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In this paper, we establish the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain $D$ in the setting of $L^2(D)$ space. The result is valid for all initial values in $L^2(D)$. The logarithmic Sobolev inequality plays an important role., 30 pages

We study the nonlinear Schrödinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large., 27 pages

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $\lambda \geq 0$ and the coefficients depend on $(\omega,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $\lambda$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$., Comment: 27 pages

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity., 68 pages, to appear in Communications in Mathematical Physics

We study both strict and mild solutions to parabolic evolution equations of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in Banach spaces. First, we explore the deterministic case. The maximal regularity of solutions has been shown. Second, we investigate the stochastic case. We prove existence of strict solutions and show their space-time regularity. Finally, we apply our abstract results to a stochastic heat equation., Comment: 42 pages

In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity $\sigma:\RR{R}\to\RR{R}$ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane(J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326). We first study the existence and uniqueness of the solution of these equations {and} under suitable conditions on the initial function, we {also} study the asymptotic behavior of the solution with respect to the parameter $\lambda$. In particular, our results are significant extensions of those in Foondun et al (M. Foondun, K. Tian and W. Liu. On some properties of a class of fractional stochastic equations. Preprint available at arxiv.org 1404.6791v1.), Foondun and Khoshnevisan (M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568.), Nane and Mijena (J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326; J. B. Mijena, and E.Nane. Intermittence and time fractional partial differential equations. Submitted. 2014)., Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1505.04615

In this paper, we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and — for the sake of simplicity — finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory, we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction, we can apply the pool of results of rough path theory, in particular we can obtain strong and weak numerical schemes of high order converging to the solution process.