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2021 Comparing the stochastic nonlinear wave and heat equations: a case study
Tadahiro Oh, Mamoru Okamoto
Electron. J. Probab. 26: 1-44 (2021). DOI: 10.1214/20-EJP575


We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha > 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha $. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$.

These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).


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Tadahiro Oh. Mamoru Okamoto. "Comparing the stochastic nonlinear wave and heat equations: a case study." Electron. J. Probab. 26 1 - 44, 2021.


Received: 12 November 2019; Accepted: 17 December 2020; Published: 2021
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.1214/20-EJP575

Primary: 35K15, 35L71, 60H15


Vol.26 • 2021
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