Carleman Estimates of Some Stochastic Degenerate Parabolic Equations and Application.

This paper is devoted to establishing global Carleman estimates for some forward and backward stochastic degenerate parabolic equations. First, two Carleman estimates for the forward equation are derived by a weighted identity method and duality technique, respectively. It is found that different from stochastic uniformly parabolic equations, both methods have their own advantages in the degenerate case. By the weighted identity method, the estimate is finer for diffusion terms. By the latter, regularity requirements on them may be reduced. Also, for both the forward and the backward equations, drift terms are allowed to belong to a Sobolev space of negative order, which guarantees well-posedness of the equations. As an application of these estimates, an insensitizing control problem is studied.