Mathematics – Probability
Scientific paper
2011-08-11
Mathematics
Probability
Scientific paper
Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in $L^1(\mathcal{O})$ on bounded domains $\mathcal{O}$. The generation of a continuous, order-preserving random dynamical system (RDS) on $L^1(\mathcal{O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^\infty(\mathcal{O})$ norm. Uniform $L^\infty$ bounds and uniform space-time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated. A pathwise Wong-Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space-time rough signals existence of solutions is proven for initial data in $L^{m+1}(\mathcal{O})$.
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