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 }, forced by an additive space-time white noise. (i) We first study SNLH for general lambda in {mathbb {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 . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case lambda >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 < 4pi . (iii) As for SdNLW in the defocusing case lambda > 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-model) and prove local well-posedness of SdNLW for the range: 0< beta ^2 < frac{32 - 16sqrt{3}}{5}pi simeq 0.86pi . 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, these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on beta 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}}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 , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.
Highlights
We reduce stochastic damped nonlinear wave equation (SdNLW) into a system of equations
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 stochastic nonlinear heat equation (SNLH) for general λ ∈ R without using the positivity of the Gaussian multiplicative chaos
In this paper, we study both SNLH (1.1) and SdNLW (1.2), which allows us to point out similarity and difference between the analysis of the stochastic heat and wave equations
Summary
On T2, it is well known that μs is a Gaussian probability measure supported on W s−1−ε,p (T2) for any ε > 0 and 1 ≤ p ≤ ∞. The key point is that, unlike [30, Theorem 1.11], this local well-posedness result yields convergence of the solution u N of the truncated sinh-Gordon equation (1.61) to some limit u Combining this local well-posedness result with Bourgain’s invariant measure argument [9,10], we obtain almost sure global well-posedness for the parabolic sinh-Gordon equation (1.13) and invariance of the renormalized Gibbs measure ρsinh in the sense of Theorem 1.9. Note that in view of (1.62), due to the exponential convergence to equilibrium for the linear stochastic heat equation, we have σNheat(t) ∼ σN as soon as t N −2+θ for some (small) θ > 0, and the regularization effect as in the wave case can only be captured at time scales t N −2+θ , which prevents us from building a local solution with deterministic initial data for arbitrary β2 > 0 in the parabolic case. In Appendix B, we present a proof of Lemma 3.5, which is crucial in establishing moment bounds for the Gaussian multiplicative chaos
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