On the Onsager-Machlup functional of the Φ⁴-measure

2025/09/24 by Gasteratos, Ioannis, Selk, Zachary
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.20471

Abstract

We investigate the existence of generalised densities for the Φ⁴_d (d=1,2,3) measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the Φ⁴₁ measure coincides with the Φ⁴ action as expected. In two dimensions, we show that OM functionals of the P(Φ)₂ measures agree with the corresponding actions, by considering ``enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension 3, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the Φ⁴₃ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

Citations

• Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime
• Singularity of solutions to singular SPDEs
• Large Deviations of the Φ⁴₃ Measure via Stochastic Quantisation
• Onsager-Machlup functional for SLE_κ loop measures
• The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit
• Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime
• Tightness of the Ising-Kac model on the two-dimensional torus
• Paracontrolled distributions and singular PDEs
• Absolute continuity and convergence of densities for random vectors on Wiener chaos
• Analysis of SPDEs arising in path sampling part II: The nonlinear case

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