A Response Calculus for Liouville Brownian Motion I: Simple Spectrum, Joint Eigenvalue Densities, and Ward Identities

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Original abstract

We develop a response calculus for Dirichlet Liouville Brownian motion under Cameron--Martin shifts of the Gaussian free field. On every bounded connected planar domain, without boundary regularity assumptions, we prove throughout the full subcritical range $0<γ<2$ that the generator has almost surely simple spectrum and that every finite vector of ordered eigenvalues has an absolutely continuous law. This resolves the open simple-spectrum problem for Dirichlet Liouville Brownian motion. The calculus combines coherent versions of Gaussian multiplicative chaos with fixed-space perturbation of the associated trace forms. The first variation of an isolated eigenvalue cluster becomes a finite-dimensional compression on its eigenspace. Every multiple cluster admits a smooth direction with simple first-order splitting, while the response measures of distinct simple eigenvalues are linearly independent. Analytic zero-set and submersion arguments on finite-dimensional Gaussian slices then give the spectral conclusions. At the operator level, in the same full subcritical range, the calculus yields differentiability of the resolvent in a fixed energy space and tested one-sided, moving-measure, and fixed-base resolvent Ward identities. We construct explicit causal tempered distributions whose Laplace transforms realize the moving-measure and fixed-base responses; no time-domain heat-semigroup difference-quotient convergence is asserted. Finally, for $0<γ<\sqrt2$, Green--Riesz potentials of response measures for spatially averaged resolvent observables realize their Gaussian Sobolev gradients. This yields absolute continuity for the occupation resolvent at every parameter and, outside deterministic null sets of resolvent parameters, for further scalar observables and finite families associated with disjoint non-negative test functions.

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