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 $1D$ nonlinear Schrodinger 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.