Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence, we obtain equivalence of existence and uniqueness results for these equations. Moreover, we prove an analogous result for stochastically perturbed Fokker-Planck-Kolmogorov equations. To do so, we particularly show that such stochastic equations for measures are, similarly to the deterministic case, intrinsically related to linearized second-order equations on the space of Borel (sub-)probability measures.Comment: 26 page

Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations

We study the incompressible hypodissipative Navier-Stokes equations with dissipation exponent $0 < \alpha < \frac{1}{2}$ on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.Comment: 32 pages, fixed several typos and notational inconsistencies, added further reference

On nonlinear Markov processes in the sense of McKean

We study nonlinear Markov processes in the sense of McKean's seminal work [30] and present a large class of new examples. Our notion of nonlinear Markov property is in McKean's spirit, but, in fact, more general in order to include examples of such processes whose one-dimensional time marginal densities solve a nonlinear parabolic PDE, more precisely, a nonlinear Fokker--Planck--Kolmogorov equation, such as Burgers' equation, the porous media equation on $\mathbb{R}^d$ for all $d \geq 1$, or variants of the latter with transport-type drift, and the $2D$ vorticity Navier--Stokes equation. We show that the associated nonlinear Markov process is given by path laws of weak solutions to a corresponding distribution-dependent stochastic differential equation where both the diffusion and drift coefficient depends singularly (i.e. Nemytskii-type) on its one-dimensional time marginals. We stress that for this the nonlinear Fokker--Planck--Kolmogorov equations do not necessarily have to be well-posed. Thus, we establish a one-to-one correspondence between solution flows of a large class of nonlinear parabolic PDEs and nonlinear Markov processes. As a particular example we obtain, for all $d \geq 1$, that the solution flow of the porous media equation on $\mathbb{R}^d$ is associated to a nonlinear Markov process, in the same way as the heat flow is associated to Brownian motion.Comment: Added Remark 3.3, Example 4.2.(v) and a slightly extended introductio

A computational study of driven micro-cavities in supersonic flat plate flow

Driven micro-cavities embedded in the wall beneath turbulent supersonic boundary layers are analyzed using two-dimensional computational fluid dynamics. This concept is a passive flow control technique in which very small cavities formed by arrays of thin vertical walls are oriented transverse to the flow direction and underlie the boundary layer. The purpose is to reduce or eliminate skin friction drag. Various micro-cavity configurations were analyzed at locations (0.1 m and 1 m) downstream of the leading edges of flat plates, for free-stream Mach numbers of 1.2, 2.0, and 3.0. Results focus on net drag reduction achieved, cavity flow-field effects, perforation effects in vertical cavity walls, cavity scale effects, mesh refinement issues, and the stability of the solutions. Skin friction drag was eliminated over micro-cavity regions for all configurations tested. Drag in these regions was due to pressure effects on vertical walls and exhibited a linear increase with downstream distance. Drag reductions as high as 18-20% (compared to a reference flat plate section) were obtained for 52-cavity geometries at Mach 2.0 and Mach 3.0 downstream of the 10 cm and 1 m flat plates, respectively. Perforation of the cavity walls showed no effect on net drag reduction for these cases. Stability issues were observed when using a fine grid mesh for the Mach 2.0 case, with significant oscillations seen in the drag. A parametric investigation in which cavity scale, number, and wall configuration were varied was also performed for two free-stream Mach numbers of 1.2 and 3.0. Drag reductions between 18-40% were seen for these cases. It is shown that drag reduction was reduced with increasing cavity length and that the steadiness of the solution increases with the number of vertical cavity walls present --Abstract, page iii

The influence of wildlife water developments and vegetation on rodent abundance in the Great Basin Desert

Rodent communities have multiple functions including comprising a majority of the mammalian diversity within an ecosystem, providing a significant portion of the available biomass consumed by predators, and contributing to ecosystem services. Despite the importance of rodent communities, few investigations have explored the effects of increasing anthropogenic modifications to the landscape on rodents. Throughout the western United States, the construction of artificial water developments to benefit game species is commonplace. While benefits for certain species have been documented, several researchers recently hypothesized that these developments may cause unintentional negative effects to desert-adapted species and communities. To test this idea, we sampled rodents near to and distant from wildlife water developments over 4 consecutive summers. We employed an asymmetrical before-after-control-impact (BACI) design with sampling over 4 summers to determine if water developments influenced total rodent abundance. We performed an additional exploratory analysis to determine if factors other than free water influenced rodent abundance. We found no evidence that water developments impacted rodent abundance. Rodent abundance was primarily driven by vegetation type and year of sampling. Our findings suggested that water developments on our study area do not represent a significant disturbance to rodent abundance and that rodent abundance was influenced by the vegetative community and temporal factors linked to precipitation and primary plant production. Our findings represent one of the 1st efforts to determine the effects of an anthropogenic activity on the rodent community utilizing a manipulation design

Evolution with Stochastic Fitness and Stochastic Migration

Migration between local populations plays an important role in evolution - influencing local adaptation, speciation, extinction, and the maintenance of genetic variation. Like other evolutionary mechanisms, migration is a stochastic process, involving both random and deterministic elements. Many models of evolution have incorporated migration, but these have all been based on simplifying assumptions, such as low migration rate, weak selection, or large population size. We thus have no truly general and exact mathematical description of evolution that incorporates migration.We derive an exact equation for directional evolution, essentially a stochastic Price equation with migration, that encompasses all processes, both deterministic and stochastic, contributing to directional change in an open population. Using this result, we show that increasing the variance in migration rates reduces the impact of migration relative to selection. This means that models that treat migration as a single parameter tend to be biassed - overestimating the relative impact of immigration. We further show that selection and migration interact in complex ways, one result being that a strategy for which fitness is negatively correlated with migration rates (high fitness when migration is low) will tend to increase in frequency, even if it has lower mean fitness than do other strategies. Finally, we derive an equation for the effective migration rate, which allows some of the complex stochastic processes that we identify to be incorporated into models with a single migration parameter.As has previously been shown with selection, the role of migration in evolution is determined by the entire distributions of immigration and emigration rates, not just by the mean values. The interactions of stochastic migration with stochastic selection produce evolutionary processes that are invisible to deterministic evolutionary theory

Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness

Rehmeier M. Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. JOURNAL OF EVOLUTION EQUATIONS. 2020.Let the coefficients a(ij) and b(i), i, j <= d, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) partial derivative(t)mu(t) = partial derivative(i)partial derivative(j)(a(ij)mu(t)) - partial derivative(i)(b(i)mu(t)) be Borel measurable, bounded and continuous in space. Assume that for every s. [0, T] and every Borel probability measure. on Rd there is at least one solution mu = (mu(t))t is an element of([s,T]) to the FPK-eq. such that mu(s) = v and t bar right arrow mu(t) is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution mu(s,v) for each pair (s,v) such that this family of solutions fulfills mu(s,v)(t) = mu(r,mu rs,v)(t) for all 0 <= s <= r <= t <= T, which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed

On Cherny's results in infinite dimensions: a theorem dual to Yamada-Watanabe

Rehmeier M. On Cherny's results in infinite dimensions: a theorem dual to Yamada-Watanabe. Stochastics and Partial Differential Equations: Analysis and Computations . 2021;9:33-70.We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of type dXt = b(t, X)dt + s(t, X)dWt, t = 0, and show that for such equations uniqueness in law is equivalent to joint uniqueness in law for deterministic initial conditions. Here W is a cylindrical Wiener process in a separable Hilbert space U and the equation is considered in a Gelfand triple V. H. E, where H is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of Cherny, who proved these statements for the case of finite-dimensional equations