Temporal increments of the KPZ equation with general initial data

We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation $\mathcal{H}^f(t,x)$ started with initial data $f$. In this article, we study the sample path properties of the KPZ temporal process $\mathcal{H}_t^f := \mathcal{H}^f(t,0)$. We show that for a very large class of initial data which includes any deterministic continuous initial data that grows slower than parabola ($f(x) \ll 1 + x^2$) as well as narrow wedge and stationary initial data, temporal increments of $\mathcal{H}_t^f$ are well approximated by increments of a fractional Brownian motion of Hurst parameter $\frac14$. As a consequence, we obtain several sample path properties of KPZ temporal properties including variation of the temporal process, law of iterated logarithms, modulus of continuity and Hausdorff dimensions of the set of points with exceptionally large increments.