Stochastic immersed boundary method incorporating thermal fluctuations

We give a brief introduction to the stochastic immersed boundary method which allows for simulation of small length-scale physical systems in which elastic structures interact with a fluid flow in the presence of thermal fluctuations. The conventional immersed boundary method is extended to account for thermal fluctuations by introducing stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a numerical method derived using stochastic calculus to overcome the stiff features of the equations. We then discuss results which indicate that the method captures physical features predicted by statistical mechanics for small length-scale systems. The stochastic immersed boundary method holds promise as a numerical approach in simulating microscopic mechanical systems in which thermal fluctuations play a fundamental role. A more detailed discussion of this work is given in [1, 2, 3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.     

Critical Homogenization of SDEs Driven by a Levy Process in Random Medium

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is, when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations.     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles

Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting particles on a surface, at a detailed atomistic level. However such algorithms are typically computationatly expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-grained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.     

Stochastic and deterministic simulations of a delayed genetic oscillation model: Investigating the validity of reductions

Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Stochastic center of systems of stochastic differential equations on the plane

We study a stochastic analogy of the famous center problem of Dulac for quadratic differential equations in the plane. We introduce the concept of center for systems of stochastic differential equations of It\^o''s type on the plane, called stochastic center. We derive a criterion for the existence of such a center. We apply it to obtain necessary and sufficient conditions for quadratic stochastic differential equations in dimension 2.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Towards mesoscopic ergodic theory Dedicated with Admiration to the Memory of Professor Shantao Liao

The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic differential equations(SDEs) with less regular coefficients and degenerate noises. These equations are often derived as mesoscopic limits of complex or huge microscopic systems. By studying the associated Fokker-Planck equation(FPE), we prove the convergence of the time average of globally defined weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions. In the case where the set of stationary measures consists of a single element, the unique stationary measure is shown to be physical.Similar convergence results for the solutions of the FPE are established as well. Some of our convergence results, while being special cases of those contained in Ji et al.(2019) for SDEs with periodic coefficients, have weaken the required Lyapunov conditions and are of much simplified proofs. Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.     

Numerical comparison of three stochastic methods for nonlinear PN junction problems

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.     

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Itô integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.     

Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: approximate invariant measures

Systems of Wick stochastic differential equations are studied. Using an estimate on the Wick product we apply Picard iteration to prove a general existence and uniqueness theorem for systems of Wick stochastic differential equations. We also show the solution is stable with respect to perturbations of the noise. This result is used to show that the solution of a linear system of Wick stochastic differential equations driven by smoothed Brownian motion tends to the solution of the corresponding It equation as the smoothed process tends to Brownian motion     

Stochastic growth of radial clusters: Weak convergence to the asymptotic profile and implications for morphogenesis

The asymptotic shape of randomly growing radial clusters is studied. We pose the problem in terms of the dynamics of stochastic partial differential equations. We concentrate on the properties of the realizations of the stochastic growth process and in particular on the interface fluctuations. Our goal is unveiling under which conditions the developing radial cluster asymptotically weakly converges to the concentrically propagating spherically symmetric profile or either to a symmetry breaking shape. We demonstrate that the long range correlations of the surface fluctuations obey a self-affine scaling and that scale invariance is achieved by means of the introduction of three critical exponents. These are able to characterize the large scale dynamics and to describe those regimes dominated by system size evolution. The connection of these results with mathematical morphogenetic problems is also outlined.     

Explicit formulas for the top Lyapunov exponents of planar linear stochastic differential equations

Our aim in this article is to establish explicit formulas for the top Lyapunov exponents of planar linear stochastic differential equations. We use these formulas to examine the sample-path stability of a linear stochastic differential equations arising in fluid dynamics and of a model of stochastic Hopf bifurcation.