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Abstract
We study the twodimensional 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 spacetime white noise. In particular, we show that the wellposedness 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 wellposedness 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 wellposedness argument for some range of α. On the other hand, when α≥1/2, we show that SNLW is illposed 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 PratoDebussche 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 wellposedness 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 language  English 

Pages (fromto)  144 
Journal  Electronic journal of probability 
Volume  26 
Issue number  9 
DOIs  
Publication status  Published  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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ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research