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We show that solutions of the stochastic heat equation driven by space-timewhite noise, although not smooth, meaningfully solve the two-dimensional Todaequations. Then by extending our arguments we show the time evolution of themultilayer process introduced by O'Connell and Warren is conjugate to a flowinduced by the stochastic heat equation. In particular this establishes aMarkov property conjectured by O'Connell and Warren. It also defines, for thefirst time, the multilayer process started from a general initial condition.

The stochastic heat equation, 2D Toda equations and dynamics for the multilayer process