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In this work, we consider the stochastic Cauchy problem driven by thecanonical $\alpha$-stable cylindrical L\'evy process. This noise naturallygeneralises the cylindrical Brownian motion or space-time Gaussian white noise.We derive a sufficient and necessary condition for the existence of the weakand mild solution of the stochastic Cauchy problem and establish the temporalirregularity of the solution.

Stable cylindrical L\'evy processes and the stochastic Cauchy problem