arXiv 0801.32572008-01-21math.PR

催化分支网络中的退化随机微分方程

Degenerate stochastic differential equations arising from catalytic branching networks

Richard F. Bass, Edwin A. Perkins

摘要：

我们建立的存在性和唯一性的鞅问题的退化系统的代表一个催化分支网络。例如，在超循环的情况下：$$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2γ_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$其中$X^{(d+1)}\equiv X^{(1)}$，当$γ$和$b$在正正交点上连续，$γ$是严格正的，并且$b_i>0$在$\{x_i=0\}$上时，证明了存在性和唯一性。特殊情况$d=2$，$b_i=θ_i-x_i$是Dawson-Greven-den Hollander-Sun-Swart关于层次群上2型分支模型块平均的平均场极限的工作中所需要的.证明中使用了一些新的方法，包括Cotlar引理来建立伴随半群的导数在不同时刻的渐近正交性，以及Dawson-Perkins的精细分部分部积分法。作为一个副产品的证明，我们得到强Feller性质的相关预解式。

英文摘要：

We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2γ_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where $X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $γ$ and $b$ are continuous on the positive orthant, $γ$ is strictly positive, and $b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=θ_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times,and a refined integration by parts technique from Dawson-Perkins]. As a by-product of the proof we obtain the strong Feller property of the associated resolvent.

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摘要：

英文摘要：

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