We study the short-time distribution P(H,L,t) of the two-point two-time height difference H=h(L,t)-h(0,0) of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for L=0 at a critical value H=H_{c}. We show that |H| and L play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when L changes sign, at supercritical H. We also determine analytically P(H,L,t) in several limits away from the second-order transition. Typical fluctuations of H are Gaussian, but the distribution tails are highly asymmetric. The tails -lnP∼|H|^{3/2}/sqrt[t] and -lnP∼|H|^{5/2}/sqrt[t], previously found for L=0, are enhanced for L≠0. At very large |L| the whole height-difference distribution P(H,L,t) is time-independent and Gaussian in H, -lnP∼|H|^{2}/|L|, describing the probability of creating a ramplike height profile at t=0.

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We study a Brownian excursion on the time interval , conditioned to stay above a moving wall such that , and . For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari–Spohn (FS) distribution and exhibit the Kardar–Parisi–Zhang (KPZ) dynamic scaling exponents 1/3 and 2/3. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of in a close vicinity of . Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous one-parameter family of scaling exponents.