随机抛物方程中漂移与扩散点源逆恢复的稳定性估计
Stability Estimates for the Inverse Recovery of Drift and Diffusion Point Sources in Stochastic Parabolic Equations
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中文总结 AI 辅助
本文针对随机抛物方程中漂移与扩散点源的联合逆恢复问题，利用边界观测建立了源位置与强度的Lipschitz或对数稳定性估计，并推广至多源情形，揭示了随机积分携带的独特时间信息。
中文摘要 AI 辅助
本文研究在Neumann边界条件下，随机抛物方程的漂移项和扩散项中同时恢复点源位置和时间强度的问题，仅利用源失效后边界非空部分上的边界观测。通过将随机方程分离为期望部分和中心化部分，漂移通道简化为具有标量Laplace矩的确定性源问题，而扩散通道由Itô等距控制，该等距保留了被积函数的完整$L^2$能量。在适当的非退化性和分离性假设下，我们建立了任一通道中单个源位置的Lipschitz稳定性、漂移强度的对数稳定性以及包括符号在内的完整扩散强度的Lipschitz稳定性。对于多个源，我们在二维和三维中获得了漂移位置和加权时间矩的置换意义下的Lipschitz稳定性，并在维数一至三中获得了扩散位置和强度的相应稳定性。证明采用了显式伴随探针，包括二维和三维中的复各向同性多项式消零子以及一维中的谱综合，并结合由Green恒等式导出的边界到源估计。这些结果通过揭示随机积分所携带的根本不同的时间信息，扩展并改进了现有的确定性稳定性理论，这一区别进一步通过数值实验加以说明。
英文摘要
This paper studies the simultaneous recovery of point-source locations and temporal strengths in both the drift and diffusion terms of a stochastic parabolic equation with Neumann boundary conditions, using only boundary observations on a nonempty portion of the boundary after the sources have become inactive. By separating the stochastic equation into its expectation and centered parts, the drift channel reduces to a deterministic source problem with scalar Laplace moments, while the diffusion channel is controlled by Itô's isometry, which preserves the full $L^2$-energy of the integrand. Under suitable nondegeneracy and separation assumptions, we establish Lipschitz stability for the location of a single source in either channel, logarithmic stability for the drift strength, and Lipschitz stability for the complete diffusion strength including its sign. For multiple sources, we obtain Lipschitz stability up to permutation for drift locations and weighted temporal moments in dimensions two and three, and for diffusion locations and strengths in dimensions one through three. The proof employs explicit adjoint probes, including complex-isotropic polynomial annihilators in dimensions two and three and spectral synthesis in one dimension, combined with boundary-to-source estimates derived from Green's identities. These results extend and improve upon existing deterministic stability theories by revealing the fundamentally different temporal information carried by stochastic integrals, a distinction that is further illustrated by numerical experiments.
发表机构
- School of Mathematics, Sichuan University（四川大学数学学院）
- School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
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