We report some recent results on Martin type representation formulas for ancient solutions of the heat equation and dimension estimates of the space of these solutions under some growth assumptions.
We will also present a new observation on the time analyticity of solutions of the heat equation under natural growth conditions. One application is a if and only if solvability condition of the backward heat equation, i.e. under what condition can one turn back the clock in a diffusion process.
Part of the results are joint work with Fanghua Lin and Hongjie Dong.