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2012 Is the stochastic parabolicity condition dependent on $p$ and $q$?
Zdzislaw Brzezniak, Mark Veraar
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Electron. J. Probab. 17: 1-24 (2012). DOI: 10.1214/EJP.v17-2186


In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $\mathbb{T} = [0,2\pi]$. The equation is considered in $L^p((0,T)\times\Omega;L^q(\mathbb{T}))$ 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 $1<p<2$ the classical stochastic parabolicity condition can be weakened.


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Zdzislaw Brzezniak. Mark Veraar. "Is the stochastic parabolicity condition dependent on $p$ and $q$?." Electron. J. Probab. 17 1 - 24, 2012.


Accepted: 22 July 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1266.60113
MathSciNet: MR2955048
Digital Object Identifier: 10.1214/EJP.v17-2186

Primary: 60H15
Secondary: 35R60

Keywords: Blow-up , gradient noise , maximal regularity , mild solution , Multiplicative noise , parabolic stochastic evolution , Stochastic parabolicity condition , Stochastic partial differential equation , Strong solution


Vol.17 • 2012
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