相关论文: Entropy dissipation of Fokker-Planck equations on …
This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first…
We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and…
We interpret a class of nonlinear Fokker-Planck equations with reaction as gradient flows over the space of Radon measures equipped with the recently introduced Hellinger-Kantorovich distance. The driving entropy of the gradient flow is not…
We propose a general method to identify nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) as gradient flows on the space of probability measures on $\mathbb{R}^d$ with a natural differential geometry. Our notion of gradient…
We derive the Fokker-Planck equation on the parametric space. It is the Wasserstein gradient flow of relative entropy on the statistical manifold. We pull back the PDE to a finite dimensional ODE on parameter space. Some analytical example…
In this paper we consider a nonlinear Fokker-Planck equation with asymptotically small parameters. It describes the diffusion of finite-size particles in the presence of a fixed distribution of obstacles in the limit of low-volume fraction.…
We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we…
In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard…
We study the Fokker-Planck equation as the hydrodynamic limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the rate functional, that characterizes the large deviations from the…
We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition…
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new…
The displacement $\lambda$-convexity of a nonstandard entropy with respect to a nonlocal transportation metric in finite state spaces is shown using a gradient flow approach. The constant $\lambda$ is computed explicitly in terms of a…
We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient…
The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we study an alternative…
In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding…
We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet…
We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric…
We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a…
The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous…
We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for…