一维Lévy驱动SDE的有限爆炸及其在SPDE中的应用
Finite Explosion for One-dimensional Lévy Driven SDEs and its Application to SPDEs
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- Beijing Institute of Technology（北京理工大学）
- Doshisha University（同志社大学）
- Fujian Normal University（福建师范大学）
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中文总结 AI 辅助
本文研究纯跳Lévy过程驱动的一维SDE的有限时间爆炸，提出三组条件刻画爆炸行为，并应用于半线性抛物型SPDE，证明其局部弱解几乎必然有限寿命。
中文摘要 AI 辅助
本文研究由纯跳Lévy过程驱动的一维随机微分方程（SDE）的有限时间爆炸问题：\\[ X_{t}^{x}=x-\int_{0}^{t}b(X_{s}^{x})\\,\dd s+L_{t}, \\] 其中漂移系数$b$是局部Lipschitz连续的，$(L_t)_{t\ge0}$是纯跳Lévy过程。我们提出了三组条件：右尾返回条件；非退化条件连同左尾Osgood界；以及右尾Osgood界。前两组条件蕴含从任意有限初始状态出发几乎必然爆炸到$-\infty$，而后两组条件蕴含平均爆炸时间的一致上界。作为应用，我们给出了非线性和Lévy测度的显式条件，在这些条件下，具有加性Lévy时空白噪声和齐次Dirichlet边界条件的半线性抛物型随机偏微分方程（SPDE）的每个局部弱解几乎必然具有有限寿命。
英文摘要
We study finite-time explosion to $-\infty$ for the one-dimensional stochastic differential equation \[ X_t^x=x-\int_0^t b(X_s^x)\,\dd s+L_t,\quad x\in\R, \] where $b$ is locally Lipschitz and $(L_t)_{t\ge0}$ is a pure-jump Lévy process. We give verifiable conditions involving Osgood-type integrability under which explosion occurs almost surely from every finite initial state. Under stronger tail conditions, we obtain a uniform bound on the mean explosion time. In particular, under suitable tail monotonicity, converse results make these criteria necessary and sufficient, and yield an explicit classification for Riccati drifts. As an application, we consider a semilinear parabolic stochastic partial differential equation (SPDE) with additive Lévy space--time white noise and homogeneous Dirichlet boundary conditions. We give explicit conditions on the nonlinearity and the Lévy measure under which every local weak solution defined up to explosion has an almost surely finite explosion time.
