1. Is the stochastic parabolicity condition dependent on $p$ and $q$?
- Author
-
Mark Veraar and Zdzislaw Brzezniak
- Subjects
Statistics and Probability ,Mathematics::Analysis of PDEs ,Type (model theory) ,01 natural sciences ,Noise (electronics) ,Omega ,Multiplicative noise ,010104 statistics & probability ,parabolic stochastic evolution ,Mathematics - Analysis of PDEs ,strong solution ,mild solution ,35R60 ,FOS: Mathematics ,0101 mathematics ,stochastic parabolicity condition ,Mathematics ,Mathematical physics ,multiplicative noise ,010102 general mathematics ,Mathematical analysis ,maximal regularity ,Probability (math.PR) ,Order (ring theory) ,Torus ,60H15, 35R60 ,gradient noise ,stochastic partial differential equation ,Functional Analysis (math.FA) ,Stochastic partial differential equation ,Gradient noise ,Mathematics - Functional Analysis ,60H15 ,Statistics, Probability and Uncertainty ,blow-up ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $\T =[0,2\pi]$. The equation is considered in $L^p(\O\times(0,T);L^q(\T))$ for $p,q\in (1, \infty)$. It is well-known that if the noise is of gradient type, one needs a stochastic parabolicity condition on the coefficients for well-posedness with $p=q=2$. In this paper we investigate whether the well-posedness depends on $p$ and $q$. It turns out that this condition does depend on $p$, but not on $q$. Moreover, we show that if $1, Comment: Revision, accepted for publication in Electronic Journal of Probability
- Published
- 2011
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