Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics

We develop a continuous-time model for the long-term dynamics of adaptive stochastic optimization, focusing on bias-corrected Adam-type methods. Starting from a finite-sum setting, we identify a canonical scaling of learning rates, decay parameters, and gradient noise that yields a coupled, time-inhomogeneous stochastic differential equation for the parameters $x_t$, first-moment tracker $z_t$, and second-moment tracker $y_t$. Bias correction persists via explicit time-dependent coefficients, and the dynamics becomes asymptotically time-homogeneous. We analyze the associated Fokker-Planck equation and, under mild regularity and dissipativity assumptions on $f$, prove existence and uniqueness of invariant measures. Noise propagation is governed by $A(x)=\mathrm{Diag}(\nabla f(x))H_f(x)$. Hypoellipticity may fail on $\mathcal D_A\times\mathbb R^m\times(\mathbb R_+)^m$, where $$\mathcal D_A=\{x\in\mathbb R^m:\exists j,\ e_j^\top A(x)=0\}\subset\{x:\det A(x)=0\}=\mathcal D_A^\dagger,$$ and critical points of $f$ lie in $\mathcal D_A$. We show $\mathcal D_A^\dagger\neq\mathbb R^m$ and use this to prove exponential convergence of the Markov semigroup $\mu_0P_t$ to a unique invariant measure, uniformly in $\mu_0$. The proof uses a Harris-type argument, minorization on Lyapunov sublevel sets, control constructions, and hypoellipticity on $(\mathbb R^m\setminus\mathcal D_A)\times\mathbb R^m\times(\mathbb R_+)^m$. This provides a transparent continuous-time view of Adam-type dynamics.