BV functions and sets of finite perimeter on configuration spaces

Abstract In this paper, we aim to develop the foundations of a theory of BV functions in the configuration space over the Euclidean space $${\mathbb {R}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> equipped with the Poisson measure $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> . We first construct the m -codimensional Poisson measure—formally written as “ $$(\infty -m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Poisson measure”—on the configuration space. We then show that our construction is consistent with potential theory induced by the infinitely many independent Brownian motions by establishing relations between the m -codimensional Poisson measure and Bessel capacities. Secondly, we introduce three different definitions of BV functions based on the variational, relaxation, and semigroup approaches, and prove the equivalence of them. In the process, we prove the p -Bakry–Émery inequality on the configuration space for any $$1&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . Thirdly, we construct perimeter measures and introduce an appropriate notion of measure-theoretic boundary, called the reduced boundary. We then prove that the perimeter measure can be expressed by the 1-codimensional Poisson measure restricted to the reduced boundary, which is a generalisation of De Giorgi’s identity to the configuration space. Finally, we construct the total variation measures for functions of bounded variation, and prove the Gauß–Green formula.