In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))\mathrm{d}t+{\sum }_{k=1}^{\infty }{g}^k(t,x)\delta {\beta }_t^k,t\in [0,T]$, with random coefficientsf and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H>1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (βk)k, we prove that the equation has a unique solution (in a Banach space of summability exponent p ≥ 2), and this solution is Hölder continuous in both time and space.

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))\mathrm{d}t+{\sum }_{k=1}^{\infty }{g}^k(t,x)\delta {\beta }_t^k,t\in [0,T]$, with random coefficients f and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H>1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (βk)k, we prove that the equation has a unique solution (in a Banach space of summability exponent p ≥ 2), and this solution is Hölder continuous in both time and space.

With an intrinsic approach on semi-simple Lie groups we find a Furstenberg–Khasminskii type formula for the limit of the diagonal component in the Iwasawa decomposition. It is an integral formula with respect to the invariant measure in the maximal flag manifold of the group (i.e. the Furstenberg boundary $B=G/MAN$). Its integrand involves the Borel type Riemannian metric in the flag manifolds. When applied to linear stochastic systems which generate a semi-simple group the formula provides a diagonal matrix...