Weighted Harris–oscillation contraction and invariant expectations for nonlinear Markov semigroups

Abstract

We establish a weighted contraction principle for sublinear Markov semigroups on noncompact state spaces. Its abstract inputs are a Lyapunov drift condition and a strict local contraction of oscillations on a Lyapunov sublevel set. For locally uniformly parabolic diffusion semigroups, the latter is obtained by applying the parabolic Harnack inequality to both the upper evolution and its conjugate lower evolution: if $u=T_tφ$, then $u-m$ and $M-u$ solve the upper and lower Bellman equations, respectively. Combining the resulting local oscillation contraction with the Lyapunov drift yields a strict contraction in a weighted oscillation seminorm, a unique invariant sublinear expectation in the finite-Lyapunov-moment class, and weighted exponential convergence for Borel observables controlled by the Lyapunov function. To treat bounded Borel data for $G$-SDEs, we use a canonical covariance-law representation, progressively measurable state maps compatible with conditioning and pasting, and a stopped dynamic programming principle. A probabilistic Krylov–Safonov argument then gives positive-time Hölder regularisation before viscosity identification, thereby avoiding a measurable-data envelope argument. The abstract result is verified for ordinary locally uniformly elliptic SDEs, multiplicative $G$-SDEs, and the full equation with quadratic-variation drift $dX_t=b(X_t)\,dt +\sum_{i,j=1}^m h_{ij}(X_t)\,d\langle B^i,B^j\rangle_t +\sum_{j=1}^mσ_j(X_t)\,dB_t^j.$ For the full equation the hypotheses are expressed through the effective drift and covariance and are required uniformly over all covariance scenarios. Finally, the contraction and stochastic continuity of the associated stationary process yield ergodicity in the sense of Feng and Zhao.