A continuum of invariant measures for the periodic KdV and mKdV equations

We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vişan-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^s(\mathbb {T})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than s greater-than 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0&gt;s&gt;1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schrödinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.