The Schwinger-Dyson Equation for a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n\in\mathbb{N}$, and for $k\in\mathbb{N}$, a law $\lambda\in\Sigma_{k}$ and $j\in[k]$ let $\partial^{\lambda}_{j}$ be the free difference quotient of $\lambda$ in the variable $x_{j}$. A law satisfies a system below if the displayed equalities hold for it; in clause 2, $m+j\in[m+n]$ for $j\in[n]$.

1. (Schwinger-Dyson equation)¶ The Schwinger-Dyson equation for a law $\lambda\in\Sigma_{n}$ is the system

$$\lambda(x_{j}p)=\partial^{\lambda}_{j}(p)\qquad(j\in[n],\ p\in\mathcal{P}_{n}).$$

2. (In the last variables)¶ The Schwinger-Dyson equation in the last $n$ variables for a law $\gamma\in\Sigma_{m+n}$ is the system

$$\gamma(x_{m+j}q)=\partial^{\gamma}_{m+j}(q)\qquad(j\in[n],\ q\in\mathcal{P}_{m+n}).$$