Stochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchets

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 behaviour 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.ou

On the Bulk Velocity of Brownian Ratchets

In this paper we study the unidirectional transport effect for Brownian ratchets modeled by Fokker--Planck-type equations. In particular, we consider the adiabatic and semiadiabatic limits for tilting ratchets, generic ratchets with small diffusion, and the multistate chemical ratchets. Having established a linear relation between the bulk transport velocity and the biperiodic solution, and using relative entropy estimates and new functional inequalities, we obtain explicit asymptotic formulas for the transport velocity and qualitative results concerning the direction of transport. In particular, we prove the conjecture by Blanchet, Dolbeault, and Kowalczyk that the bulk velocity of the stochastic Stokes' drift is nonzero for every nonconstant potential

Numerical methods for computing effective transport properties of flashing Brownian motors

We develop a numerical algorithm for computing the effective drift and diffusivity of the steady-state behavior of an overdamped particle driven by a periodic potential whose amplitude is modulated in time by multiplicative noise and forced by additive Gaussian noise (the mathematical structure of a flashing Brownian motor). The numerical algorithm is based on a spectral decomposition of the solution to the Fokker-Planck equation with periodic boundary conditions and the cell problem which result from homogenization theory. We also show that the numerical method of Wang, Peskin, Elston (WPE, 2003) for computing said quantities is equivalent to that resulting from homogenization theory. We show how to adapt the WPE numerical method to this problem by means of discretizing the multiplicative noise via a finite-volume method into a discrete-state Markov jump process which preserves many important properties of the original continuous-state process, such as its invariant distribution and detailed balance. Our numerical experiments show the effectiveness of both methods, and that the spectral method can have some efficiency advantages when treating multiplicative random noise, particularly with strong volatility.Comment: 49 pages, 8 figure

Asymptotic Behaviour and Derivation of Mean Field Models

This thesis studies various problems related to the asymptotic behaviour and derivation of mean field models from systems of many particles. Chapter 1 introduces mean field models and their derivation, and then summarises the following chapters of this thesis. Chapters 2, 3 and 4 directly study systems composed of many particles. In Chapter 2 we prove quantitative propagation of chaos for systems of interacting SDEs with interaction kernels that are merely Hölder continuous (the usual assumption being Lipschitz). On the way we prove the existence of differentiable stochastic flows for a class of degenerate SDEs with rough coefficients and a uniform law of large numbers for SDEs. Chapters 3 and 4 study the asymptotic behaviour of the Arrow-Hurwicz-Uzawa gradient method, which is a dynamical system for locating saddle points of concave-convex functions. This method is widely used in distributed optimisation over networks, for example in power systems and in rate control in communication networks. Chapter 3 gives an exact characterisation of the limiting solutions of the gradient method on the full space for arbitrary concave-convex functions. In Chapter 4 we extend this result to the subgradient method where the dynamics of the gradient method are restricted to an arbitrary convex set. Chapters 5, 6 and 7 study the stability of mean field models. Chapters 5 and 6 prove an instability criterion for non-monotone equilibria of the Vlasov-Maxwell system. In Chapter 5 we study a related problem in approximation of the spectra of families of unbounded self adjoint operators. In Chapter 6 we show how the instability problem for Vlasov-Maxwell can be reduced to this spectral problem. In Chapter 7 we give a proof of well-posedness of a class of solutions to the Vlasov-Poisson system with unbounded spatial density. Chapters 8 and 9 change track and study the dynamics of a solute in a fluid background. In Chapter 8 we study a simple model for this phenomena, the kinetic Fokker-Planck equation, and show contraction of its semi-group in the Wasserstein distance when the spatial variable lies on the torus. Chapter 9 studies a more complex model of passive transport of a solute under a large and highly oscillatory fluid field. We prove a homogenisation result showing convergence to an effective diffusion equation for the transported solute profile