No abstract

We consider a system of real-valued spins interacting with each other through a meanfield Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time t c ∈ [0, ∞] from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness (t c = 0), short-time conservation and large-time loss of Gibbsianness (t c ∈ (0, ∞)), and preservation of Gibbsianness (t c = ∞). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time t = t c .MSC 2010. 60F10, 60K35, 82C22, 82C27.

We consider the parabolic Anderson model (PAM) ∂ t u = 1 2 ∆u + ξu in R 2 with a Gaussian (space) white-noise potential ξ. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t, written U (t), is given by log U (t) ∼ χt log t, with the deterministic constant χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue λ 1 (Q t ) of the Anderson operator on the box Keywords and phrases. parabolic Anderson model, AndersonHamiltonian, white-noise potential, singular SPDE, paracontrolled distribution, regularization in two dimensions, intermittency, almost-sure large-time asymptotics, principal eigenvalue of random Schrödinger operator.

We consider the stochastic differential equation on $\mathbb {R}^{d}$ ℝ d given by $$ \begin{array}{@{}rcl@{}} \mathrm{d} X_{t} = b(t,X_{t}) \mathrm{d} t + \mathrm{d} B_{t}, \end{array} $$ d X t = b ( t , X t ) d t + d B t , where B is a Brownian motion and b is considered to be a distribution of regularity $ > -\frac 12$ > − 1 2 . We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat-kernel estimates for Γt with explicit dependence on t and the norm of b.

We consider the stochastic differential equation on R d given bywhere B is a Brownian motion and b is considered to be a distribution of regularity > − 1 2 . We show that the martingale solution of the SDE has a transition kernel Γ t and prove upper and lower heat kernel bounds for Γ t with explicit dependence on t and the norm of b.