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We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in \cite{yuan}. The DG methods in \cite{yuan} can maintain positivity and high order accuracy, but they rely both on the scaling limiter in \cite{zhang1} and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax-Wendroff type theorem is proved in \cite{yuan}, guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $P^k$ of piecewise polynomials of degree at most $k$. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in \cite{zhang1} can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $P^k$ or $Q^k$ spaces (piecewise $k$-th degree polynomials or piecewise tensor-product $k$-th degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in \cite{zhang1}. Computational results are provided to demonstrate the good performance of our DG schemes.

An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [54], [10]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy1 and Airy2 processes. The process takes values in real valued functions which look locally like Brownian motion, and is Hölder 1/3−in time. Both the KPZ fixed point and TASEP are shown to be stochastic integrable systems in the sense that the time evolution of their transition probabilities can be linearized through a new Brownian scattering transform and its discrete analogue.