Nonlinear equations in diffusion theory
The paper is devoted to the study of some connections between nonlinear differential equations in diffusion theory. As is known, the Cole-Hopf substitution reduces the Burgers equation to a linear differential equation. Another example is known as the inverse scattering problem, which is based on a connection between the KdV and a linear Schrödinger equation. The connection between some nonlinear equations of soliton theory and absolute continuous transformation of diffusion measures is studied. In particular, using a multidimensional extension of the KdV equation the existence of the logarithmic derivative of a one-dimensional family of smooth measures generated by a diffusion process, is established. Also, some other connections between nonlinear equations of soliton theory and diffusion processes are studied.
scientific article; zbMATH DE number 2182724