Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInvariant Measure for Stochastic Functional Differential Equations in Hilbert Spaces
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.
rate research
Read More
Using the Maslowski and Seidler method, the existence of invariant measure for 2-dimensional stochastic Cahn-Hilliard-Navier-Stokes equations with multiplicative noise is proved in state space $L_x^2times H^1$, working with the weak topology. Also, the existence of global pathwise solution is investigated using the stochastic compactness argument.
This paper is concerned with solution in H{o}lder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach spaces of random (vector) processes. We define suitable functional H{o}lder spaces for them and give some inequalities among these H{o}lder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of $mathcal{O}(e)$ instead of $mathcal{O}(sqrt{e})$ attained in previous averaging.
Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrodinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a purification condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast towards this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}$, for a dispersion parameter $alpha>d$, some coupling constant $beta>0$, and $gamma eq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $gamma>0$, the measure is well-defined in the whole regime $alpha>d$ and $beta>0$ (Theorem 1.1 (i)), while in the focusing case $gamma<0$ its partition function is always infinite for any $alpha>d$ and $beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $alpha>d$ and $0<beta < beta^star_alpha$ for some natural parameter $0<beta^star_alphasim (alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<beta llbeta^star_alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $mathcal{M}=mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+frac{sqrt{2}}2<alpha leq 2$, locally for arbitrary $beta>0$ and globally for $0<beta ll beta^star_alpha$ (Theorem 1.5).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
