Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form [Formula: see text] We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, [Formula: see text], [Formula: see text], and [Formula: see text] are allowed to be multivalued, as required by the modelization of solid–liquid phase transitions. In this regard, the equation corresponds to a nonlinear-diffusion version of the classical two-phase Stefan problem with stochastic perturbation. The existence of martingale solutions is proved via regularization and passage-to-the-limit. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô’s formula. Under some more restrictive assumptions on the nonlinearities, existence and uniqueness of strong solutions follows. Besides the relation above, the theory covers equations with nonlocal terms as well as systems.

中文翻译：

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双非线性随机演化方程

以随机效应为特征的非线性扩散问题可以通过以下形式的随机偏微分方程来描述[公式：见正文] 我们在非线性的一般单调性假设下提出了此类方程的存在理论。特别是，[公式：见正文]、[公式：见正文]和[公式：见正文]允许为多值，这是固液相变模型化的要求。在这方面，该方程对应于具有随机扰动的经典两相 Stefan 问题的非线性扩散版本。通过正则化和通过极限证明了鞅解的存在。极限的识别是通过基于适当推广的伊藤公式的下半连续性论证获得的。在对非线性的一些更严格的假设下，强解的存在性和唯一性如下。除了上述关系之外，该理论还包括具有非局部项的方程以及系统。