We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the [Formula: see text]th moment of solutions depends on the integrability of the initial datum, in the whole range [Formula: see text]. Lipschitz continuity of the solution map in [Formula: see text]th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range [Formula: see text]. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.

中文翻译：

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一类半线性耗散SPDE的细化存在性和正则性结果

我们证明了一类具有单调非线性漂移项和乘性噪声的随机半线性演化方程解的存在性和唯一性，大大扩展了我们之前工作中获得的相应结果。特别是，我们假设初始数据只能测量，并且我们允许扩散系数是局部 Lipschitz 连续的。此外，我们以定量的方式展示了[公式：见文本]解的有限性如何取决于初始数据在整个范围内的可积性[公式：见文本]。在扩散系数的 Lipschitz 连续性假设下，在更大的范围内 [公式：见正文]，建立了 [公式：见正文] 中解图的 Lipschitz 连续性。Itô 公式在满足最小可积性条件的过程的变分设置中的范数平方发挥了关键作用，它产生了解决方案的路径连续性。此外，我们展示了初始数据和扩散系数的规律性如何提高解的规律性，如果适用的话，不变测度的规律性。