My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \frac{\delta}{\delta \phi} \cdot \Delta \cdot \frac{\delta}{\delta \phi} S[\phi] - \frac{\delta}{\delta \phi}S[\phi]\cdot \Delta \cdot \frac{\delta}{\delta \phi}S[\phi]$, where $S[\phi]$ is the action, $\phi$ is the field, $\Delta$ contains a cut-off version of parametrix associated with the classical equation of motion of the field (for example Klein Gordon equation for scalar bosonic fields). In addition, t is the renormalization "time" which is the log of energy scale, and $\Delta$ depends on t since it involves high-energy cut-off.

Note that Polchinski's equation is meant to be a non-perturbative field-theoretic formulation of the Wilsonian renormalization.

By now, I think there are several expositions of mathematical aspects of renormalization. For example, Costello's formulation of perturbative renormalization in BV formalism. We also have Kreimer-Connes formulation of BPHZ renormalization emphasizing on Hopf algebra structure and non-commutative aspects. However neither seems to work with non-perturbative aspects of renormalization as in the sense of Polchinski.

Therefore, my question is whether there is an attempt to study Polchinski equation in a rigorous mathematical setting. If we write $S[\phi]$ in terms of formal power series in $\phi$, and study the corresponding equations term by term, an important point is that this set of infinitely many equations hold "within path integral" which means they are to be understood as equations of operators. Therefore preferably as a first step, I want to ask if there are formulations of differential equations in operator algebras.

To be more concrete, as a toy model, if we consider finite dimensional self-adjoint matrices $A$ and perturb it by self-adjoint matrices $A_s = A + sK$, where $K$ is self adjoint, and we consider $f \in B^1_{\infty,1}(\mathbb{R})$ (Besov space), it is known that we have a formulation of differentiation in terms of double operator integral. Then what can we say about solution to the ODE: $\frac{d}{ds}f(A_s) = B$ where B is a fixed self adjoint matrix.

Furthermore, suppose we keep the above setting and take $f\in B^2_{\infty,1}(\mathbb{R})$, we can now consider second order derivative in terms of a multi-operator integral. Then following the usual Laplacian on $\mathbb{R}^n$ and let $\{ E_{ij} \}$ be a basis of n-dim Hermitian matrices, consider the analog of Laplacian $\Sigma_{ij}\frac{d^2}{ds^2}f(A^{ij}_s)$ where $A^{ij}_s = A + sE_{ij}$. Can we formulate and prove a maximum principle for such an analog of Laplacian?

Anyhow, this is a super long question. Any thoughts are helpful. The comment on operator differential equation is just my very pre-mature thoughts. Any direction regarding the Polchinski's equation (not necessarily related to operator algebra) is greatly appreciated. Thanks a lot for the help.