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On the Onsager-Machlup functional of the $Φ^4$-measure

Ioannis Gasteratos, Zachary Selk

Abstract

We investigate the existence of generalised densities for the $Φ^4_d$ $(d=1,2,3)$ 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 $Φ^4_1$ measure coincides with the $Φ^4$ 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 $Φ^4_3$ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

On the Onsager-Machlup functional of the $Φ^4$-measure

Abstract

We investigate the existence of generalised densities for the 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 measure coincides with the action as expected. In two dimensions, we show that OM functionals of the 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 , two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

Paper Structure

This paper contains 12 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. $\Phi_d^4$ theory
  3. Onsager-Machlup functional
  4. Contributions
  5. Notations and preliminaries
  6. Wick powers and renormalised approximations
  7. Background on OM functionals
  8. Onsager-Machlup for $\Phi_1^4$
  9. Onsager-Machlup for $\Phi_2^4$
  10. Onsager-Machlup for $\Phi_3^4$
  11. Restoring finiteness in $3$ dimensions
  12. Outlook and conclusion

Key Result

Theorem 1.1

Let $\alpha\in[0, 1/2),$$\mu$ be the $\Phi_1^4$ measure supported on $C^{\alpha}(\mathbb{T})$. For any $z_1,z_2\in H_0^1(\mathbb T^1)$ we have where $B^1_r(z_i),$$i=1,2,$ are norm-open balls of radius $r>0$ and centered at $z_i.$ In particular,

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.2: Duality estimate in Hölder-Besov spaces
  • proof
  • Lemma 2.3: Uniform convergence of Fourier series
  • proof
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 20 more