关于热与随机heat equations的基本解
On the fundamental solution of heat and stochastic heat equations
摘要：
考虑一类二阶抛物型方程的一般发散性，其系数在空间变量上是正则的，在时间上是可测的。我们证明了它的基本解的空间导数的上界与Aronson型估计一致，并且只依赖于方程的椭圆性常数和其系数的空间导数的L $\infty$范数.我们还研究了相应的随机偏微分方程，并证明了在自然假设的噪声方程承认一个温和的解决方案，预计随机积分。
英文摘要：
We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $\infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.
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