Curvature Bound of Dyson Brownian Motion

Abstract We construct a strongly local symmetric Dirichlet form on the configuration space $$\varvec{\Upsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Υ</mml:mi> </mml:mrow> </mml:math> whose symmetrising (thus also invariant) measure is $$\textsf{sine}_\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sine</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> , which is the law of the sine $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> ensemble for every $$\beta &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . For every $$\beta &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , this Dirichlet form satisfies the Bakry–Émery gradient estimate $$\textsf {BE} (K, \infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>BE</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$K=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . This implies various functional inequalities, including the local Poincaré inequality, the local log–Sobolev inequality and the local hyper-contractivity. We then introduce an $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -transportation-type extended distance $$\bar{\textsf {d} }_{\varvec{\Upsilon }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mrow> <mml:mi>Υ</mml:mi> </mml:mrow> </mml:msub> </mml:math> on $$\varvec{\Upsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Υ</mml:mi> </mml:mrow> </mml:math> , and prove the dimension-free Harnack inequality and several Lipschitz regularisation estimates of the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -semigroup associated with the Dirichlet form in terms of $$\bar{\textsf {d} }_{\varvec{\Upsilon }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mrow> <mml:mi>Υ</mml:mi> </mml:mrow> </mml:msub> </mml:math> . As a result of $$\textsf {BE} (0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>BE</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we obtain that the dual semigroup on the space of probability measures over $$\varvec{\Upsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Υ</mml:mi> </mml:mrow> </mml:math> , endowed with a Benamou–Brenier-like extended distance $$\textsf {W} _{\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>E</mml:mi> </mml:msub> </mml:math> , satisfies the evolutional variation inequality with respect to the Bolzmann–Shannon entropy $$\textsf {Ent} _{\textsf{sine}_\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ent</mml:mi> <mml:msub> <mml:mi>sine</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:msub> </mml:math> associated with $$\textsf{sine}_\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sine</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> . Furthermore, the dual semigroup is characterised as the unique $$\textsf {W} _{\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>E</mml:mi> </mml:msub> </mml:math> -gradient flow in the space of probability measures with respect to $$\textsf {Ent} _