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Stochastic very weak solution to parabolic equations with singular coefficients

Snežana Gordić, Tijana Levajković, Ljubica Oparnica

Abstract

A class of stochastic parabolic equations with singular potentials is analysed in the chaos expansion setting where the Wick product is used to give sense to the product of generalized stochastic processes. For the analysis of such equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions from the theory of partial differential equations. The stochastic very weak solution to the stochastic parabolic evolution problem is defined and its existence and uniqueness are shown. For regular enough potentials and data we prove consistency of stochastic very weak solution with a stochastic weak solution. We give an example to illustrate the method and possible applications.

Stochastic very weak solution to parabolic equations with singular coefficients

Abstract

A class of stochastic parabolic equations with singular potentials is analysed in the chaos expansion setting where the Wick product is used to give sense to the product of generalized stochastic processes. For the analysis of such equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions from the theory of partial differential equations. The stochastic very weak solution to the stochastic parabolic evolution problem is defined and its existence and uniqueness are shown. For regular enough potentials and data we prove consistency of stochastic very weak solution with a stochastic weak solution. We give an example to illustrate the method and possible applications.
Paper Structure (17 sections, 10 theorems, 146 equations)

This paper contains 17 sections, 10 theorems, 146 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Notations and classical results
  3. White noise analysis
  4. Multi-indices of arbitrary lenght
  5. Chaos expansions in $L^2(\mu)$
  6. Kondratiev spaces
  7. $X$-valued generalized stochastic processes of Kondratiev-type
  8. Singular GSPs of Kondratiev-type
  9. Stochastic parabolic equations with bounded potentials
  10. Stochastic parabolic equations with singular potentials
  11. Regularization of the singular GSPs of Kondratiev-type
  12. Stochastic very weak solution
  13. Uniqueness of the stochastic very weak solution
  14. Consistency
  15. Examples and concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

Given the force term $f\in AC([0,T];L^2(\mathbb R^d))$, the initial condition $g\in D$ and the potential $q \in L^{\infty} (\mathbb{R}^d)$, the (deterministic) parabolic initial value problem has a unique bounded solution $u\in AC([0,T];L^2(\mathbb R^d))$ which satisfies where

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Remark 1
  • Lemma 1.4
  • Lemma 1.5
  • Example 1.1
  • Example 1.2
  • Definition 1.1
  • Example 1.3
  • ...and 10 more