Title: Time evolution of the Kardar-Parisi-Zhang equation
Description: The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. The Kardar-Parisi-Zhang (KPZ) equation is well-known for its applications in describing various statistical mechanical models including randomly growing surfaces, directed polymers and interacting particle systems. We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation. We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between super-exponential decay with exponent $\tfrac{5}{2}$ (and leading pre-factor $\frac{4}{15\pi} T^{1/3}$) for tail depth greater than $T^{2/3}$ (deep tail), and exponent $3$ (with leading pre-factor at least $\frac{1}{12}$) for tail depth less than $T^{2/3}$ (shallow tail).

We also consider the case when the initial data is drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $\frac{5}{2}$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $\frac{3}{2}$ at all depths in the tail. We study the correlation of fluctuations of the narrow wedge solution to the KPZ equation at two different times. We show that when the times are close to each other, the correlation approaches one at a power-law rate with exponent $\frac{2}{3}$, while when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent $-\frac{1}{3}$.