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The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states.
Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness of nonnegative strong
In this thesis we present few theoretical studies of the models of self-organized criticality. Following a brief introduction of self-organized criticality, we discuss three main problems. The first problem is about growing patterns formed in the abe
We consider a nonlinear stochastic heat equation $partial_tu=frac{1}{2}partial_{xx}u+sigma(u)partial_{xt}W$, where $partial_{xt}W$ denotes space-time white noise and $sigma:mathbf {R}to mathbf {R}$ is Lipschitz continuous. We establish that, at every
We consider a porous media type equation over all of $R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equati
From the starting point of the well known Reynolds number of fluid turbulence we propose a control parameter $R$ for a wider class of systems including avalanche models that show Self Organized Criticality (SOC) and ecosystems. $R$ is related to the