In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (\frac 14,1)$. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.

中文翻译：

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具有线性乘法分数噪声的 SPDE：关于 Hurst 指数的法律连续性

在本文中，我们考虑由线性乘法高斯噪声驱动的一维随机波和热方程，该噪声在时间上为白色，在空间中表现为具有 Hurst 指数 $H\in (\frac 14,1) 的分数布朗运动$. 我们证明了上述每个方程的解就指数$H$而言是连续的，关于连续函数空间中的定律收敛性。该证明基于平面上的紧密性标准和 Malliavin 微积分技术，以识别极限定律。