Dynamics generated by spatially growing derivations on quasi-local algebras

We prove global existence and uniqueness of dynamics on the quasi-local algebra $\mathcal{A}$ of a quantum lattice system for spatially growing derivations ${\mathcal{L}}_{\mathrm{\Phi }}={\sum }_x[{\mathrm{\Phi }}_x,\cdot ]$. Existing results assume that the local terms ${\mathrm{\Phi }}_x\in \mathcal{A}$ of the generator are uniformly bounded in space with respect to appropriate weighted norms $\parallel {\mathrm{\Phi }}_x{\parallel }_{G,x}$. Analogous to the global existence result for first order ODEs, we show that global existence and uniqueness persist if the size of the local terms $\parallel {\mathrm{\Phi }}_x{\parallel }_{G,x}$ grows at most linearly in space. This considerably enlarges the class of derivations known to have well-defined dynamics. Moreover, we obtain Lieb-Robinson bounds with exponential light cones for such dynamics. For the proof, we assume Lieb-Robinson bounds with linear light cones for dynamics, whose generators have uniformly bounded local terms. Such bounds are known to hold, for example, if the local terms are of finite range or exponentially localized.