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Chaos expansion solutions of a class of magnetic Schrödinger Wick-type stochastic equations on $\mathbb{R}^d$

Sandro Coriasco, Stevan Pilipović, Dora Seleši

TL;DR

This work studies Schrödinger-type SPDEs on $R^d$ with magnetic structure and stochastic data, utilizing white-noise analysis, Wiener chaos expansions, Wick renormalization, and SG pseudodifferential calculus. It develops a rigorous framework to handle singular stochastic multiplications by transforming SPDEs into deterministic chaos hierarchies and proving local-in-time existence/uniqueness in Sobolev-Kato-Kondratiev spaces, with an explicit chaos expansion that yields unbiased moments and a pathway to numerical truncation. The paper provides results for linear, semilinear, and Wick-square nonlinearities, including a detailed recursive scheme for chaos coefficients and energy estimates to ensure convergence. The methods open avenues for moment computations, potential extensions to curved spaces, and applications in quantum dynamics under randomness, while offering a robust renormalization-based approach to stochastic Schrödinger equations.

Abstract

We treat some classes of linear and semilinear stochastic partial differential equations of Schrödinger type on $\mathbb{R}^d$, involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-Itô chaos expansions and pseudodifferential operator methods. The initial data and potential term of the Schrödinger operator are assumed to be generalized stochastic processes that have spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate (intersections of weighted) Sobolev-Kato-Kondratiev spaces.

Chaos expansion solutions of a class of magnetic Schrödinger Wick-type stochastic equations on $\mathbb{R}^d$

TL;DR

This work studies Schrödinger-type SPDEs on with magnetic structure and stochastic data, utilizing white-noise analysis, Wiener chaos expansions, Wick renormalization, and SG pseudodifferential calculus. It develops a rigorous framework to handle singular stochastic multiplications by transforming SPDEs into deterministic chaos hierarchies and proving local-in-time existence/uniqueness in Sobolev-Kato-Kondratiev spaces, with an explicit chaos expansion that yields unbiased moments and a pathway to numerical truncation. The paper provides results for linear, semilinear, and Wick-square nonlinearities, including a detailed recursive scheme for chaos coefficients and energy estimates to ensure convergence. The methods open avenues for moment computations, potential extensions to curved spaces, and applications in quantum dynamics under randomness, while offering a robust renormalization-based approach to stochastic Schrödinger equations.

Abstract

We treat some classes of linear and semilinear stochastic partial differential equations of Schrödinger type on , involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-Itô chaos expansions and pseudodifferential operator methods. The initial data and potential term of the Schrödinger operator are assumed to be generalized stochastic processes that have spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate (intersections of weighted) Sobolev-Kato-Kondratiev spaces.
Paper Structure (9 sections, 12 theorems, 104 equations)

This paper contains 9 sections, 12 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Solutions of linear magnetic stochastic Schrödinger equations on $\mathbb{R}^d$
  3. Solutions of semilinear magnetic stochastic Schrödinger equations on $\mathbb{R}^d$
  4. Wick-product nonlinearities
  5. White noise analysis
  6. Chaos expansions and the Wick product
  7. Estimates on functions of multiindeces
  8. Stochastic operators and differential operators with stochastic coefficients
  9. The calculus of $SG$ pseudodifferential operators

Key Result

Theorem 2.5

Under Assumptions hyp:P, the solution $u(t)$ to the associated deterministic Cauchy problem eq:SPDE with $u_0\in\mathcal{H}_{z,\zeta}(\mathbb{R}^d)$, $\mathbf{F}\equiv 0$ and $P_\gamma\equiv0$, $\gamma\not=(0,0,\cdots)$, satisfies the estimate for $T_0\in(0,T]$ and a positive constant $C_{z,\zeta}$ depending only on $z,\zeta\in\mathbb{N}$.

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.4
  • Theorem 2.5: craig
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • Definition 3.1
  • Remark 3.2
  • ...and 18 more