  In this paper, we study the stochastic nonlinear heat equations (SNLH) and
stochastic nonlinear wave equations (SNLW) on two-dimensional torus driven by a
subordinate cylindrical Brownian noise, which we define by the time-derivative
of a cylindrical Brownian motion subordinated to a nondecreasing cadlag
stochastic process. To construct the solution, we introduce a suitable
renormalization. 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.
