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The deterministic KPZ equation has been recently formulated as a gradient flow. Its nonequilibrium analog of a free energy---the "nonequilibrium potential'' Φ[h], providing at each time the landscape where the stochastic dynamics of h(x ,t) takes place---is however unbounded, and its exact evaluation involves all the detailed histories leading from some initial configuration h(x ,0) to a final one h(x ,t). After pinpointing some implications of these facts, we study the time behavior of 〈Φ[h] 〉 (the average of Φ[h] over noise realizations at time t) and show the interesting consequences of its structure when an external driving force F is included (i.e. the KPZ behavior as an activation-like process). The asymptotic form of the time derivative Φ [h] is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d>1.

d-Dimensional KPZ Equation as a Stochastic Gradient Flow in an Evolving Landscape: Interpretation and Time Evolution of Its Generating Functional