Fokker-Planck Equation and Its Application in Production Function

A one-dimensional Fokker-Planck equation (FPE) with drift and diffusion coefficients depending on space variables is identified by a semigroup approach. The stationary solution u_s of the FPE induces a Hilbert space X, i.e., L^2 (a,b) with an inner product weighted by u_s. The backward Fokker-Planck operator A generates a C_0-semigroup in X. The well-posedness for the FPE follows the well-posedness for the Cauchy problem generated by A. The solution u is asymptotically stable to u_s as t→∞. Furthermore, if the ratio of the drift to diffusion coefficients is nondecreasing, then u is a nonnegative classical solution. As an application, the backward Fokker-Planck operatorAconfirms the well-posedness for production function equations. In case X=L^2 (a,∞), operator Ahas a continuous spectrum generating the Gaus-Weierstrass semigroup.