Noise-intensity bifurcations of transition paths by Morse index formula

Abstract

We study most probable transition paths (MPTPs) of a gradient diffusion system in ${\mathbb{R}}^n$ with fixed endpoints, in the framework of Onsager--Machlup (OM) theory. The existence theory for such paths is developed within Lagrangian variational theory: above the Mañé critical value $c_u(L)$ the energy-penalized action admits global minimizers, while below the endpoint-dependent critical value $k_0(L;x_{\pm })$ only time-truncated minimizers exist; at regular levels between them, free-time extremals exist. The main result is a $\{0,1\}$-index theorem: at a nondegenerate free-time extremal, the Morse index of the full Hessian equals the fixed-time Morse index plus a correction $\nu \in \{0,1\}$, equal to $1$ exactly when the reduced (minimal fixed-time) action satisfies $S^{\prime \prime }(T)<0$. The loss of local minimality along a branch is thereby split into two mechanisms: a conjugate-point mechanism and a duration mechanism. Combined with a Hamiltonian spectral-flow formula, this yields a bifurcation criterion in the noise intensity $\sigma$ and a sharp local stability criterion expressed in terms of the crossing instants. As an example, the one-dimensional quartic double well is analyzed in detail.