A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Facilitation shifts paradigms and can amplify coastal Restoration efforts

Restoration has been elevated as an important strategy to reverse the decline of coastal wetlands worldwide. Current practice in restoration science emphasizes minimizing competition between outplanted propagules to maximize planting success. This paradigm persists despite the fact that foundational theory in ecology demonstrates that positive species interactions are key to organism success under high physical stress, such as recolonization of bare substrate. As evidence of how entrenched this restoration paradigm is, our survey of 25 restoration organizations in 14 states in the United States revealed that >95% of these agencies assume minimizing negative interactions (i.e., competition) between outplants will maximize propagule growth. Restoration experiments in both Western and Eastern Atlantic salt marshes demonstrate, however, that a simple change in planting configuration (placing propagules next to, rather than at a distance from, each other) results in harnessing facilitation and increased yields by 107% on average. Thus, small adjustments in restoration design may catalyze untapped positive species interactions, resulting in significantly higher restoration success with no added cost. As positive interactions between organisms commonly occur in coastal ecosystems (especially in more physically stressful areas like uncolonized substrate) and conservation resources are limited, transformation of the coastal restoration paradigm to incorporate facilitation theory may enhance conservation efforts, shoreline defense, and provisioning of ecosystem services such as fisheries production.