带不规则漂移的超布朗运动的紧支集性质
Compact Support Property of Super-Brownian Motion with Irregular Drift
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中文总结 AI 辅助
该研究针对带不规则漂移的一维超布朗运动对应的随机偏微分方程，通过层分解方法证明其弱解的紧支集性质，推广了适用漂移类，并证明解具有正灭绝概率。
中文摘要 AI 辅助
我们研究一维随机偏微分方程：d_t X_t(x)=1/2ΔX_t(x) +b₁𝟙_{X_t(x)>0} +√(X_t(x))Ẇ(t,x)，其中b₁>0，Ẇ为时空白噪声，初始条件为非负、具紧支集的连续函数。我们证明其唯一弱解具有紧支集性质。该漂移项不满足Dawson–Girsanov定理通常的正则性假设，因此证明基于层分解方法：将解构造为若干超布朗运动之和的单调极限，这些超布朗运动的随机迁入率由前一层的正集递归确定。通过比较论证，紧支集性质可推广到一类更广泛的、在原点处为零的有界非负漂移。最后，支集半径估计给出了X的总质量与平方贝塞尔过程的比较，由此我们证明X具有正的灭绝概率。
英文摘要
We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}ΔX_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove that its unique weak solution has the compact support property. The drift term lies outside the usual regularity assumptions for the Dawson--Girsanov theorem. Instead, the proof is based on a layer decomposition in which the solution is constructed as the monotone limit of sums of super-Brownian motions with random immigration rates determined recursively by the positivity sets of the preceding layers. By comparison, the compact support property extends to a broader class of bounded nonnegative drifts vanishing at the origin. Finally, support-radius estimates yield a comparison of the total mass of $X$ with a squared Bessel process, which allows us to show that $X$ has a positive extinction probability.
