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We derive a multiscale generalisation of the Bakry‐Émery criterion for a measure to satisfy a log‐Sobolev inequality. Our criterion relies on the control of an associated PDE well‐known in renormalisation theory: the Polchinski equation. It implies the usual Bakry‐Émery criterion, but we show that it remains effective for measures that are far from log‐concave. Indeed, using our criterion, we prove that the massive continuum sine‐Gordon model with  β  < 6 π  satisfies asymptotically optimal log‐Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors.  Communications on Pure and Applied Mathematics  published by Wiley Periodicals LLC.

Log‐Sobolev Inequality for the Continuum Sine‐Gordon Model