We consider Hamilton–Jacobi equations which characterize optimal controlled partial differential equations of the following types: the Allen–Cahn equation, the Cahn–Hilliard equation, a nonlinear Fokker–Planck equation, and aVlasov–Fokker–Planck equation. In each of the examples, the optimal control problem and its associated cost functional can be derived as limit from a microscopically define...

The effect of an external perturbation on the energy spectrum of a mesoscopic quantum conductor can be described by a Brownian motion model developed by Dyson who wrote a Fokker–Planck equation for the evolution of the joint probability distribution of the energy levels. For weakly disordered conductors, which can be described by a Gaussian random matrix ensemble, the solution of the Fokker–Pla...

Using exact expressions for the persistence probability and for the leading eigenvalue of the Fokker-Planck operator of a random walk in a random environment, we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many exact results are obtained from this correspondence, including th...

Two conceptually different approaches to incorporate growth uncertainty into sizestructured population models have recently been investigated. One entails imposing a probabilistic structure on all the possible growth rates across the entire population, which results in a growth rate distribution model. The other involves formulating growth as a Markov stochastic diffusion process, which leads t...

By treating the bulk–quantized Yang–Mills theory as a constrained system we obtain a consistent gauge–fixed BRST hamiltonian in the minimal sector. This provides an independent derivation of the 5–d lagrangian bulk action. The ground state is independent of the (anti)ghosts and is interpreted as the solution of the Fokker–Planck equation, thus establishing a direct connection to the Fokker–Plan...

In this article, we establish a general formula for higher order linear functional derivatives the composition of an arbitrary smooth on 1-Wasserstein space with solution Fokker-Planck PDE. This has important links theory propagation chaos and mean-field games. Dans cet nous établissons une formule générale pour les dérivées fonctionnelles linéaires d'ordre supérieur la d'une fonctionnelle régu...

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator techniques, we derive a Fokker-Planck equation describing the relaxation of a "test" vortex in a bath of "field" vortices at statistical equilibrium. The relaxation is due to the combined effect of a diffusion and a drift. The drift is shown to be responsible for the organization ...

We establish a new and intriguing connection between the FokkerPlanck equation with gradient drift term and an associated free energy functional. Namely, we demonstrate that such a Fokker-Planck equation may be interpreted as a gradient flux, or a steepest descent, of a free energy functional with respect to a certain metric. This is accomplished through the construction of a time-discrete iter...

We consider an evolving network of a fixed number of nodes. The allocation of edges is a dynamical stochastic process inspired by biological reproduction dynamics, namely by deleting and duplicating existing nodes and their edges. The properties of the degree distribution in the stationary state is analysed by use of the Fokker-Planck equation. For a broad range of parameters exponential degree...

For the characterization of surface height profiles we present a new stochastic approach which is based on the theory of Markov processes. With this analysis we achieve a characterization of the complexity of the surface roughness by means of a Fokker-Planck or Langevin equation, providing the complete stochastic information of multiscale joint probabilities. The method was applied to different...