Integrable matrix probabilistic diffusions and the matrix stochastic heat equation

We introduce a matrix version of the stochastic heat equation, the MSHE, and obtain its explicit invariant measure in spatial dimension D=1. We show that it is classically integrable in the weak-noise regime in terms of the matrix extension of the imaginary-time one-dimensional (1D) nonlinear Schrödinger equation, which allows us to study its short-time large deviations through inverse scattering. The MSHE can be viewed as a continuum limit of the matrix log-Gamma polymer on the square lattice introduced recently. We also show classical integrability of that discrete model, as well as of other extensions such as of the semi-discrete matrix O'Connell-Yor polymer and the matrix strict-weak polymer. For all these models, we obtain the Lax pairs of their weak-noise regime, as well as the invariant measure, using a fluctuation-dissipation transformation on the dynamical action.