Stochastic wave equation with heavy-tailed noise: uniqueness of solutions and past light-cone property (Q6635681)

The author considers a stochastic wave equation \N\[\N\frac{\partial^2u}{\partial t^2}=\Delta u+\sigma(u)\dot\Lambda \tag{1}\N\]\Non \(\mathbb R^d\), \(d\le 2\) with \(u(0)=u_0\), \(\frac{\partial u}{\partial t}(0)=v_0\), where \(\sigma\) is a globally Lipschitz continuous real-valued function, \(u_0\) and \(v_0\) are deterministic functions and \(\Lambda\) is a pure-jump Lévy space-time white noise with a Lévy measure \(\nu\). If \(d=1\) then \(u_0\) is assumed to be continuous and \(v_0\) locally bounded and measurable. If \(d=2\) then \(u_0\) is assumed to be continuously differentiable and \(v_0\) locally \(L^{q_0}\)-integrable for some \(q_0>2\).\N\NExistence and uniqueness of global mild solutions are proved under various assumptions upon moments of the Lévy measure \(\nu\). Moreover some uniform moment estimates are established for the solutions.\N\NThe results are then firstly extended to equations (1) driven by noises of type \(\Lambda+aW\), where \(a>0\), \(W\) is a space-time Gaussian white noise and \(\Lambda\) is the same as above. Then, secondly, the results are extended to equations \N\[\N\frac{\partial^2u}{\partial t^2}(t,x)=\Delta u(t,x)+f(t,x,u(t,x))+\sigma(t,x,u(t,x))\dot\Lambda(t,x),\tag{2}\N\]\Nwhere \(f\) and \(\sigma\) are predictable functions globally Lipschitz continuous in the third variable and \(\Lambda\) is the same as above.