Symposium on Stochastic Analysis, Kyoto University, October 25–27, 2001.

Pathwise Regularity of Some Measure-Valued Processes

Isamu DÔKU

Department of Mathematics, Saitama University

Saitama 338-8570, Japan e-mail : idoku@ math.edu.saitama-u.ac.jp

Let X = {X_{t}} be a super-Brownian motion (or Dawson-Watanabe superprocess), which is a typical example of measure-valued stochastic processes. Based upon a heuristic interpretation coming from the construction of particle systems as its background, X can be described as a solution of the symbolic stochastic differential equation

(1)  \{ \begin{matrix}dX_{t} & = \frac{1}{2}ΔX_{t}dt + \sqrt{2ρX_{t}}dW_{t}, t > 0, x ∈ ℝ^{d} \\ X_{0} & = δ_{x}, \end{matrix}

where Δ is the Laplacian, ρ > 0 is a branching rate, and Ẇ is a space-time white noise. As you can easily imagine, the above (1) suggests that independently at each space point the population grows according to the Feller branching diffusion, and also that additionally the population mass is smeared out by the heat flow.

Furthermore, according to characterization via Laplace functional, the super-Brownian motion is naturally associated with a certain type of nonlinear reactiondiffusion equation. As a matter of fact, there exists a M_{F} (ℝ^{d}) -valued Markov process

X = [ X_{t}, P_{s,μ}; s > 0, μ ∈ M_{F} (ℝ^{d}) ]

whose Laplace transition functional is given by

(2)  P_{s,μ} exp⟨X_{t}, −φ⟩ = exp⟨μ, −v^{[φ]}(s, t, ·)⟩,
0 ≤ s ≤ t, φ ∈ bC_{+}(ℝ^{d}) ∩ Dom(Δ)

where the function v^{[φ]}(s, t, ·) solves uniquely the nonlinear parabolic partial differential equation

(3)  \{ \begin{matrix}−\frac{∂v}{∂s} & = \frac{1}{2}Δv − ρv^{2} \\ v|_{s=t} & = φ. (terminal condition) \end{matrix}

We consider a more general situation of (3) where the generator of standard Brownian motion is replaced by L^{κ}. Actually L^{κ} is a uniformly elliptic second order differential operator with continuous coefficients satisfying Hölder condition. We denote by ξ = (ξ_{t}, Π_{s,x}) the corresponding L^{κ} -diffusion. Also we are interested in the special case of singular branching rate γ instead of ρ. We may assume without loss of generality that the singular point c = 0. We say that γ(x), x ∈ ℝ, is a singular branching rate if γ : ℝ → ℝ^{+} is continuous except x = 0, such that

0 < γ(x) < γ(1) for |x| > 1

lim_{x→±∞} γ(x) = 0, γ(x) = O(x^{−2}), (x → 0)

c_{0}e^{−|x|} < γ(x) < c_{1}|x|e^{−|x|}, (∃c_{0}, c_{1} > 0) for ∃M > |x| > 2.

Suppose that for t > 0 fixed, γ(ξ_{st}) ∼ t^{α}, (s > 0) for α > −1.

Theorem 1. There exists a non-degenerate, finite measure-valued superprocess

X^{κ} = [ X_{t}^{κ}, P_{s,μ}; s ≥ 0, μ ∈ M_{F} (ℝ) ]

on ℝ with singular branching rate γ and L^{κ} -diffusion process ξ^{c} killed at c = 0 as motion law.

Theorem 2. There exists a cadlag version of X^{κ} if it is realized as a M_{η} -valued process, where η is a proper weight function.

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