$L^2$-Theory of Linear Degenerate SPDEs and $L^p$ ($p>0$) Estimates for the Uniform Norm of Weak Solutions

Abstract

In this paper, we are concerned with possibly degenerate stochastic partial differential equations (SPDEs). An $L^2$-theory is introduced, from which we derive the Hörmander theorem with an analytical approach. With the method of De Giorgi iteration, we obtain the maximum principle which states the $L^p$ ($p>0$) estimates for the time-space uniform norm of weak solutions.

21 pages

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