SPDEs driven by standard symmetric $\alpha$-stable cylindrical L\'evy processes: existence, Lyapunov functionals and It\^{o} formula

We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $\alpha$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric $\alpha$-stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an It\^{o} formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an It\^{o} formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical $\alpha$-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.