On exponential growth for a certain class of linear systems

We consider a class of stochastic growth models on the integer lattice which includes various interesting examples such as the number of open paths in oriented percolation and the binary contact path process. Under some mild assumptions, we show that the total mass of the process grows exponentially in time whenever it survives. More precisely, we prove that there exists an open path, oriented in time, along which the mass grows exponentially fast.