Is the stochastic parabolicity condition dependent on $p$ and $q$?

TY - JOUR

T1 - Is the stochastic parabolicity condition dependent on $p$ and $q$?

AU - Brzezniak, Zdzislaw

AU - Veraar, M. C.

PY - 2012/7

Y1 - 2012/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84864829013&partnerID=8YFLogxK

U2 - 10.1214/EJP.v17-2186

DO - 10.1214/EJP.v17-2186

M3 - Article

SN - 1083-6489

VL - 17

SP - 1

EP - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 56

ER -