Stochastic Analysis and its Applications / Sep. 11-13, 2005, Osaka University

On Convergence of a Class of Interactive Superprocesses

Isamu Dôku Department of Mathematics, Saitama University

The superprocess with dependent spatial motion (SDSM) is a measurevalued diffusion process, which is obtained by a high density limit from a family of interactive branching particle systems. A new class of superprocesses, named immigration superprocesses associated with SDSM (simply written ”IMS” in what follows), was constructed by Dawson-Li (2003). Our purpose is to discuss a convergence problem for rescaled processes of the above type. Let M_{F} (R) (resp. M_{a}(R)) be the space of all finite Borel (resp. purely-atomic) measures on R, and the symbol ⟨f, μ⟩ denotes an integral of measurable function f relative to measure μ. For h ∈ C^{1}(R) and h, h′ ∈ L^{2}(R), define

ρ(x) = ∫ h(y − x)h(y)dy, for x ∈ R.

Furthermore, σ is a positive constant, m is a σ-finite Borel measure on R (as a reference measure for immigration), and q is a Borel function on M_{F} (R) × R satisfying a regularity condition, which is introduced in connection with the immigration. For simplicity, we put C_{M}(R_{+}) = C([0, ∞), M_{F} (R)).

Let X = (X_{t}) be a {ρ(0), ρ, σ, q}-immigration superprocess associated with SDSM. Then for each μ ∈ M_{F} (R), there exists a unique Borel probability measure on C_{M}(R_{+}) such that for any φ ∈ bC^{2}(R),

M_{t}(φ) = ⟨φ, X_{t}⟩ − ⟨φ, μ⟩ − ∫_{0}^{t} ⟨\frac{ρ(0)}{2}φ′′, X_{s}⟩ ds − ∫_{0}^{t}⟨φq(X_{s}, ·), m⟩ds

is a continuous martingale, and its quadratic variation process is given by

⟨M(φ)⟩_{t} = ∫_{0}^{t}⟨σφ^{2}, X_{s}⟩ds + ∫_{0}^{t} ds ∫_{R}⟨h(z − ·)φ′, X_{s}⟩^{2}dz.

For any θ > 0, we set g_{θ}(x) = g(θx), q_{θ}(ν, x) = q(ν, θx) (x ∈ R), and Λ_{θ}ν(B) = ν{θx; x ∈ B} for ν ∈ M_{F} (R). Let us consider the following conditions:

(C.1) {μ_{θ}} ⊂ M_{a}(R) and μ_{θ} → μ ∈ M_{a}(R) as θ → ∞;

(C.1)′ For {μ^{(θ)}} ⊂ M_{F} (R), we set μ_{θ} = θ^{−2}Λ_{θ}μ^{(θ)} and μ_{θ} → μ ∈ M_{F} (R) as θ → ∞;

(C.2) ρ(x) → 0 as |x| → ∞;

(C.3) For each ν, q(ν, x) → 0 as |x| → ∞.

Theorem . (a) When X^{(θ)} = (X_{t}^{(θ)}, P_{μ_{θ}}) is a {ρ(0), ρ, σ, q}-IMS, then X_{t}^{θ} = θ^{−2}Λ_{θ}X_{θ^{2}t}^{(θ)} becomes a {ρ(0), ρ_{θ}, σ, q_{θ}}-IMS.

(b) Moreover, for μ ∈ M_{a}(R) (resp. M_{F} (R)), under the conditions (C.1) (resp. (C.1)′), (C.2) and (C.3), X^{θ} converges as θ → ∞ to {ρ(0), σ, μ}- superprocess with coalescing spatial motion of Dawson-Li-Zhou (2004) in the sense of probability distribution on C_{M}(R_{+}).

References

1. D. A. Dawson and Z.-H. Li : Probab. Theory Related Fields, 127 (2003), 37–61.

2. D. A. Dawson, Z. Li and X. Zhou : J. Theoret. Probab. 17 (2004), 673–692.

3. I. Dôku : Acta Appl. Math. 63 (2000), 101–117.

4. I. Dôku : Sci. Math. Japonic. 54 (2001), 161–169.

5. I. Dôku : Quant. Inform. III (2001), 197–217.

6. I. Dôku : Far East J. Theo. Stat. 9 (2003), 1–80.

7. I. Dôku : J. SU Math. Nat. Sci. 53 (2004), 1–16.