Explicit solution of the generalised Langevin equation

Generating an initial condition for a Langevin equation with memory is a non trivial issue.We introduce a generalisation of the Laplace transform as a useful tool for solving thisproblem, in which a limit procedure may send the extension of memory effects to arbitrarytimes in the past. This method allows us to compute average position, work, their variancesand the entropy production rate of a particle dragged in a complex fluid by an harmonicpotential, which could represent the effect of moving optical tweezers. For initial conditionsin equilibrium we generalise the results by van Zon and Cohen, finding the variance of thework for generic protocols of the trap. In addition, we study a particle dragged for a longtime captured in an optical trap with constant velocity in a steady state. Our formulas openthe door to thermodynamic uncertainty relations in systems with memory