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Mild solutions to semilinear rough partial differential equations

Stefan Tappe

TL;DR

The paper develops existence and uniqueness results for mild solutions to rough partial differential equations in a semigroup framework with time-inhomogeneous coefficients, by defining rough convolution directly through the Gubinelli integral on controlled rough paths in $D(A^2)$. It presents a fixed-point approach to obtain local and global mild solutions, and extends the theory to SPDEs driven by infinite-dimensional Wiener processes and infinite-dimensional fractional Brownian motion. A key contribution is removing the need for a mild Sewing Lemma by working within $D(A^2)$ and providing explicit estimates for compositions, convolutions, and their remainders. The work also constructs a rough-path realization of infinite-dimensional Wiener processes and proves the equivalence of rough and Itô integrals in that setting, linking the RPDE framework to classical stochastic PDE theory and broadening the applicability to high-dimensional stochastic dynamics.

Abstract

We provide an existence and uniqueness result for mild solutions to rough partial differential equations in the framework of the semigroup approach. Applications to stochastic partial differential equations driven by infinite dimensional Wiener processes and infinite dimensional fractional Brownian motion are presented as well.

Mild solutions to semilinear rough partial differential equations

TL;DR

The paper develops existence and uniqueness results for mild solutions to rough partial differential equations in a semigroup framework with time-inhomogeneous coefficients, by defining rough convolution directly through the Gubinelli integral on controlled rough paths in . It presents a fixed-point approach to obtain local and global mild solutions, and extends the theory to SPDEs driven by infinite-dimensional Wiener processes and infinite-dimensional fractional Brownian motion. A key contribution is removing the need for a mild Sewing Lemma by working within and providing explicit estimates for compositions, convolutions, and their remainders. The work also constructs a rough-path realization of infinite-dimensional Wiener processes and proves the equivalence of rough and Itô integrals in that setting, linking the RPDE framework to classical stochastic PDE theory and broadening the applicability to high-dimensional stochastic dynamics.

Abstract

We provide an existence and uniqueness result for mild solutions to rough partial differential equations in the framework of the semigroup approach. Applications to stochastic partial differential equations driven by infinite dimensional Wiener processes and infinite dimensional fractional Brownian motion are presented as well.
Paper Structure (23 sections, 76 theorems, 364 equations)

This paper contains 23 sections, 76 theorems, 364 equations.

Table of Contents

  1. Introduction
  2. Frequently used notation
  3. Rough path theory
  4. Strongly continuous semigroups
  5. Compositions of regular paths with functions
  6. Compositions of controlled rough paths with functions
  7. Compositions with time-dependent smooth functions
  8. Compositions with bilinear operators
  9. Compositions with time-dependent linear operators
  10. Regular convolution integrals
  11. Rough convolution integrals
  12. Rough partial differential equations
  13. Solution concepts
  14. The space for the fixed point problem
  15. Auxiliary results
  16. ...and 8 more sections

Key Result

Lemma 3.4

Let $\beta \in (\frac{1}{3},\frac{1}{2}]$ and $\alpha \in (0,\beta)$ be arbitrary. Then for each rough path $\mathbf{X} = (X,\mathbb{X}) \in \mathscr{C}^{\beta}([0,T],V)$ we also have $\mathbf{X} \in \mathscr{C}^{\alpha}([0,T],V)$ and the estimate

Theorems & Definitions (153)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Definition 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Definition 3.9
  • ...and 143 more