  Path integrals play a crucial role in describing the dynamics of physical
systems subject to classical or quantum noise. In fact, when correctly
normalized, they express the probability of transition between two states of
the system. In this work, we show a consistent approach to solve conditional
and unconditional Euclidean (Wiener) Gaussian path integrals that allow us to
compute transition probabilities in the semi-classical approximation from the
solutions of a system of linear differential equations. Our method is
particularly useful for investigating Fokker-Planck dynamics, and the physics
of string-like objects such as polymers. To give some examples, we derive the
time evolution of the d-dimensional Ornstein-Uhlenbeck process, and of the Van
der Pol oscillator driven by white noise. Moreover, we compute the end-to-end
transition probability for a charged string at thermal equilibrium, when an
external field is applied.
