The stochastic tamed MHD equations: existence, uniqueness and invariant measures

We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space $${\mathbb {R}}^{3}$$ R 3 and on the torus $${\mathbb {T}}^{3}$$ T 3 . In a first step, existence of a unique strong solution are established by constructing a weak solution, proving that pathwise uniqueness holds and using the Yamada–Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.

Stability of stochastic differential equations driven by multifractional Brownian motion

Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

Approximation of solutions of mean-field stochastic differential equations

Our aim in this paper is to establish some strong stability properties of solutions of mean-field stochastic differential equations. These latter are stochastic differential equations where the coefficients depend not only on the state of the unknown process but also on its probability distribution. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.