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Existence of Martingale Solutions to Stochastic Constrained Heat Equation

Javed Hussain, Abdul Fatah, Saeed Ahmed

Abstract

This article extends the work on stochastic constrained heat equation in \cite{brzezniak2020global}. We will show the existence of Martingale solutions to the stochastic-constrained heat equations. The proof is based on compactness, tightness of measure, quadratic variations, and Martingale representation theorem.

Existence of Martingale Solutions to Stochastic Constrained Heat Equation

Abstract

This article extends the work on stochastic constrained heat equation in \cite{brzezniak2020global}. We will show the existence of Martingale solutions to the stochastic-constrained heat equations. The proof is based on compactness, tightness of measure, quadratic variations, and Martingale representation theorem.

Paper Structure

This paper contains 10 sections, 17 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Important Function Spaces
  3. Important Definitions, lemmas and Theorems
  4. Martingale Representation Theorem
  5. Faedo Galerkin Approximation
  6. Some Important Estimates
  7. Tightness of Measures
  8. Convergence of Quadratic Variations
  9. Convergence of Quadratic Variations
  10. Existence of Martingale Solution

Key Result

Theorem 3.3

dhariwal2017study The sequence $Y_k$of random variables with values in $\mathcal{S}$satisfies [T] if and only if

Theorems & Definitions (30)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Definition 3.5
  • Proposition 3.6
  • Lemma 3.7
  • Theorem 3.8
  • Theorem 4.1
  • proof
  • ...and 20 more