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A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

Antonio Agresti, Max Sauerbrey, Mark Veraar

TL;DR

This work advances the regularity theory for parabolic SPDEs with transport noise by establishing Hölder continuity of weak solutions under spatial regularity assumptions on the noise and a uniform parabolicity condition enforced by $\nu>0$, via Kunita's stochastic flow of diffeomorphisms and a novel Itô–Wentzell formula for rough random fields. The authors transform the SPDE into a random PDE using stochastic characteristics, derive sharp a priori estimates for the inverse flow, and apply deterministic De Giorgi–Nash–Moser theory to obtain Hölder continuity, with both qualitative and quantitative versions. They further demonstrate the applicability of these regularity results by proving global, regular solutions for certain quasilinear SPDEs with transport noise, thereby connecting linear regularity theory to nonlinear, parabolic stochastic dynamics. The work also analyzes the limitations of the flow-based approach (potentially random Hölder exponents) and identifies conditions under which a uniform Hölder exponent can be achieved, offering a pathway toward broader well-posedness results in stochastic regularity theory.

Abstract

The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key open problem in the regularity theory of parabolic SPDEs. In this work, we resolve this question under the assumption that the noise coefficients are sufficiently regular in space. Our approach relies on Kunita's stochastic method of characteristics, which allows us to transform the original SPDE-via a stochastic flow of diffeomorphisms-into a random PDE to which the classical De Giorgi-Nash-Moser estimates apply. This program is accomplished through new a-priori estimates for the inverse of stochastic flows of diffeomorphisms, and a novel version of the Itô-Wentzell formula adapted to rough random fields. To demonstrate the applicability of our results, we establish the existence of global, regular solutions to quasilinear SPDEs with transport noise.

A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

TL;DR

This work advances the regularity theory for parabolic SPDEs with transport noise by establishing Hölder continuity of weak solutions under spatial regularity assumptions on the noise and a uniform parabolicity condition enforced by , via Kunita's stochastic flow of diffeomorphisms and a novel Itô–Wentzell formula for rough random fields. The authors transform the SPDE into a random PDE using stochastic characteristics, derive sharp a priori estimates for the inverse flow, and apply deterministic De Giorgi–Nash–Moser theory to obtain Hölder continuity, with both qualitative and quantitative versions. They further demonstrate the applicability of these regularity results by proving global, regular solutions for certain quasilinear SPDEs with transport noise, thereby connecting linear regularity theory to nonlinear, parabolic stochastic dynamics. The work also analyzes the limitations of the flow-based approach (potentially random Hölder exponents) and identifies conditions under which a uniform Hölder exponent can be achieved, offering a pathway toward broader well-posedness results in stochastic regularity theory.

Abstract

The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key open problem in the regularity theory of parabolic SPDEs. In this work, we resolve this question under the assumption that the noise coefficients are sufficiently regular in space. Our approach relies on Kunita's stochastic method of characteristics, which allows us to transform the original SPDE-via a stochastic flow of diffeomorphisms-into a random PDE to which the classical De Giorgi-Nash-Moser estimates apply. This program is accomplished through new a-priori estimates for the inverse of stochastic flows of diffeomorphisms, and a novel version of the Itô-Wentzell formula adapted to rough random fields. To demonstrate the applicability of our results, we establish the existence of global, regular solutions to quasilinear SPDEs with transport noise.

Paper Structure

This paper contains 14 sections, 18 theorems, 215 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof strategy for Theorem \ref{['t:DNM_intro']} and challenges
  3. Open problems
  4. Organization of the manuscript
  5. Preparatory results
  6. Stochastic flows on ${\mathbb R}^d$
  7. Periodic stochastic flows
  8. An Itô--Wentzell formula
  9. Stochastic De Giorgi--Nash--Moser estimates
  10. Proofs of Theorems \ref{['t:DeGiorgiNashMoser']}--\ref{['t:DeGiorgiNashMoser_quantitative']}
  11. Flow method and uniform Hölder exponent
  12. Proofs of Propositions \ref{['prop:blow_up_elliptic_ratio']} and \ref{['prop:sufficient_uniform_gamma']}
  13. Global smooth solutions to quasilinear SPDEs
  14. Proof of Theorem \ref{['t:global_quasilinear']}

Key Result

Theorem 1.1

Let the coefficient fields $a\colon \Omega \times [0,T]\times \mathbb{T}^d \to {\mathbb R}^{d\times d}$ and $b_n \colon \Omega \times [0,T]\times \mathbb{T}^d \to {\mathbb R}^{d}$ for $n\ge 1$ be progressively measurable and suppose that for some $\gamma_0>0$, $u_0\colon \Omega \to C^{\gamma_0}(\m holds almost surely. Then, the weak solution $u$ to eq:SPDE_lin_intro satisfies

Theorems & Definitions (36)

  • Theorem 1.1: Stochastic De Giorgi--Nash--Moser estimate -- simplified version
  • Remark 1.2
  • Theorem 1.3: Global regular solutions to quasilinear SPDEs -- simplified version
  • Theorem 2.2: Kolmogorov--Chentsov
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 26 more