Wiener Path Integral Formalism

This chapter introduces the path integral as a mathematical tool with the potential to address some of the current challenges in stochastic engineering dynamics. This is done while the concept of Wiener path integral as a functional integral over the space of paths is presented. Notably, it is shown that the Wiener path integral formulation can account, in a direct manner, not only for Markovian but also for non-Markovian response processes corresponding to systems with a history-dependent state. Furthermore, a functional series expansion is employed and a most probable path approximation is derived based on a variational principle for determining approximately the system response joint transition probability density function. The chapter concludes with an encapsulation of the Wiener path integral formalism.