The value function of mean field control with common noise is the unique viscosity solution of a fully second-order HJB equation in the Wasserstein space.
Convergence Rate of Particle System for Second-order PDEs On Wasserstein Space
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abstract
In this paper, we provide a convergence rate for particle approximations of a class of second-order PDEs on Wasserstein space. We show that, up to some error term, the infinite-dimensional inf(sup)-convolution of the finite-dimensional value function yields a super (sub)-viscosity solution to the PDEs on Wasserstein space. Hence, we obtain a convergence rate using a comparison principle of such PDEs on Wasserstein space. Our argument is purely analytic and relies on the regularity of value functions established in \cite{DaJaSe23}.
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Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space
The value function of mean field control with common noise is the unique viscosity solution of a fully second-order HJB equation in the Wasserstein space.