Gradual convergence for Langevin dynamics on a degenerate potential
Gerardo Barrera, Conrado da Costa, Milton Jara
TL;DR
The authors analyze a one-dimensional degenerate Langevin dynamics with a global attractor at $0$, perturbed by small Gaussian noise. They prove exponential convergence to a unique invariant measure for fixed noise, and reveal a gradual convergence in the zero-noise limit under a precise time-scale $a_\varepsilon = \varepsilon^{-\alpha/(2+\alpha)}$, with the limit profile given by the TV distance between the time-$t$ marginal of a limit SDE that comes down from infinity and its invariant measure. This limit dynamics is explicitly characterized by $dY_t - C_0|Y_t|^{1+\alpha}\operatorname{sgn}(Y_t)\,dt + dW_t$, and the TV distance profile is continuous, strictly decreasing, and bounded away from $0$ and $1$ for $t>0$, ruling out cut-off and enabling mixing-time asymptotics. The analysis combines multiscale scaling, tight coupling near the origin, and convergence of invariant measures to provide a complete classification of TV-distance behaviors for one-dimensional convex potentials in the zero-noise regime, including a coming-down-from-infinity phenomenon for the limit SDE.
Abstract
In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed $t\geq 0$ the total variation distance between the accelerated random dynamics at time $t$ and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in $(0,1)$. Moreover, we prove that this limit function for each fixed $t \geq 0$ corresponds to the total variation distance between the marginal, at time $t$, of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.