A Stochastic Control Verification Theorem for the Dequantized Schrödinger Equation Not Requiring a Duration Restriction

A stochastic control representation for solution of the Schrödinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case.