On the Parabolic and Hyperbolic Liouville Equations

Abstract

Abstract We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>β</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range $$0&lt; \beta ^2 &lt; \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>β</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfrac> <mml:mo>≃</mml:mo> <mml:mn>1.37</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: $$0&lt; \beta ^2 &lt; 4\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>β</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mn>4</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> . (iii) As for SdNLW in the defocusing case $$\lambda &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical $$\Phi ^4_3$$ <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> -model) and prove local well-posedness of SdNLW for the range: $$0&lt; \beta ^2 &lt; \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>β</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>32</mml:mn> <mml:mo>-</mml:mo> <mml:mn>16</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> <mml:mn>5</mml:mn> </mml:mfrac> <mml:mi>π</mml:mi> <mml:mo>≃</mml:mo> <mml:mn>0.86</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When $$\lambda &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general $$\lambda \in {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0&lt; \beta ^2 &lt; \frac{4}{3} \pi \simeq 1.33 \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>β</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mi>π</mml:mi> <mml:mo>≃</mml:mo> <mml:mn>1.33</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.