Long-Time Behavior of Mean Field Games on ℝd with Common White Noise and Quadratic Growth
Abstract
We study finite-horizon mean field game systems on ℝd with common white noise and quadratic control cost. The value function is allowed to have quadratic growth, and the state space is noncompact. Under Lasry--Lions monotonicity and uniform confinement and contraction assumptions on the running and terminal costs, we first establish existence and uniqueness of strong finite-horizon equilibria. We develop a weighted analytic framework for the associated stochastic forward--backward system, based on weighted stability estimates for stochastic Fokker--Planck equations, well-posedness and regularity estimates for backward stochastic Hamilton--Jacobi--Bellman equations, and a two-sided Gronwall argument adapted to the forward--backward structure. We then construct a stationary adapted regime for the common-noise MFG dynamics. Our main long-time result is a quantitative turnpike estimate: finite-horizon equilibria converge, away from the initial and terminal layers, to this stationary regime in weighted norms, both for the population density and for the gradient of the value function. We also identify the associated ergodic constant as the averaged stationary cost.
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BibTeX
@misc{yan2026longtime,
title = {{Long-Time Behavior of Mean Field Games on $\mathbbR^d$ with Common White Noise and Quadratic Growth}},
author = {Wenbin Yan},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05783310v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long-Time Behavior of Mean Field Games on ℝ^d with Common White Noise and Quadratic Growth
AU - Wenbin Yan
PY - 2026
DA - 2026/10/07
PB - HAL
UR - https://hal.science/hal-05783310v1
AB - We study finite-horizon mean field game systems on ℝ^d with common white noise and quadratic control cost. The value function is allowed to have quadratic growth, and the state space is noncompact. Under Lasry--Lions monotonicity and uniform confinement and contraction assumptions on the running and terminal costs, we first establish existence and uniqueness of strong finite-horizon equilibria. We develop a weighted analytic framework for the associated stochastic forward--backward system, based on weighted stability estimates for stochastic Fokker--Planck equations, well-posedness and regularity estimates for backward stochastic Hamilton--Jacobi--Bellman equations, and a two-sided Gronwall argument adapted to the forward--backward structure. We then construct a stationary adapted regime for the common-noise MFG dynamics. Our main long-time result is a quantitative turnpike estimate: finite-horizon equilibria converge, away from the initial and terminal layers, to this stationary regime in weighted norms, both for the population density and for the gradient of the value function. We also identify the associated ergodic constant as the averaged stationary cost.
ER -