White noise driven SPDEs with reflection
White noise driven SPDEs with reflection
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DOI：
10.1007/bf01197335
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发表时间：
1993-03
影响因子：
2
通讯作者：
C. Donati-Martin;Etienne Pardoux
中科院分区：
文献类型：
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作者：
C. Donati-Martin;Etienne Pardoux
研究了时空白色噪声驱动下空间区间[0，1]上一类非线性热方程Dirichlet边界条件的反射解.漂移和扩散系数都表现出非线性。粗略地说，在任何点（t，x），解u（t，x）是严格正的，它服从方程，在点（t，x），其中u（t，x）是零，我们添加一个力，以防止它成为负的。这可以被看作是一个扩展的一维SDEs反映在0，和确定性变分不等式。证明了最小解的存在性。该结构使用了惩罚参数，一个新的SPDE的存在定理，其系数取决于过去的解决方案，和白噪声驱动的SPDE的解决方案的比较定理。
We study reflected solutions of a nonlinear heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by space-time white noise. The nonlinearity appears both in the drift and in the diffusion coefficient. Roughly speaking, at any point (t, x) where the solutionu(t, x)is strictly positive it obeys the equation, and at a point (t, x) whereu(t, x)is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. Existence of a minimal solution is proved. The construction uses a penalization argument, a new existence theorem for SPDEs whose coefficients depend on the past of the solution, and a comparison theorem for solutions of white-noise driven SPDEs.