Three problems that are linked by way of motivation are addressed in this work.
In the first part of the thesis, we study the generalised Langevin equation for simulated
annealing with the underlying goal of improving continuous-time dynamics for the problem of global optimisation of nonconvex functions. The main result in this part is on
the convergence to the global optimum, which is shown using techniques from hypocoercivity given suitable assumptions on the nonconvex function. Alongside, we investigate
numerically the problem of parameter tuning in the continuous-time equation.
In the second part of the thesis, this last problem is addressed rigorously for the underdamped Langevin dynamics. In particular, a systematic procedure for finding the optimal
friction matrix in the sampling problem is presented. We give an expression for the gradient of the asymptotic variance in terms of solutions to Poisson equations and present a
working algorithm for approximating its value.
Lastly, regularity of an associated semigroup, twice differentiable-in-space solutions to
the Kolmogorov equation and weak numerical convergence rates of order one are shown
for a class of stochastic differential equations with superlinearly growing, non-globally
monotone coefficients. In the relation to the previous part, the results allow the use of
Poisson equations for variations of Langevin dynamics not permissible before.