On the Parabolic and Hyperbolic Liouville Equations

Abstract

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β e β u , forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R . 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 $$0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$ 0 < β 2 < 8 π 3 + 2 2 ≃ 1.37 π . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda >0$$ λ > 0 , 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: $$0< \beta ^2 < 4\pi $$ 0 < β 2 < 4 π . (iii) As for SdNLW in the defocusing case $$\lambda > 0$$ λ > 0 , 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 $$\Phi ^4_3$$ Φ 3 4 -model) and prove local well-posedness of SdNLW for the range: $$0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$ 0 < β 2 < 32 - 16 3 5 π ≃ 0.86 π . 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 $$\lambda > 0$$ λ > 0 , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$ β 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 $$\lambda \in {\mathbb {R}}$$ λ ∈ R without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi $$ 0 < β 2 < 4 3 π ≃ 1.33 π , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

“…In addition to the nonlinear wave and Schrödinger equations with odd power-type and Hartree nonlinearities discussed above, invariant Gibbs measures have also been studied in several other settings. For instance, there has been research on invariant Gibbs measures for derivative nonlinearities [BLS21,Den15,Tzv10], quadratic nonlinearities [GKO18,OOT21], exponential nonlinearities [ORTW20, ORW21,ST20b], radially-symmetric settings [BB14a,BB14b,BB14c,Den12,Tzv06], KdV and generalized KdV [Bou94, CK21, ORT16, Ric16], fractional dispersion relations [ST20a,ST21], and lattice models [AKV20].…”

Hyperbolic $P(Φ)_2$-model on the plane

“…where the noise ζ is primarily taken to be a space-time white noise ξ. Here, N (u) denotes a nonlinearity which may be of a power-type [34,35,36,65,57,80,58,13,59,70,15] and trigonometric and exponential nonlinearities [63,66,64]. We also mention the works [69,61,60,76,71] on the (deterministic) nonlinear wave equations (1.4) with rough random initial data and [24,25,56] on SNLW with more singular (both in space and time) noises.…”

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

“…Stochastic nonlinear wave equations have been studied extensively in various settings; see [15,Chapter 13] for the references therein. In particular, over the last few years, we have witnessed a rapid progress in the theoretical understanding of nonlinear wave equations with singular stochastic forcing and/or rough random initial data; see [56,24,25,26,50,52,46,49,64,18,53,44,55,19,54,47,12,48]. In [25], Gubinelli, Koch, and the first author studied the quadratic SNLW on T 3 :…”