Space-time fractional stochastic Burgers-type equation

Abstract

We establish the existence and uniqueness of solutions to a (1+1)-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equations of this type describe non-Markovian dynamics with long-range temporal and spatial dependencies. The main difficulty is that the solution to the linearized equation has low spatial Hölder regularity, rendering the singular nonlinearity G(u)∂_x u classically ill-defined. The proof relies on two main ingredients: first, combining the Da Prato-Debussche decomposition with rough path theory to rigorously interpret the nonlinear product; and second, deriving Schauder-type estimates for fractional heat kernels given in terms of slowly decaying Mittag-Leffler functions. The nonlocality of the fractional time derivative requires a strong assumption on the initial condition.

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BibTeX RIS

BibTeX

@misc{kuo2026spacetime,
  title = {{Space-time fractional stochastic Burgers-type equation}},
  author = {Ming-Wei Kuo and Konstantin Matetski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20599},
  eprint = {2609.20599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20599},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Space-time fractional stochastic Burgers-type equation
AU  - Ming-Wei Kuo
AU  - Konstantin Matetski
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20599
UR  - https://arxiv.org/abs/2609.20599
AB  - We establish the existence and uniqueness of solutions to a (1+1)-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equations of this type describe non-Markovian dynamics with long-range temporal and spatial dependencies. The main difficulty is that the solution to the linearized equation has low spatial Hölder regularity, rendering the singular nonlinearity G(u)∂_x u classically ill-defined. The proof relies on two main ingredients: first, combining the Da Prato-Debussche decomposition with rough path theory to rigorously interpret the nonlinear product; and second, deriving Schauder-type estimates for fractional heat kernels given in terms of slowly decaying Mittag-Leffler functions. The nonlocality of the fractional time derivative requires a strong assumption on the initial condition.
ER  -

Abstract

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