The stochastic fractional transport heat equation
Abstract
We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caputo derivative of order α∈ (0,2] and we assume that the Hurst coefficient H=(H0,H1,H2, ...,Hd) of the time-space fractional white noise is in (1/2,1)1+d. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we derive the correct space--time convolution equation and its Wiener chaos expansion. For 1<α≤2 a second initial condition is required. The endpoint α=2 is treated separately and yields a stochastic wave equation with finite propagation speed. A solution Y(t,x) is called \emph{mild} if E[Y2(t,x)] < ∞ for all t,x. In the multiplicative case the random-field condition is expressed by the square integrability of the forcing kernel. For the classical Laplacian and space--time white noise, the endpoint α=2 admits a random-field solution only in space dimension d=1. This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.
Related
- Beyond the Reduced Temperature Equation: Energy–Flux Dynamics in Fractional Heat Transport
- Exact temporal variations and parameter estimation for space--time fractional stochastic heat equations driven by multiplicative noise
- Solutions of the space-fractional heat equation with polynomial growth: Liouville theorems and Schauder estimates
- Singular Stochastic Heat Equations with Fractional Memory
- Backward Homogeneous Heat Equation with Ψ-Caputo Time Derivative and Spectral Fractional Laplacian
BibTeX
@misc{draouil2026stochastic,
title = {{The stochastic fractional transport heat equation}},
author = {Olfa Draouil and Rahma Yasmina Moulay Hachemi and Bernt Øksendal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08369},
eprint = {2610.08369},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08369},
}
RIS
TY - UNPB
TI - The stochastic fractional transport heat equation
AU - Olfa Draouil
AU - Rahma Yasmina Moulay Hachemi
AU - Bernt Øksendal
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08369
UR - https://arxiv.org/abs/2610.08369
AB - We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caputo derivative of order α∈ (0,2] and we assume that the Hurst coefficient H=(H_0,H_1,H_2, ...,H_d) of the time-space fractional white noise is in (1/2,1)^1+d. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we derive the correct space--time convolution equation and its Wiener chaos expansion. For 1<α≤2 a second initial condition is required. The endpoint α=2 is treated separately and yields a stochastic wave equation with finite propagation speed. A solution Y(t,x) is called \emph{mild} if E[Y^2(t,x)] < ∞ for all t,x. In the multiplicative case the random-field condition is expressed by the square integrability of the forcing kernel. For the classical Laplacian and space--time white noise, the endpoint α=2 admits a random-field solution only in space dimension d=1. This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.
ER -