On the stochastic \(p\)-Laplace equation
The \(p\)-Laplace equation with random perturbation \[ dX_t=\left[\text{div}\left(|\nabla X_t|^{p-2}\nabla X_t\right) -\gamma |X_t|^{p-2}\right]dt + BdW_t, \quad X_0=x \] in the singular case \(1<p\leq 2\) where \(\gamma\geq 0\), \(B\) is a Hilbert-Schmidt operator and \(W\) a cylindrical Brownian motion is studied. Some ergodicity, ultraboundedness and compactness properties for the associated transition semigroups are obtained. Moreover, uniqueness of the invariant measure and convergence rate of the semigroup are derived. The main tool is a dimension-free Harnack inequality which is established by a coupling argument. Results are applied to stochastic reaction-diffusion equations and stochastic \(p\)-Laplace equations in Hilbert space.
scientific article; zbMATH DE number 5626948
scientific article