Justification of the transformed labelled Dyson equation

Justify the labelled stochastic differential equation for the transformed infinite-dimensional Dyson Brownian motion with parameters \(\gamma_1\) and \(\gamma_2\), including its principal-value interaction and the time-dependent drift term, for \(0<t<1/\gamma_2\).

Background

The paper derives a transformation of the determinantal process associated with nonzero linear and quadratic parameters γ1\gamma_1 and γ2\gamma_2. Assuming that the corresponding zero-defect process has a suitable labelling satisfying the principal-value Dyson equation, the authors formally obtain a transformed labelled equation containing a common time-dependent drift and a linear term involving γ2Xi(t)\gamma_2 X_i(t). The determinantal construction establishes the process at the level of configurations and kernels, but does not justify this labelled stochastic equation.

References

We leave justifying eq ISDE gamma_2 as an open problem.

eq ISDE gamma_2:

$\difX_i\left(t\right)=\dif\widetilde{B}_i\left(t\right) +\left\{\operatorname{PV}\sum_{j\ne i} \frac1{X_i\left(t\right)-X_j\left(t\right)} -\frac{\gamma_1-p_b\left(\right)+\gamma_2X_i\left(t\right)} {1-\gamma_2t}\right\}\dif t. $

— Infinite-dimensional Dyson Brownian motion(s)  (2609.26386 - Assiotis et al., 22 Sep 2026) in Remark immediately following Theorem A, after equation (ISDE gamma_2)

The construction at $\beta=2$ leaves open the problem of identifying labelled equations for very dense initial configurations and for $\gamma_2>0$. This requires an appropriate summation or renormalisation of the interaction.

— Infinite-dimensional Dyson Brownian motion(s)  (2609.26386 - Assiotis et al., 22 Sep 2026) in Section 1, subsection “Future directions”

A related question is to find an intrinsic analytic class for the Stieltjes equation and prove initial-value uniqueness which also identifies its evolving poles, without assuming the regular particle representation and continuation properties of Proposition~\ref{prop:pole-recovery}.

— Infinite-dimensional Dyson Brownian motion(s)  (2609.26386 - Assiotis et al., 22 Sep 2026) in Section 1, subsection “Future directions”

It would be interesting to develop this and we leave this as an open problem.

— Infinite-dimensional Dyson Brownian motion(s)  (2609.26386 - Assiotis et al., 22 Sep 2026) in Section 1, subsection “Future directions”