We consider a system of $d$ non-linear stochastic fractional heat equations in spatial dimension $1$ driven by multiplicative $d$-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of $(u(s, y), u (t, x))$. From this result, we deduce optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualart [\textit{Probab. Theory Related Fields} \textbf{144} (2009) 371--424] for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

中文翻译：

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随机分数热方程非线性系统的命中概率的最优下界

我们考虑由乘法$d$维时空白噪声驱动的$d$空间维度$1$非线性随机分数热方程系统。我们在 $(u(s, y), u (t, x))$ 的两点概率密度函数上建立了一个尖锐的高斯型上限。从这个结果，我们推导出过程 $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$非高斯情况，就牛顿容量而言，与高斯情况一样尖锐。这也改善了 Dalang、Khoshnevisan 和 Nualart [\textit{Probab. 理论相关领域} \textbf{144} (2009) 371--424] 用于经典随机热方程系统。我们还根据 Hausdorff 度量建立了解决方案命中概率的上限。