We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the γ-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.

In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n⩽9. This result, that was only known to be true for n⩽4, is optimal: log(1/|x|^2) is a W^{1,2} singular stable solution for n⩾10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n⩽9, stable solutions are bounded in terms only of their L^1 norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are W^{1,2} in every dimension and they are smooth in dimension n⩽9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.

We study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed. We develop this representation into a computational tool for computing the front speeds without resorting to closure approximations. We demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner.

We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.