[Paper Review] The stochastic logarithmic Schrödinger equation

Viorel Barbu, Michael Röckner+1 more|arXiv (Cornell University)|Nov 2, 2015

Stochastic processes and financial applications10 references3 citations

TL;DR

This paper establishes global existence and uniqueness of solutions to the stochastic logarithmic Schrödinger equation with linear multiplicative noise using a rescaling approach and maximal monotone operator theory. It proves uniform estimates in the energy space $H^1(\mathbb{R}^d)$ and a specialized Orlicz space, overcoming non-Lipschitz nonlinearities via quasi-monotonicity and compact embedding arguments in stochastic settings.

ABSTRACT

In this paper we prove global existence and uniqueness of solutions to the stochastic logarithmic Schrödinger equation with linear multiplicative noise. Our approach is mainly based on the rescaling approach and the method of maximal monotone operators. In addition, uniform estimates of solutions in the energy space $H^1(\mathbb{R}^d)$ and in an appropriate Orlicz space are also obtained here.

Motivation & Objective

• To establish global existence and uniqueness of solutions to the stochastic logarithmic Schrödinger equation with linear multiplicative noise.
• To overcome the challenge of non-Lipschitz nonlinearity $y \mapsto y \log|y|^2$ in the stochastic setting.
• To derive uniform a priori estimates in the energy space $H^1(\mathbb{R}^d)$ and an appropriate Orlicz space $V$.
• To extend deterministic well-posedness results to the stochastic case using operator-theoretic and probabilistic techniques.

Proposed method

• Employing a rescaling transformation to reduce the stochastic equation to a form amenable to maximal monotone operator analysis.
• Applying the method of maximal monotone operators to handle the nonlinear drift term $X \log|X|^2$ in the infinite-dimensional stochastic PDE framework.
• Using a mollification technique with $\psi_\delta$ to regularize the solution and justify the application of Itô’s formula in the Itô-Skorokhod sense.
• Establishing continuity in space variable $\xi$ via uniform bounds on mollified solutions and applying the generalized dominated convergence theorem.
• Introducing a truncated logarithmic nonlinearity $L_{1/m}$ to control singularities and enabling interchange of integrals via stochastic Fubini theorem.
• Deriving a priori estimates in $H^1(\mathbb{R}^d)$ and Orlicz space $V$ to ensure tightness and passage to the limit in approximating sequences.

Experimental results

Research questions

• RQ1 Can global well-posedness be established for the stochastic logarithmic Schrödinger equation with linear multiplicative noise despite the non-Lipschitz nature of the logarithmic nonlinearity?
• RQ2 How can the method of maximal monotone operators be adapted to stochastic PDEs where the generator $i\Delta$ is not coercive?
• RQ3 What a priori estimates in $H^1(\mathbb{R}^d)$ and Orlicz spaces are necessary and sufficient to ensure convergence of approximating solutions?
• RQ4 Can compact embedding arguments be extended from deterministic to stochastic settings to pass to the limit in nonlinear terms?

Key findings

• Global existence and uniqueness of solutions are established for the stochastic logarithmic Schrödinger equation with initial data in $H^1(\mathbb{R}^d)$.
• Uniform a priori estimates are obtained in the energy space $H^1(\mathbb{R}^d)$, ensuring boundedness of the $H^1$-norm of solutions over time.
• The solution satisfies uniform bounds in a specific Orlicz space $V$, which controls the logarithmic nonlinearity near zero and infinity.
• The method successfully overcomes the lack of local Lipschitz continuity of $y \mapsto y \log|y|^2$ via quasi-monotonicity and regularization techniques.
• The norm square $|X(t)|^2_{L^2}$ is shown to be a continuous local martingale, implying mean norm square conservation and enabling the definition of a physical probability law in open quantum systems.
• The convergence of mollified solutions is justified via generalized dominated convergence and continuity arguments, allowing passage to the limit in the nonlinear term.