We establish a regularity result for stochastic heat equations in probabilistic evolution spaces of Besov type and we use it to prove a global in time existence and uniqueness of solution to a stochastic magnetohydrodynamics equation. The existence result holds with a positive probability which can be made arbitrarily close to one. The work is carried out by blending harmonic analysis tools such as Littlewood–Paley decomposition, Jean–Micheal Bony paradifferential calculus and stochastic calculus. The law of large numbers is a key tool in our investigation. Our global existence result is new in three-dimensional spaces.

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The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; −3/2 is the spectral slope in Kraichnan–Iroshnikov–Dobrowolny (KID) theory, −5/3 in Marsch–Matthaeus–Zhou and Goldreich–Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the −5/3 and −3/2 scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426–438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to −5/3 is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.

The spectral slope of Magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; −3/2 is the spectral slope in Kraichnan-Iroshnikov-Dobrowolny (KID) theory, −5/3 in Marsch-Matthaeus-Zhou's and Goldreich-Sridhar theories also called Kolmogorov-like (K-41 like) MHD theory, combination of the −5/3 and −3/2 scales in Biskamp and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig [Physica D 241(2012) 426-438], we establish inertial range bounds for K-41 like phenomenon in MHD turbulent flow through a mathematical rigour; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to −5/3 is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.