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

TL;DR

This paper investigates the dependence of the stochastic parabolicity condition on the parameters p and q in the well-posedness of second order SPDEs with multiplicative noise, revealing a dependence on p but not on q.

Contribution

It demonstrates that the stochastic parabolicity condition depends on p but is independent of q, and shows that for 1<p<2, the classical condition can be relaxed.

Findings

The parabolicity condition depends on p but not on q.

For 1<p<2, the classical condition can be weakened.

Well-posedness results are established for the SPDE in L^p and L^q spaces.

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 $1<p<2$ the classical stochastic parabolicity condition can be weakened.