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Stochastic Porous Media Equation on General Measure Spaces with Increasing Lipschitz Nonlinearties

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Abstract

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $$(E, \mathscr{B}(E), \mu)$$, and the Laplacian replaced by a self-adjoint operator $$L$$. In the case of Lipschitz nonlinearities $$\Psi$$, we in particular generalize previous results for open $$E\subset \mathbb{R}^d$$ and $$L\!\!=$$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $$L$$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $$\Psi$$ at infinity for $$L^2(\mu)$$-initial data.

Most cited references7

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A harmonic calculus on the Sierpinski spaces

(1989)
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Dirichlet forms on fractals and products of random matrices

(1989)
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Stochastic generalized porous media and fast diffusion equations

(2007)
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Author and article information

Journal
2016-06-09
1606.03001