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 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：本文研究了一类涉及随机抛物型方程的不适定柯西问题。首先建立了该方程的卡尔曼估计，进而利用该估计推导了上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度。为补充理论分析，本文采用基于核的学习理论对吉洪诺夫正则化方法进行了数值实现，并给出了若干算例。