SPDE games driven by a Brownian sheet with applications to pollution minimization

This paper investigates a nonzero-sum stochastic differential game with $ N \ge 2 $ players in the context of shared spatial-domain pollution control. The pollution dynamics are modeled by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing spatial and temporal environmental randomness. Each player, representing a distinct region or decision-maker, minimizes an individual cost functional balancing pollution penalties against control implementation costs.Using a stochastic maximum principle in the space-time setting, we derive necessary and sufficient conditions for Nash equilibrium strategies. A novel family of adjoint variables, $ (L_1, \ldots, L_N) $, is introduced; these variables have no counterpart in formulations driven by the classical one-parameter Brownian motion. Two illustrative examples demonstrate how spatial interactions and stochastic perturbations shape the equilibrium strategies and collective environmental outcomes.