We shall discuss a class of noncommutative probabilistic evolutions, understood as Markov semigroups acting on von Neumann algebras and associated L^p-spaces. The focus will be placed on quantum counterparts of classical convolution semigroups, i.e. translation invariant evolutions associated to locally compact quantum groups. We will describe their generators via quantum Dirichlet forms with specific invariance properties, discuss some examples and indicate certain natural further research directions. Based on joint work with Ami Viselter (University of Haifa).