In this paper, we study the stochastic nonlinear heat equations (SNLH) and stochastic nonlinear wave equations (SNLW) on two-dimensional torus \({\mathbb {T}}^2 = ({\mathbb {R}}/2\pi {\mathbb {Z}})^2\) driven by a subordinate cylindrical Brownian noise, which we define by the time-derivative of a cylindrical Brownian motion subordinated to a nondecreasing càdlàg stochastic process. To construct the solution, we introduce a suitable renormalization similarly to Da Prato and Debussche (Ann Probab 31(4):1900–1916, 2003) and Gubinelli et al. (Trans Am Math Soc 370(10):7335–7359, 2018). For SNLH, we cannot expect the time-continuity for the solutions because the noise is jump-type. Moreover, due to the low time-integrability of the solutions, we could establish a local well-posedness result for SNLH only with a quadratic nonlinearity. On the other hand, for SNLW, the solutions have time-continuity and we can show the local well-posedness for general polynomial nonlinearities. Through this example, we can see that the heat case behaves worse than the wave case in the singular noise of jump-type cases.

在本文中，我们研究了二维环面上的随机非线性热方程（SNLH）和随机非线性波动方程（SNLW） \ ({\mathbb {T}}^2 = ({\mathbb {R}}/2\ pi {\mathbb {Z}})^2\)由从属的圆柱形布朗噪声驱动，我们将其定义为从属于非递减 càdlàg 随机过程的圆柱形布朗运动的时间导数。为了构造解决方案，我们引入了类似于 Da Prato 和 Debussche (Ann Probab 31(4):1900–1916, 2003) 和 Gubinelli 等人的合适重整化。（Trans Am Math Soc 370(10):7335–7359, 2018）。对于 SNLH，我们不能期望解的时间连续性，因为噪声是跳跃型的。此外，由于解的时间可积性低，我们只能为具有二次非线性的 SNLH 建立局部适定性结果。另一方面，对于 SNLW，解具有时间连续性，我们可以证明一般多项式非线性的局部适定性。通过这个例子，