This thesis builds on recent advances using Picard-Lefschetz (PL) theory to perform semiclassical expansions of path integrals in quantum field theory. We show that PL theory also provides a rigorous framework for evaluating stochastic path integrals in the weak-noise limit (D approaching zero). In particular, it resolves long-standing issues with multi-instanton contributions, especially the role of instanton-anti-instanton (IA) pairs in the thermal escape rate across a potential barrier.
Traditional approaches focus on real-valued IA configurations, but the integral over the quasi-zero-mode (QZM) direction diverges due to the mutual attraction between instantons and anti-instantons. A standard analytic workaround continues the noise parameter from D to minus D, known as the Bogomolny-Zinn-Justin (BZJ) procedure. While operationally effective, this procedure lacks conceptual clarity and introduces ambiguity. Instead, we apply PL theory to uncover a complex saddle-point solution, which we call the stochastic complex bounce (CB). The CB dynamics takes place in a tilted effective potential. In the Markovian stochastic setting, the tilt is a feature of the discretisation scheme, whereas in quantum field theory, an analogous tilt can arise from fermionic effects.
The CB is a genuine saddle of the complexified action and is associated with a convergent integral over a special descent manifold known as a Lefschetz thimble. It contains a composite instanton-anti-instanton structure characterised by a complex separation. In this framework, the QZM integral becomes convergent and physically transparent when defined over the appropriate thimble.
We present an algorithm using PL theory to evaluate stochastic path integrals in the weak-noise limit, including quadratic fluctuations about complex saddles, and apply it to compute escape rates in two canonical systems: the stochastic cubic potential and the stochastic sine-Gordon potential. The thesis concludes by outlining broader implications of complex stochastic saddles for non-equilibrium stochastic systems and possible connections to resurgence theory.
