Global Strong Well-Posedness of the stochastic bidomain equations with FitzHugh-Nagumo transport
Matthias Hieber, Amru Hussein, Martin Saal
TL;DR
This work addresses the global well-posedness of stochastic bidomain equations with FitzHugh–Nagumo transport under current noise modeled by a cylindrical Wiener process $W$. The authors decompose the solution into a stochastic convolution $Z$ and a deterministic remainder $V$, applying stochastic maximal regularity to $Z$ and deterministic maximal regularity in an interpolation–extrapolation framework to $V$ within critical Besov-type spaces. They establish local existence in multiple regularity regimes and then prove global existence in both $d=2$ and $d=3$ by deriving energy-type a priori bounds and utilizing parabolic regularization to upgrade regularity, yielding a unique global strong pathwise solution $U=V+Z$. The results provide a rigorous mathematical foundation for stochastic electrophysiology models under current noise, enabling robust analysis and simulation in two and three dimensions.
Abstract
Consider the bidomain equations from electrophysiology with FitzHugh--Nagumo transport subject to current noise, i.e., subject to stochastic forcing modeled by a cylindrical Wiener process. It is shown that this set of equations admits a unique global, strong pathwise solution within the setting of critical spaces. The proof is based on combining methods from stochastic and deterministic maximal regularity. In addition, the method of extrapolation spaces from deterministic evolution equations is transferred to the stochastic setting.