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Normalized Gaussian Path Integrals

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.