论文标题
在无限的空间结构域上漂移随机热方程的温和溶液的存在和独特性
Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain
论文作者
论文摘要
我们证明了在无界空间结构域定义的非线性随机热方程的温和溶液的存在和独特性。非线性不被认为是全球，甚至在本地的Lipschitz连续的。相反，假定非线性满足单方面的Lipschitz条件。首先，引入了kolmogorov连续定理的增强版本，以证明热方程式的基本解决方案和空间均匀噪声的随机卷积速度不比多一项差异快。其次，确定性的映射将随机卷积映射到随机热方程的溶液被证明是在连续函数的多项式加权空间上连续的Lipschitz。这两种成分使PICARD迭代方案的制定能够证明温和溶液的存在和独特性。
We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.
