of (2.1) and (2.13). Same as the forward case, since the solution of (2.2) depends on the initial value φ(·), we can rewrite the solution of (2.2) as ψs,t (x) = ψs,t (x, φ). Based on this new notation, we say that ψs,t (x, φ), the solution of the backward equation (2.2), defines a backward ψ-semigroup if there T exists a N such that P(N ) = 0 and for any φ ∈ Cb (R)+ H1 (R) and 0 ≤ r ≤ s ≤ t, ψr,t (x, φ) = ψr,s (x, ψs,t (·, φ)) holds for all ω ∈ / N . Based on this definition, we have following theorem. 8

(2.15)

T Theorem 2.4 Suppose that the basic condition (A) holds. Then, for any φ ∈ {Cb (R)+ T H1 (R)}, equation (2.2) has a unique Cb (R) H1 (R)-valued, non-negative, strong solution {ψr,t : t ≥ r ≥ 0}. Moreover, the solution of (2.2) defines a backward ψ-semigroup. T Proof: Based on the Theorem 2.2, we only need to prove that the Cb (R)+ H1 (R)-valued strong solution {ψr,t : t ≥ r ≥ 0} defines a backward ψ-semigroup. Let ψr,t (x) = ψr,t (x, φ) be the unique strong solution of (2.2). Then, for any nonnegative t, s, v, we have · ¸ Z t 1 1 2 2 ψt−s−v,t (x) = φ(x) + a(x)∂xx ψu,t (x) − σ(x)ψu,t (x) du 2 t−s−v 2 Z t Z + h(y − x)∂x ψu,t (x) · W (du, dy), t ≥ s + v, (2.16)

R

t−s−v

and Z ψt−s,t (x) = φ(x) + Z

t

+ t−s

·

t

t−s

Z

R

¸ 1 1 2 2 a(x)∂xx ψu,t (x) − σ(x)ψu,t (x) du 2 2

h(y − x)∂x ψu,t (x) · W (du, dy),

t ≥ s ≥ 0.

(2.17)

That (2.16) minus (2.17) gives Z ψt−s−v,t (x) − ψt−s,t (x) =

t−s

·

t−s−v Z t−s

+

¸ 1 1 2 2 a(x)∂xx ψu,t (x) − σ(x)ψu,t (x) du 2 2 Z h(y − x)∂x ψu,t (x) · W (du, dy), t ≥ v + s, (2.18)

t−s−v

R

In (2.18), for fixed t and s, let v change in t − s ≥ v ≥ 0. Then, ψt−s,t (x) is treated as the initial value. Since the upper limit of the integrals in the right hand side is t − s and the stochastic integral is backward, t − s must be the backward initial time for the solution. Then, without loss any generality, we can reform (2.18) to get ¸ Z t−s · 1 1 2 2 a(x)∂xx ψu,t−s (x) − σ(x)ψu,t−s (x) du ψt−s−v,t−s (x) − ψt−s,t (x) = 2 t−s−v 2 Z t−s Z + h(y − x)∂x ψu,t−s (x) · W (du, dy), t ≥ v + s,(2.19) t−s−v

R

By the uniqueness of the strong solution of (2.2), for any fixed s ≥ 0 and t − s ≥ 0, ψt−s−v,t (x, φ) is the unique strong solution of (2.18). On the other hand, for the same fixed s ≥ 0 and t − s ≥ 0, just following the same idea to prove Theorem 2.2 we can prove that (2.19) has a unique strong solution which is just ψt−s−v,t−s (x, ψt−s,t (·, φ)) since the initial value is ψt−s,t (·, φ). This obviously gives that for any fixed s ≥ 0 and t − s ≥ 0 ψt−s−v,t (x, φ) = ψt−s−v,t−s (x, ψt−s,t (·, φ)),

t − s ≥ v ≥ 0,

(2.20)

holds for all ω ∈ / N with P(N ) = 0. The existence of the set N comes from the continuity of the unique strong solution of (2.2) and (2.19).

9

3

Conditional log-Laplace functionals

In order to construct conditional entrance laws and conditional excursion laws, we need some conclusions and notations from conditional log-Laplace functional. Same as log-Laplace functional for Dawson-Watanabe processes, it will be demonstrated that conditional log-Laplace functional is a very powerful tool to handle the models in which Brownian branching particles move in a random medium. First, let us characterize the SDSM as a unique weak solution of a SPDE. T Theorem 3.1 Assume that c ∈ Cb2 (R), c2 (x) ≥ ² > 0, σ ∈ Cb (R), h ∈ Cb2 (R) H2 (R). Then, for any given µ ∈ M (R) and any φ ∈ C 2 (R), the following SPDE has a unique, continuous weak solution {Xt : t ≥ 0} : Z 1 t hφ, Xt i = hφ, µi + haφ00 , Xs ids 2 0 Z tZ Z tZ + φ(y)Z(ds, dy) + hh(y − ·)φ0 , Xs iW (ds, dy), (3.1)

R

0

0

R

where W (ds, dx) is a space-time white noise and Z(ds, dy) is an orthogonal martingale measure which is orthogonal to W (ds, dy) and has covariation measure σ(y)Xs (dy)ds. Proof: See the proofs of Theorem 1.2 of [9] or Theorem 3.1 of [15] for the construction of the weak solution of the equation (3.1). The uniqueness follows from the duality argument and the continuity of the solution follows from the Proposition 2 of Bakry-Emery [2] It is obvious that above {Xt : t ≥ 0} also solves the (L, δµ )-martingale problem. Let (Gt )t≥0 denote the filtration generated by {W (ds, dy)}, {Z(ds, dy)} and {Xs (dy)}. By T Theorem 2.4, for any φ ∈ Cb (R)+ H1 (R) the backward equation (2.2) has a unique strong solution ψr,t . The following are two main results of the conditional log-Laplace functionals. Theorem 3.2 Suppose that the basic condition (A) holds. Then, for any t ≥ r ≥ 0 and φ ∈ T Cb (R)+ H1 (R), we have a.s. Z tZ Z 1 t 2 hφ, Xt i = hψr,t , Xr i + ψs,t (x)Z(ds, dx) + hσψs,t , Xs ids. (3.2) 2 r R r Proof: See the proof of the Lemma 5.3 of [15]. Theorem 3.3 Suppose that the basic condition (A) holds. Let EW denote the conditional expectation of {Xt : t ≥ 0} given the space-time white noise W (ds, dy). Then, for t ≥ r ≥ 0 and T φ ∈ Cb (R)+ H1 (R), we have a.s. ½ ¾ W −hφ,Xt i E {e |Gr } = exp − hψr,t , Xr i . (3.3) In particular, if φ ≡ 1, we have a.s. ½ W

−h1,Xt i

E {e

|Gr } = exp 10

¾ − hΥr,t (x), Xr i ,

(3.4)

where Υr,t (x) is the unique, nonnegative, Cb (R)-valued solution of (2.2) with initial data φ ≡ 1. Consequently, {Xt : t ≥ 0} is a diffusion process with Feller transition semigroup (Qt )t≥0 given by ½ ¾ Z −hφ,νi e Qt (µ, dν) = E exp − hψ0,t , µi . (3.5)

R

M( )

T Proof: For φ ∈ Cb (R)+ H1 (R), see the proof of Theorem 5.1 of [15]. For φ ≡ 1, the conclusion follows from a monotone convergence sequence method.

4

Conditional Generalized Super-Brownian Motion

In this section, we derive some new properties of the conditional SDSM which are similar to that of super-Brownian motion. These properties are necessary for the conditional excursion representation of the SDSMI. First, let us consider the following backward SPDE: ¸ Z t· 1 2 Tr,t (x) = φ(x) + a(x)∂xx Ts,t (x) ds 2 r Z tZ + h(y − x)∂x Ts,t (x) · W (ds, dy), t ≥ r ≥ 0, (4.1) r

R

where “·” denotes the backward stochastic integral. (4.1) is just (2.2) with σ(·) ≡ 0. Thus, T according to Theorem 2.4, for any φ ∈ Cb (R)+ H1 (R), the backward equation (4.1) has a T unique Cb (R)+ H1 (R)-valued strong solution {Tr,t : t ≥ r ≥ 0} which is continuous in t. Moreover, the solution of (4.1) defines a backward ψ-semigroup. In order to give a better estimate of the solution of (4.1) by using the results of Rozovskii [19], here we introduce some new notations. Let {hj : j = 1, 2, · · ·} be a complete orthonormal system of L2 (R). Then Z tZ Wj (t) =

0

R

hj (y)W (ds, dy),

t≥0

defines a sequence of independent standard Brownian motions {Wj : j = 1, 2, · · ·}. For ² > 0 let [1/²] ²

W (dt, dx) =

X

hj (x)Wj (dt)dx,

s ≥ 0, x ∈ R.

j=1

Let L2 ([0, T ], P, Hk ) denote the space of Hk -valued, predictable, square-integrable stochastic processes and C([0, T ], P, Hk ) denote the space of Hk -valued, strongly continuous stochastic T T processes. Assume that c ∈ Cb2 (R), c2 (x) ≥ ² > 0, h ∈ Cb2 (R) H2 (R). For any φ ∈ {Cb (R)+ H1 (R)}, by Rozovskii ([19] p133, Theorem 2), the equation ¸ Z t· 1 ² 2 ² Tr,t (x) = φ(x) + a(x)∂x Tr,s (x) ds 2 r Z tZ ² + h(y − x)∂x Tr,s (x) · W ² (ds, dy), t ≥ r ≥ 0, (4.2) r

R

11

² (x) ∈ L2 ([0, T ], P, H2 ) has a unique solution Tr,t

T

C([0, T ], P, H1 ) and the following inequality

² 2 E sup kTr,s k2 ≤ KEkφk22 s∈[r,T ]

holds. By a limit argument similar to the proof of Rozovskii ([19] p111, Theorem 2), we can get that Tr,s (x), the solution of (4.1), satisfies E sup kTr,s k22 ≤ KEkφk22 .

(4.3)

s∈[r,T ]

In the following, in order to emphasize some special points, we use Tr,t (x) = Tr,t (x, φ) and ψr,t (x) = ψr,t (x, φ) to denote the unique solution of (4.1) and (2.2), respectively. Similar to the T super-Brownian motion case, for any φ ∈ Cb (R)+ H1 (R) we consider the following stochastic equation: Z 1 t Ψr,t (x) = Tr,t (x, φ) − Tr,s [σ(x)(Ψs,t (x))2 ]ds, (4.4) 2 r where Tr,t (x, φ) is the unique strong solution of (4.1). From the inequality (4.3), we can prove that the equation (4.4) has a unique solution by the Picard iterative scheme. Then, we have following theorem. T Theorem 4.1 Suppose that the basic condition (A) holds. Then, for any φ ∈ Cb (R)+ H1 (R), (4.4) has a unique strong solution that defines a ψ-semigroup for all ω ∈ / N with P(N ) = 0. Let {ψr,t : 0 ≤ r ≤ t} denote the unique strong solution of the equation (4.4). Then, for each µ ∈ M (R) and given space-time white noise W there is a unique conditional probability measure E QW µ on W ≡ C([0, ∞), M (R)) such that ½ ¾ Z −hφ,wt i W e Qµ (dw) = exp − hψ0,t , µi ω∈ / N, (4.5) WE

holds and the coordinate process {wt : t ≥ 0} on W E under the system {QW µ : µ ∈ M (R)} defines a conditional diffusion process, called conditional generalized super-Brownian motion, with transition semigroup {(QW r,t ) : t ≥ r ≥ 0} given by ½ ¾ Z e−hφ,νi QW (µ, dν) = exp − hψ , µi , ω∈ / N. (4.6) r,t r,t

R

M( )

Furthermore, (4.4) is equivalent to (2.2). Thus, {ψr,t : 0 ≤ r ≤ t} in (4.6) is also the unique strong solution of (2.2). E and the conditional transition semiProof: The existence of the unique measure QW µ on W W group {(Qr,t ) : t ≥ r ≥ 0} such that (4.5) and (4.6) hold follows from the construction of the conditional log-Laplace functional for SDSMI with b = 0 = m given in the section 5 in [15]. Let Tr,t (x) = Tr,t (x, φ) be the unique strong solution of (4.1), which is just the strong solution of (2.2) with σ(·) ≡ 0. By Theorem 2.4, we know that {Tr,t : t ≥ r ≥ 0} is a backward ψsemigroup. To complete the proof of the theorem, it suffices to prove that (4.4) is equivalent to (2.2). To this end, in the following we prove that given a solution of (4.4), we can change the form

12

of (4.4) into that of (2.2) by a stochastic Fubini theorem (See Theorem 2.6 of Walsh [20]) as T follows: For any φ ∈ Cb (R)+ H1 (R), let ψr,t (x) be a solution of (4.4). Thus, we have Z 1 t Tr,s [σ(x)(ψs,t (x))2 ]ds (4.7) ψr,t (x) = Tr,t (x) − 2 r Z t Z tZ 1 2 = [ a(x)∂xx Tu,t (x)]du + h(y − x)∂x Tu,t (x) · W (du, dy) r 2 r R Z Z ½Z s 1 1 t 1 t 2 2 − [σ(x)(ψs,t (x)) ]ds − a(x)∂xx Tu,s [σ(x)(ψs,t (x))2 ]du 2 r 2 r 2 r ¾ Z sZ 2 + h(y − x)∂x Tu,s [σ(x)(ψs,t (x)) ] · W (du, dy) ds + φ(x) r R ¾ Z t Z ½Z t 1 1 1 t 2 2 2 = [ a(x)∂xx Tu,t (x)]du − a(x)∂xx Tu,s [σ(x)(ψs,t (x)) ]ds du 2 r r 2 u 2 Z Z tZ 1 t 2 +φ(x) − [σ(x)(ψs,t (x)) ]ds + h(y − x)∂x Tu,t (x) · W (du, dy) 2 r r R ¾ Z t Z ½Z t 1 2 h(y − x)∂x Tu,s [σ(x)(ψs,t (x)) ]ds · W (du, dy) − 2 r R u Z t Z 1 1 t 2 = [ a(x)∂xx ψu,t (x)]du − [σ(x)(ψs,t (x))2 ]ds 2 r r 2 Z tZ h(y − x)∂x ψu,t (x) · W (du, dy) + φ(x), + r

R

which gives (2.2) and the proof is complete. T Theorem 4.2 Suppose that the basic condition (A) holds. For any φ ∈ Cb (R)+ H1 (R), let QW r,t (µ, dν) be the transition semigroup constructed from theorem 4.1 and Tr,t (x) = Tr,t (x, φ) be the unique strong solution of (4.1). Then, we have Z hφ, νiQW ω∈ /N (4.8) r,t (µ, dν) = hTr,t , µi,

R

M( )

and

Z

Z 2

R

M( )

hφ, νi

QW r,t (µ, dν)

2

= hTr,t , µi +

r

t

hTr,s [σ(x)(Ts,t (x))2 ], µids,

ω∈ /N

(4.9)

λ (x) be the unique strong solution of the Proof: For any non-negative real number λ, let ψr,t following equation: Z 1 t λ λ ψr,t (x) = Tr,t (λφ(x)) − Tr,s [σ(x)(ψs,t (x))2 ]ds (4.10) 2 r λ (x) is the which is just the equation (4.4) with initial value λφ(x) and where Tr,t (λφ(x)) or Tr,t unique strong solution of the equation ¸ Z t· 1 λ 2 λ Tr,t (x) = λφ(x) + a(x)∂xx Ts,t (x) ds 2 r Z tZ λ + h(y − x)∂x Ts,t (x) · W (ds, dy), t ≥ r ≥ 0, (4.11) r

R

13

λ (x)| where “·” denotes the backward stochastic integral. From (4.11) and (4.10), we have ψr,t λ=0 λ (x) and ψ λ (x) are differentiable with respect to λ in the norm ≡ 0. Now we prove that Tr,t r,t λ (x) = λT (x) is a solution of (4.11). Since (4.11) has k · k0 . First, we can directly check that Tr,t r,t λ (x) is differentiable with respect to λ and ∂ T λ (x) = T (x) uniqueness, it is obvious that Tr,t r,t ∂λ r,t ∂2 λ λ −1 λ and ∂λ2 Tr,t (x) ≡ 0. Let Zs,t (x) = λ ψs,t (x) − Ts,t (x). According to (4.10) and (4.11), we have

Z λ Zr,t (x)

−1

= −(2λ) 1 =− 2 1 =− 2

r

Z

t

r

Z

t

r

t

λ Tr,s [σ(x)ψs,t (x)2 ]ds

(4.12)

λ λ λ Tr,s [σ(x)λ−1 ψs,t (x)2 − σ(x)ψs,t (x)Ts,t (x) + σ(x)ψs,t (x)Ts,t (x)]ds λ λ Tr,s [σ(x)ψs,t (x)Zs,t ]ds

1 − 2

Z r

t

λ Tr,s [σ(x)ψs,t (x)Ts,t (x)]ds

λ (x)k ≤ λkφk , kT (x)k ≤ kφk , Gronwall’s inequality yields Since kψs,t a a r,s a a λ 2 EkZr,t k0 → 0

Let

Z ur,t (x) = −

t

r

as λ → 0.

Tr,s [σ(x)Ts,t (x)2 ]ds

and 2λ λ uλs,t (x) = λ−2 [ψs,t (x) − 2ψs,t (x)] − us,t (x).

According to (4.10) and (4.11), we have ½ Z 1 t λ −2 2λ ur,t (x) = λ − Tr,s [σ(x)ψs,t (x)2 ]ds 2 r ¾ Z t Z t λ + Tr,s [σ(x)ψs,t (x)2 ]ds + Tr,s [σ(x)Ts,t (x)2 ]ds r

r

 #2 #2  " " Z t  λ 2λ ψs,t (x) ψs,t (x)   ds = Tr,s σ(x) Ts,t (x)2 + −2   λ 2λ r 

λ (x)k ≤ λkφk , kT (x)k ≤ kφk , and Since kψs,t a a r,s a a λ 2 EkZr,t k0 → 0

as λ → 0,

Ekuλr,t k20 → 0

as λ → 0.

we get

Therefore, we have

λ (x) ∂ψr,t ∂λ |λ=0

= Tr,t (x) and

λ (x) ∂ 2 ψr,t |λ=0 = − ∂λ2

Z

t

r

Tr,s [σ(x)(Ts,t (x))2 ]ds.

14

(4.13)

Then, the conclusion follows from taking derivative with respect to λ in the following equation ½ ¾ Z −hλφ,νi W λ e Qr,t (µ, dν) = exp − hψr,t , µi , ω∈ / N, (4.14)

R

M( )

and then set λ = 0.

5

Conditional Entrance Laws for SDSM

Before we start to construct the conditional entrance law for SDSM, we first give a required lemma which gives the monotonicity of the solution of the SPDE (2.1) in the coefficient σ(·). T Lemma 5.1 Suppose that h ∈ Cb2 (R) H2 (R), c ∈ Cb2 (R), σi (x) ∈ Cb2 (R), i = 1, 2, and 0 ≤ σ1 (x) ≤ σ2 (x). For i = 1, 2, let ψti (x) be the unique strong solution of the following equation: ¶ Z tµ 1 1 ψti (x) = φ(x) + a(x)∂x2 ψsi (x) − σi (x)ψsi (x)2 ds 2 2 0 Z tZ + h(y − x)∂x ψsi (x)W (dsdy). 0

R

Then ψt1 (x) ≥ ψt2 (x). Proof: Let ut (x) = ψt1 (x) − ψt2 (x). Then ¶ Z tµ 1 1 2 a(x)∂x us (x) − ds (x)us (x) ds ut (x) = 2 2 0 Z tZ + h(y − x)∂x us (x)W (dsdy) 0 R Z t + cs (x)ds,

(5.1)

0

where ds (x) = σ1 (x)(ψs1 (x) + ψs2 (x)) ≥ 0, and

1 cs (x) = (σ2 (x) − σ1 (x))ψs2 (x)2 ≥ 0. 2 Similar to (2.8) we can show that (5.1) has at most one solution. For φ ≥ 0, we consider equation ¶ Z tµ 1 1 2 yt (x) = φ(x) + a(x)∂x ys (x) − ds (x)ys (x) ds 2 2 r Z tZ + h(y − x)∂x ys (x)W (dsdy). r

R

15

(5.2)

c (x, φ). By Kurtz-Xiong [13], we know that T c (x, φ) ≥ 0. Denote the solution to (5.2) by Tr,t r,t Now we claim that Z t

θt (x) =

0

c Tr,t (x, cr )dr ≥ 0

(5.3)

is a solution to (5.1). From (5.2), we have ¶ Z tµ 1 1 c 2 c c Tr,t (x, cr ) = cr (x) + a(x)∂x Tr,s (x, cr ) − ds (x)Tr,s (x, cr ) ds 2 2 r Z tZ c + h(y − x)∂x Tr,s (x, cr )W (dsdy). r

R

Hence Z θt (x) =

Z tZ tµ

t

¶ 1 1 2 c c a(x)∂x Tr,s (x, cr ) − ds (x)Tr,s (x, cr ) dsdr 2 2

cr (x)dr + 0 r Z tZ tZ c + h(y − x)∂x Tr,s (x, cr )W (dsdy)dr 0 r R ¶ Z t Z tZ sµ 1 1 2 c c a(x)∂x Tr,s (x, cr ) − ds (x)Tr,s (x, cr ) drds = cr (x)dr + 2 2 0 0 0 Z tZ Z s c + h(y − x)∂x Tr,s (x, cr )drW (dsdy) 0 R 0 ¶ Z t Z tµ 1 1 = cr (x)dr + a(x)∂x2 θs (x) − ds (x)θs (x) ds 2 2 0 0 Z tZ + h(y − x)∂x θs (x)W (dsdy). 0

0

R

This finishes the proof of the claim, and hence, the proof of the lemma. W ◦ Let {Q◦,W r,t : t ≥ r ≥ 0} be the restriction of {Qr,t : t ≥ r ≥ 0} on M (R) := M (R) \ {0}.

T Theorem 5.2 Suppose that the basic condition (A) holds. For any φ ∈ Cb (R)+ H1 (R), let {ψr,t : 0 ≤ r ≤ t} be the unique strong solution of the backward equation (2.2) which defines a ψ-semigroup for all ω ∈ / N with P(N ) = 0. Then, for each x ∈ R, any t > r ≥ 0, and for all ◦ ω∈ / N , there is a unique finite random measure LW r,t (x, dν) on M (R) such that Z (1 − e−hφ,νi )LW (5.4) r,t (x, dν) = ψr,t (x) = ψr,t (x, φ)

R

M ( )◦

◦,W holds. Furthermore, LW r,t (x, dν) is an entrance law for {Qr,t : t ≥ r ≥ 0}, i.e. for each x ∈ R, any t > s > r ≥ 0, and for all ω ∈ / N , we have ◦,W W LW r,t (x, dν) = Lr,s (x, ·) ◦ Qs,t (·, dν),

where the right hand side is the convolution measure defined by Z ◦,W W W Lr,s (x, ·) ◦ Qs,t (·, dν) = Q◦,W s,t (µ, dν)Lr,s (x, dµ).

R

M ( )◦

16

(5.5)

Proof: Recall that {Q◦,W r,t (µ, dν) : t ≥ r ≥ 0} is the transition semigroup of the conditional generalized super-Brownian motion given space-time white noise W and restricted on M (R)◦ . First, we prove the tightness of { 1ε Q◦,W r,t (εδx , ·)}. Note that, for any real positive number η > 0, {µ ∈ M (R) : h1 + |x|, µi ≤ η} is a compact subset in M (R). We only need to show that 1 E lim sup Q◦,W (²δx , {µ ∈ M (R) : h1 + |x|, µi > η}) = 0 η→∞ ε>0 ε r,t

(5.6)

and 1 ◦ sup Q◦,W r,t (²δx , M (R) ) < ∞, ²>0 ²

for t > r ≥ 0.

(5.7)

Let {γn } be a sequence of functions in Cb (R)+ ∩ H1 (R), which increasingly converge to 1 + |x|. From Theorem 4.2, we have LHS of (5.6) = ≤ = = =

1 lim E sup Q◦,W r,t (²δx , {µ ∈ M (R) : h1 + |x|, µi > η}) ε>0 ε Z 1 1 (²δx , dµ) lim E sup h1 + |x|, µi Q◦,W η→∞ ε>0 η M (R) ε r,t Z 1 1 lim lim E sup (²δx , dµ) hγn , µi Q◦,W η→∞ n→∞ ε>0 η M (R) ε r,t 1 lim lim E Tr,t (x, γn ) η→∞ n→∞ η 1 lim Ex (1 + |ξt |) = 0, η→∞ η η→∞

where the fourth equality follows from (4.1) and ξt is the driftless diffusion process with diffusion p coefficient a(x). Recall that we have assumed that 0 < σa ≤ σ(x) ≤ σb < ∞, where σa and σb are constants. a (x) and ψ b (x) denote the unique, non-negative solution of (2.2) with σ(x) replaced by Let ψr,t r,t b (x) ≤ ψ (x) ≤ ψ a (x). Let σa and σb , respectively. By Lemma 5.1, we know that 0 ≤ ψr,t r,t r,t W,σa Qr,t (µ, dν) denote the conditional transition probability of the SDSM with branching rate σa given Brownian sheet W . Since 0 < σa ≤ σ(x), we have Z Z a e−λh1,νi QW (µ, dν) ≤ e−λh1,νi QW,σ 0 ≤ λ. (5.8) r,t r,t (µ, dν),

R

M( )

R

M( )

σa Let ψr,t (λ) be the unique solution of (2.2) with φ(·) ≡ λ and ψr,t (λ) be the unique solution of (2.2) with φ(·) ≡ λ and σ(·) ≡ σa . Then, Z W Qr,t (µ, {0}) = lim e−λh1,νi QW r,t (µ, dν)

R

λ→∞ M ( )

= lim e

−hψr,t (λ),µi

λ→∞

σa

≥ lim e−hψr,t (λ),µi =

λ→∞ a QW,σ r,t (µ, {0})

17

(5.9)

σa since σa ≤ σ(x) and ψr,t (λ) ≤ ψr,t (λ). From (5.8), we get Z Z a e−λh1,νi QW (µ, dν) ≤ e−λh1,νi QW,σ r,t r,t (µ, dν),

R

R

M ( )0

0 ≤ λ.

(5.10)

M ( )0

Since the right hand side of (5.8) is the Laplace transformation of a continuous state branching process with generator σa ∂ 2 f. Gf = 2 ∂x2 From the theory of continuous state branching process (see [1]), we can get that Z Z Z −λh1,νi W,σa −λh1,νi W,σa a e Qr,t (µ, dν) = e Qr,t (µ, dν) − e−λh1,νi QW,σ r,t (µ, dν)

R

R

M ( )◦

M( )

= exp{−

{0}

h1, µiλ 2h1, µi } − exp{− }. 1 + σa λ(t − r)/2 σa (t − r)

(5.11)

Thus, we have Z 1 ²λ 2² 1 a lim } − exp{− }] e−λh1,νi QW,σ r,t (²δx , dν) = lim [ exp{− ²↓0 M (R)◦ ²↓0 ² ² 1 + σa λ(t − r)/2 σa (t − r) −²λ 2² 1 }+ ] = lim [ ²↓0 ² 1 + σa λ(t − r)/2 σa (t − r) λ 2 =− + . (5.12) 1 + σa λ(t − r)/2 σa (t − r) This proves (5.7) and hence, the desired tightness. For any convergence sequence ²1k Q◦,W r,t (²k δx , ·), we have Z 1 (²k δx , dν) (1 − e−hφ,νi ) Q◦,W ²k r,t M (R)◦ Z 1 = (²k δx , dν) (1 − e−hφ,νi ) QW ²k r,t M (R) Z 1 1 = − e−hφ,νi QW r,t (²k δx , dν) ²k ²k M (R) =

(5.13)

1 1 − exp{−²k ψr,t (x)} ²k ²k

−→ ψr,t (x)

as k → ∞,

and especially (5.13) implies the limit is independent of choice of the convergent subsequences. Thus, by Theorem 25.10 and the Corollary of [4], the unique limit of (1/²)Q◦,W r,t (²δx , dν) W (x, dν) is a finite (x, dν). Especially, for any t > r ≥ 0, L exists, which is denoted by LW r,t r,t random measure by ([3] p.94 Theorem 2). Then, for any x ∈ R, t ≥ r ≥ 0, and ω ∈ / N, o W Lr,t (x, dν) is a σ-finite measure on M (R) such that Z (1 − e−hφ,νi )LW (5.14) r,t (x, dν) = ψr,t (x) = ψr,t (x, φ).

R

M ( )◦

18

The uniqueness of LW r,t (x, dν) comes from (5.14). ◦,W Now we are going to check that LW r,t (x, dν) is an entrance law of Qr,t . Since for any t ≥ s > r ≥ 0, and for all ω ∈ / N, Z Z W (1 − e−hφ,νi )Q◦,W (5.15) s,t (µ, dν)Lr,s (x, dµ) M (R)◦ M (R)◦ Z Z W = (1 − e−hφ,νi )QW s,t (µ, dν)Lr,s (x, dµ) ◦ M (R) M (R) Z (1 − e−hψs,t ,µi )LW = r,s (x, dµ)

R

M ( )◦

= ψr,s (x, ψs,t (·, φ)) = ψr,t (x, φ) Z (1 − e−hφ,µi )LW = r,t (x, dµ).

R

M ( )◦

Thus, (5.5) is proved. Remark: Another way to prove this theorem is using the property that {QW µ : µ ∈ M (R)} is conditionally infinitely divisible. Then, by the canonical representation (See Dawson((1993) Theorem 3.3.1 and Section 11.5), for each x ∈ R there exists a set of σ-finite random measures ◦ LW r,t (x, dν) for t ≥ r ≥ 0 on M (R) and there exists a set N such that P(N ) = 0 and Z (1 − e−ν(f ) )LW t ≥ r ≥ 0, x ∈ R, f ∈ Cb (R)+ ∩ H1 (R) (5.16) r,t (x, dν) = ψr,t (x),

R

M ( )◦

holds for each ω ∈ / N.

6

Excursion representation for the SDSM with immigration

Immigration processes associated with the SDSM were studied in [17, 15]. Based on the results on conditional entrance laws developed in the last section, we here give a representation for the sample paths of the immigration SDSM. Let W denote the totality of continuous path w ∈ C((0, ∞), M (R)) that takes values from M (R)◦ := M (R)\{0} in some interval (α(w), β(w)) ⊂ (0, ∞) and takes the value zero elsewhere. Let (B(W), Bt (W)) stand for the natural filtration of W. For a ≥ 0 let Wa denote the subset of W consisting of path w with α(w) = a. Let us fix a typical sample of W out of a null set so that W the corresponding semigroup (QW s,t )t≥r is defined by (4.6). By Theorem 5.2, (La,t (x, ·))t>a is an ◦,W entrance law for the Markov semigroup (Qs,t )t≥r at a. Then there is a unique σ-finite Borel measure Qxa on C((a, ∞), M (R)) such that ◦,W ◦,W Qxa (wt1 ∈ dν1 , · · · , wtn ∈ dνn ) = LW a,t1 (x, dν1 )Qt1 ,t2 (ν1 , dν2 ) · · · Qtn−1 ,tn (νn−1 , dνn )

(6.1)

for a < t1 < t2 < · · · < tn and ν1 , ν2 , · · · , νn ∈ M (R)◦ . The existence of this measure follows from a result proved in [11] in the setting of right processes. In particular, for t ≥ a we have Z (1 − e−hφ,wt i )Qxa (dw) = ψa,t (x, f ). (6.2)

Wa

19

Roughly speaking, under Qxa the coordinate process {wt : t > a} is a conditional diffusion process W with transition semigroup (QW r,t )t≥r and one-dimensional distributions (La,t (x, ·))t>a . x We may and do regard Qa as a σ-finite measure on W supported by Wa . Now we fix µ, m ∈ M (R). By considering an extension of the original probability space, we may deW (ds, dx, dw) that, conditioned upon the white fine the random objects NµW (dx, dw) and Nm noise W , are independent Poisson random measures with intensity measures µ(dx)Qx0 (dw) and m(dx)Qxs (dw)ds, respectively. For t ≥ 0, let Ft0 be the σ-algebra generated by the P-null sets and the random variables {NµW (J × A) : J ∈ B(R), A ∈ Bt (W)},

(6.3)

and let Ft1 be the σ-algebra generated by the P-null sets and the random variables W {Nm (J × A) : J ∈ B([0, r] × R), A ∈ Bt (Wr ), 0 ≤ r ≤ t}.

(6.4)

Define G¯t := Ft0 ∨Ft1 which is the σ-algebra generated by Ft0 ∪Ft1 . We define the measure-valued processes Z Z Xt := wt NµW (dx, dw), t ≥ 0, (6.5)

R W

and

Z It :=

Z Z

(0,t]

R W

W wt Nm (ds, dx, dw),

t ≥ 0.

(6.6)

T Theorem 6.1 Let Yt = Xt + It . Then, for any h ∈ Cb (R)+ H1 (R) we have ½ ¾ Z t ¯ ´ ³ ¯¯ W E exp (− hh, Yt i) ¯Gr = exp − hψr,t (·, h), Yr i − hψs,t (·, h), mids .

(6.7)

Therefore, {Yt : t ≥ 0} is a diffusion process with transition semigroup (Ut )t≥0 given by ½ ¾ Z Z t −hh,νi e Ut (µ, dν) = E exp − hψ0,t (·, h), µi − hψs,t (·, h), mids

(6.8)

r

R

M( )

0

and {Yt : t ≥ 0} is just the immigration SDSM constructed as (J , D(J )-martingale problem in [17]. Proof: Let t ≥ r ≥ 0 and EW denote the conditional expectation given W . For nonnegative f ∈ B(R) × Br (W), and nonnegative g ∈ Kr , where Kr is the σ-algebra σ{J × A : J ∈ B([0, s] × T R), A ∈ Br (Ws ), 0 ≤ s ≤ r}, and h ∈ Cb (R)+ H1 (R) we can use an expression for the Laplace transforms of Poisson random measures and Theorem 4.1 to get ½ Z Z ¾ Z Z Z W W W E exp − f (x, w)Nµ (dx, dw) − g(s, x, w)Nm (ds, dx, dw) − hh, Yt i R½ W (0,r] R W Z Z = EW exp − [f (x, w) + hh, wt i]NµW (dx, dw) ZR WZ Z W − [g(s, x, w) + hh, wt i]Nm (ds, dx, dw) (0,r]

R W

20

Z

Z Z

− ½

Z

(r,t]

Z

R W

¾ W hh, wt iNm (ds, dx, dw)

µ(dx) (1 − exp[−f (x, w) − hh, wt i])Qx0 (dw) R Z WZ Z r − ds m(dx) (1 − exp[−g(s, x, w) − hh, wt i])Qxs (dw) 0 R W ¾ Z t Z Z x − ds m(dx) (1 − exp[−hh, wt i])Qs (dw) r W ½ Z Z R = exp − µ(dx) (1 − exp[−f (x, w) − hψr,t (·, h), wr i])Qx0 (dw) R Z WZ Z r − ds m(dx) (1 − exp[−g(s, x, w) − hψr,t (·, h), wr i])Qxs (dw) 0 R W ¾ Z t Z − ds ψs,t (x, h)m(dx) r R ½ Z Z = EW exp − [f (x, w) + hψr,t (·, h), wr i]NµW (dx, dw) ZR WZ Z W − [g(s, x, w) + hψr,t (·, h), wr i]Nm (ds, dx, dw) = exp

−

(0,r] t

= EW

R W

¾ − hψs,t (·, h), mids ½ Z rZ Z exp − f (x, w)NµW (dx, dw) − Z

R W

−hψr,t (·, h), Yr i −

Z r

(0,r]

t

Z Z

R W

W g(s, x, w)Nm (ds, dx, dw)

¾ hψs,t (·, h), mids ,

which yields (6.7) and (6.8). By (6.8), Theorem 5.1 of [15], and the uniqueness of immigration SDSM, {Yt : t ≥ 0} is just the unique solution of (5.1) of [15] with b ≡ 0, which is just the immigration SDSM constructed as (J , D(J )-martingale problem in [17] and the proof is complete.

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