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

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.Comment: 18page

Limit theorems of Hilbert valued semimartingales and Hilbert valued martingale measures

AbstractIn this paper, we study tight criteria of càdlàg Hilbert valued processes and prove the tightness of Hilbert valued square integrable martingales and Hilbert valued semimartingales by using their characteristics. These extend appropriate results of Jacod and Shiryaev (1987). We also discuss the property of Hilbert valued martingale measure and introduce the concept of convergence of martingale measures in distribution. The sufficient and necessary conditions are provided for strongly orthogonal martingale measures with independent increments. The conditions are given for convergence of martingale measures

Vague convergence of locally integrable martingale measures

AbstractIn this paper, we introduce the concept of the vague convergence of locally integrable martingale measures in distribution, which is an organic combination of the vague convergence of Radon measures and the weak convergence of martingales in distribution. The conditions are provided for vague convergence of martingale measures. We also study the convergence of stochastic integrale with respect to martingale measures in distribution