| Abstract: |
| We consider Hamilton-Jacobi equations on the Wasserstein space of probability measures, arising in deterministic and stochastic dynamics. We develop a unified viscosity solution framework for both first-order equations and semilinear equations with idiosyncratic noise, based on an appropriate choice of subdifferential compatible with comparison and stability properties.
Within this framework, we establish a vanishing viscosity limit for semilinear Hamilton-Jacobi equations, showing convergence to the corresponding first-order equation as the noise intensity tends to zero, together with an optimal convergence rate. Our results provide a PDE-level description of the zero-noise transition. |
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