In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we are able to derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

翻译：本文研究涉及随机抛物型方程的不适定柯西问题。我们首先建立了该方程的卡尔曼估计，并利用这一估计推导出前述不适定柯西问题的条件稳定性以及吉洪诺夫正则化方法的收敛速率。为补充理论分析，我们采用基于核的学习理论实现了完备的吉洪诺夫正则化方法，并给出了若干数值算例。