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Register a new userExistence of invariant measures for the stochastic damped Schrodinger equation
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In this paper, we address the long time behaviour of solutions of the stochastic Schrodinger equation in $mathbb{R}^d$. We prove the existence of an invariant measure and establish asymptotic compactness of solutions, implying in particular the existence of an ergodic measure.
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We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (partial^3_x u +upartial_x u +lambda u)dt = f dt+Phi dW_t$ on ${mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.
In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions $f$ and $sigma$ and elliptic operators $A$ for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $A$ is the Shr{o}dinger-type operator $A = frac{1}{rho}(text{div} rho abla u)$ where $rho = e^{-|x|^2}$ is the Gaussian weight.
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrodinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
In this paper we consider the inhomogeneous nonlinear Schrodinger equation $ipartial_t u +Delta u=K(x)|u|^alpha u,, u(0)=u_0in H^s({mathbb R}^N),, s=0,,1,$ $Ngeq 1,$ $|K(x)|+|x|^s| abla^sK(x)|lesssim |x|^{-b},$ $0<b<min(2,N-2s),$ $0<alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $alpha_0(b)<alpha<(4-2b)/N$. The value $alpha_0(b)$ is the positive root of $Nalpha^2+(N-2+2b)alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=mu|x|^{-b}$, $muin mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.
Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrodinger equation in one-dimension and as an application the exit problem is investigated.
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