#### Vol. 2, No. 3, 2009

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Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

Vol. 2 (2009), No. 3, 281–304
##### Abstract

We prove the local well-posedness of the periodic stochastic Korteweg–de Vries equation with the additive space-time white noise. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space ${\stackrel{̂}{b}}_{p,\infty }^{s}\left(\mathbb{T}\right)$ for $s=-\frac{1}{2}+$, $p=2+$ such that $sp<-1$. In establishing local well-posedness, we use a variant of the Bourgain space adapted to ${\stackrel{̂}{b}}_{p,\infty }^{s}\left(\mathbb{T}\right)$ and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in $M\left(\mathbb{T}\right)$, the space of finite Borel measures on $\mathbb{T}$.

##### Keywords
stochastic KdV, white noise, local well-posedness
##### Mathematical Subject Classification 2000
Primary: 35Q53, 60H15
##### Milestones
Received: 2 January 2009
Revised: 28 August 2009
Accepted: 19 October 2009
Published: 9 February 2010
##### Authors
 Tadahiro Oh Department of Mathematics University of Toronto 40 St. George Street, Rm 6290 Toronto, ON  M5S 2E4 Canada