In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for the forward stochastic degenerate parabolic equation with regular weighted function, based on which we obtain the Lipschitz stability for an inverse problem of determining a random source depending only on time in the forward stochastic degenerate parabolic equation.

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随机退化抛物线方程的卡尔曼估计及其在零可控性和逆随机源问题中的应用

在本文中，我们建立了一个随机退化抛物线方程的两个 Carleman 估计。第一个是具有奇异权函数的后向随机退化抛物线方程。结合这个 Carleman 估计和一个近似参数，我们证明了具有梯度项的前向随机退化抛物线方程的零可控性。第二个是具有正则加权函数的前向随机退化抛物线方程，在此基础上我们得到了前向随机退化抛物线方程中仅依赖于时间确定随机源的逆问题的Lipschitz稳定性。