一些非线性s.p.d.e.在时间上是第二顺序的
Some non-linear s.p.d.e.'s that are second order in time
摘要：
我们扩展了Walsh的鞅测度理论，以处理时间上为二阶的双曲型随机偏微分方程，如波动方程和梁方程，并由空间均匀高斯噪声驱动。对于这样的方程，基本解可以是Schwartz意义下的分布，它在s.p.d.e.的重新表述中表现为被积函数。作为随机积分方程。我们的方法提供了一个替代的希尔伯特空间积分的Hilbert-Schmidt算子。我们给出了几个例子，包括梁方程和波动方程，与非线性乘性噪声项。
英文摘要：
We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.
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