We introduce a new stochastic heat equation with a mixed operator, which is a combination of the standard Laplacian, a fractional Laplacian and the gradient operator, driven by an additive Gaussian noise which is white in time and in space. We establish the existence of the solution and we study its behavior with respect to the time variable. In particular, we establish sharp two-sided interesting estimates of its variance increments, which allow us to get some regularity characteristics of the solution sample paths, among many other possible applications.

中文翻译：

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具有分数拉普拉斯算子和梯度扰动的混合随机热方程

我们引入了一个新的带有混合算子的随机热方程，它是标准拉普拉斯算子、分数拉普拉斯算子和梯度算子的组合，由时间和空间上都是白色的加性高斯噪声驱动。我们建立了解决方案的存在性，并研究了它相对于时间变量的行为。特别是，我们对其方差增量建立了尖锐的双边有趣估计，这使我们能够获得解样本路径的一些规律性特征，以及许多其他可能的应用。