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Coarse-graining techniques play a central role in reducing the complexity of stochastic models and are typically characterized by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyze an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example. [ABSTRACT FROM AUTHOR]

We prove two new functional inequalities of the forms JG \varphi(\psi -- \psi) < a JG \psi ln(ψ-ψ} Jg 1/a and fG \ψ < ψ/ψ fG llikp)² f°r anY finitely connected, bounded C2-domain G C R2, a constant ßo > 0, any a > 0, and sufficiently regular functions p, ß>. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties for certain generalized solutions to the chemotaxis-Navier--Stokes system {nt + u ΔVn = Δn -- V • (nS'(æ,n,c)Vc);Δt + u Δc = Δc -- nf(c);ut + (« • Δ)u = Δu + ΔP + ΔΔo. V • u = 0} on a smooth, bounded, convex domain Q C R² with no-flux boundary conditions for n and c as well as a Dirichlet boundary condition for u. We further allow for a general chemotactic sensitivity S attaining values in R2x2 as opposed to a scalar one. [ABSTRACT FROM AUTHOR]

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 diffusions of Langevin--Smoluchowski type, the Fokker--Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backwards direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker--Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I). [ABSTRACT FROM AUTHOR]

We study nonlinear degenerate parabolic equations of Fokker-Planck type that can be viewed as gradient flows with respect to the recently introduced spherical Hellinger-Kantorovich distance. The driving entropy is not assumed to be geodesically convex. We prove solvability of the problem and the entropy-entropy production inequality, which implies exponential convergence to the equilibrium. As a corollary, we obtain some related results for the Wasserstein gradient flows. We also deduce transportation inequalities in the spirit of Talagrand, Otto, and Villani for the spherical and conic Hellinger-Kantorovich distances. [ABSTRACT FROM AUTHOR]

A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates towards equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes' drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behavior of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in self-similar variables attached to the center of mass to a stationary solution of a Fokker--Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes' drift with given potential flow. [ABSTRACT FROM AUTHOR]