title: First-passage times for pattern formation in nonlocal partial differential equations.
abstract: We describe the lifetimes associated with the stochastic evolution from an unstable uniform state to a patterned one when the time evolution of the field is controlled by a nonlocal Fisher equation. A small noise is added to the evolution equation to define the lifetimes and to calculate the mean first-passage time of the stochastic field through a given threshold value, before the patterned steady state is reached. In order to obtain analytical results we introduce a stochastic multiscale perturbation expansion. This multiscale expansion can also be used to tackle multiplicative stochastic partial differential equations. A critical slowing down is predicted for the marginal case when the Fourier phase of the unstable initial condition is null. We carry out Monte Carlo simulations to show the agreement with our theoretical predictions. Analytic results for the bifurcation point and asymptotic analysis of traveling wave-front solutions are included to get insight into the noise-induced transition phenomena mediated by invading fronts.
authors.affiliation: Centro Atómico Bariloche, CNEA, Instituto Balseiro, Universidad Nacional de Cuyo, and CONICET, Avenida E. Bustillo 9500, CP 8400, Bariloche, Argentina.
authors.forename: Manuel O
authors.initials: MO
authors.lastname: Cáceres
authors.affiliation: Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.|Instituto de Investigaciones Filosóficas, Bulnes 642, Buenos Aires 1428, Argentina.|Universidad San Sebastián, Lota 2465, Santiago 7500000, Chile.
authors.forename: Miguel A
authors.initials: MA
authors.lastname: Fuentes