On the Onsager-Machlup Functional of the $$\Phi ^4$$-Measure

Abstract We investigate the existence of generalised densities for the $$\Phi ^4_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Φ</mml:mi> <mml:mi>d</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> $$(d=1,2,3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\Phi ^4_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Φ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> measure coincides with the $$\Phi ^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Φ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> action as expected. In two dimensions, we show that OM functionals of the $$P(\Phi )_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Φ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> 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 $$\Phi ^4_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Φ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.