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Temporal random noises in stochastic parabolic equations
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Stochastic parabolic equations are different from deterministic ones in many angles and the ways of approaching them are to be different. In this talk we discuss such differences with emphasis on the regularity of solutions.
A stochastic parabolic equation describes diffusion with random noises. In the first part of the talk we briefly discuss a modeling of temporal noises. In the second part we focus on the Sobolev regularity of the solution with white noises. Unlike the case with deterministic convection, the stochastic convection reduces diffusion. Moreover, the way that the stochastic inhomogeneous term affects the regularity of the solution is different from the one by deterministic inhomogeneous term. By the nature of the problem, in particular by the bad contribution of the white noises in the inhomogeneous part, the second derivatives of the solution may blow up on the boundary even with $C^1$ space domain. This makes us need help of appropriate weights near the boundary to describe the regularity of solutions. Also, in the case of systems, the theory of stochastic ones sometimes fails. We discuss this with an example.