We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an \(L^1\)-method. Using this result, we are able to show existence and uniqueness of strong solutions.

中文翻译：

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具有随机扰动的双重非线性PDE

我们考虑一个带有乘性噪声的双重非线性演化方程，并显示了一个强解的存在性和路径唯一性。使用半隐式时间离散化，我们得到具有单调性参数的近似解。我们为近似解建立先验估计，并显示由近似解序列引起的图像度量序列的紧度。作为Prokhorov和Skorokhod定理的结果，我们得到了一个子序列在一个新概率空间上的收敛性，该概率空间可以显示show问题解的存在。路径唯一性是通过\（L ^ 1 \）方法获得的。使用此结果，我们能够显示强解决方案的存在性和唯一性。