Comparing the stochastic nonlinear wave and heat equations: a case study

Tadahiro Oh, Mamoru Okamoto

Research output: Contribution to journalArticlepeer-review

Abstract

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 α>0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α=1/2 for SNLW and at α=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<α<1/2. We first revisit this argument and establish multilinear smoothing of order 1/4 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 α. On the other hand, when α≥1/2, 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 α≥1/2. (ii) As for SNLH, we establish analogous results with a threshold given by α=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 (α=3/4 in the wave case and α=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).
Original languageEnglish
Pages (from-to)1-44
JournalElectronic journal of probability
Volume26
Issue number9
DOIs
Publication statusPublished - 12 Jan 2021

Keywords

  • stochastic nonlinear wave equation
  • nonlinear wave equation
  • stochastic nonlinear heat equation
  • nonlinear heat equation
  • stochastic quantization equation
  • renormalization
  • white noise

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