Regenerative processes and applications to stochastic partial differential equations

This thesis focuses on additive functionals of regenerative processes and their applications to critical Stochastic Partial Differential Equations (SPDEs). In the first part, we extend the well-known Kallianpur-Robbins laws [KR53] for Brownian motion to the setting of general regenerative processes by deriving the limiting distribution of the functional ∫t 0 R(Xs)ds, as t → ∞, under appropriate normalization. In the second part, we study the stochastic heat equation with multiplicative noise: ∂tu(t, x) = 1/2 Δu(t, x) + βu(t, x)V(t, x), in the critical dimension d = 2, where the potential V is the space-time mollification of the two-dimensional space-time white noise. By leveraging the extended Kallianpur-Robbins laws for regenerative processes, we demonstrate that, after proper renormalization, the fluctuations of the solution converge to the Edwards-Wilkinson equation.