6 problems

Consider the stochastic Brusselator system with noise intensity $\sigma>0$. A random equilibrium is a random state invariant under the cocycle dynamics, namely a measurable point…

Consider the stochastic Brusselator with parameters $a$, $b$, and $\sigma$, and let $\varphi^t_\omega$ denote the cocycle map generated by a common noise realization $\omega$. Two…

Let $a,b,\sigma>0$ and let $A(\omega)$ be the random equilibrium of the stochastic Brusselator, satisfying … almost surely. Newman's conjecture. The top Lyapunov exponent is negati…

Let $a,b,\sigma>0$ and consider the stochastic Brusselator as a smooth random dynamical system $(\theta,\varphi)$ on $\mathbb{R}_+^2$, with stationary distribution $\rho$, invarian…

Random Milnor attractor conjecture. Both relative attractors (or attracting regions) in the multiply degenerated subspaces $M_{wv}$ and the optimal attractors in…

Let $X = X_0 + \sum X_i \circ \dfrac{dB_t^i}{dt}$ be an SDS on a manifold $M$. Suppose that $X$ is integrable and diffusion invariant with respect to an action of a compact Lie gro…