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Mild Solution of Semilinear Rough Stochastic Evolution Equations

Jiahao Liang, Shanjian Tang

Abstract

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $α$-Hölder rough path with $α\in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family.

Mild Solution of Semilinear Rough Stochastic Evolution Equations

Abstract

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term , where is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, is a two-step -Hölder rough path with and is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family.
Paper Structure (14 sections, 14 theorems, 137 equations)

This paper contains 14 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional analytic framework
  4. Increment operators and Hölder type spaces
  5. Rough paths
  6. Convolution and composition
  7. Stochastic controlled rough paths according to a monotone family
  8. Rough stochastic convolutions
  9. Stochastic controlled operator families
  10. Mild solution of semilinear rough SPDEs
  11. Existence and uniqueness
  12. Continuity of the mild solution map
  13. Spatial regularity of the mild solution away from the initial time
  14. An example

Key Result

Proposition 2.1

For $\gamma_{1} \leq \gamma_{2} \leq \gamma_{1}+1$, we have

Theorems & Definitions (29)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 19 more