On the convection of a passive scalar by a turbulent Gaussian velocity field

Through the use of the Novikov-Furutsu formula for Gaussian processes an equation is obtained for the diffusion of the ensemble average of a passive scalar in an incompressible turbulent velocity field in terms of the two-point, two-time correlator of this field. The equation is valid for turbulence which is not necessarily homogeneous or stationary and thus generalizes previous work.     

Numerical simulations of stochastic differential equations

A simple, very accurate algorithm for numerical simulation of stochastic differential equations is described. Its relationship to colored noise is elucidated and exhibited by explicit results. The especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown.     

Numerical methods for stochastic partial differential equations with multiple scales

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.     

Numerical strategies for filtering partially observed stiff stochastic differential equations

In this paper, we present a fast numerical strategy for filtering stochastic differential equations with multiscale features. This method is designed such that it does not violate the practical linear observability condition and, more importantly, it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach. The proposed filtering algorithm comprises of a “macro-filter” that borrows ideas from the Heterogeneous Multiscale Methods and a “micro-filter” that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro-filter and macroscopic observations at asynchronous times. We will show that the proposed micro-filter is equivalent to solving an inverse problem for parameterizing differential equations. Numerically, we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions, especially when the microscopic dynamics is not mixing at all.     

A stochastic return map for stochastic differential equations

A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O()] additive Gaussian colored noise. The construction is valid provided the correlation time isO() orO(1). The effective noise in the return map has nonzeroO( ²) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.     

微分方程的部分Hamilton化与积分

将微分方程部分地表示为Hamilton系统的方程并写成逆变代数形式.用动力学代数建立方程的Poisson积分理论，并举例说明结果的应用. 关键词： 微分方程 部分Hamilton化 逆变代数形式 Poisson积分理论     

An inverse problem for stochastic differential equations

We discuss the problem of reconstructing the drift coefficient of a diffusion from the knowledge of the transition probabilities outside a given bounded region in ^d ,d>1. We also give an interpretation of the solution of this inverse problem in the framework of stochastic mechanics.This paper is dedicated to the dear memory of Paola Calderoni.     

Symmetry and integrability for stochastic differential equations

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A 43 (2010) & 44 (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility of a system of SDEs to a lower dimensional one. We consider both “deterministic” symmetries and “random” ones, in the sense introduced recently by Gaeta and Spadaro [J. Math. Phys. 58 (2017)].     

Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Abstract

Necessary and sufficient conditions for the linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations are given. Stochastic integrating factor is introduced to solve the linear jump- diffusion stochastic differential equations. Closed form solutions to certain linearizable jump-diffusion stochastic differential equations are obtained.     

Numerical integration of the fluctuating hydrodynamic equations

An approach to numerically integrate the Landau-Lifshitz fluctuating hydrodynamic equations is outlined. The method is applied to one-dimensional systems obeying the nonlinear Fourier equation and the full hydrodynamic equations for a dilute gas. Static spatial correlation functions are obtained from computer-generated sample trajectories (time series). They are found to show the emergence of long-range behavior whenever a temperature gradient is applied. The results are in very good agreement with those obtained from solving the correlation equations directly.     

Narrow bounds for numerical integration of differential equations

Through a detailed analysis of the properties of a system of differential equations, bounds are given for the error affecting the final result of a numerical integration. These bounds appear to be narrower than those obtained with other methods. The key procedure is to consider carefully the linear part of the system and to bound it taking account of all possible errors. No very significant restriction is made on the system.This work was partially supported by the Ministero della Pubblica Istruzione.     

On causal stochastic equations for log-stable multiplicative cascades

We reformulate various versions of infinitely divisible cascades proposed in the literature using stochastic equations. This approach sheds a new light on the differences and common points of several formulations that have been recently provided by several teams. In particular, we focus on the simplification occurring when the infinitely divisible noise at the heart of such model is stable: an independently scattered random measure becomes a stable stochastic integral. In the last section we discuss the D-dimensional generalization.     

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.     

Reduction and noncommutative integration of linear differential equations

A new method is proposed for derivation of exactly integrable linear differential equations based on the theory of noncommutative integration. The equations are obtained by reduction from original equations which are integrable in the noncommutative sense, with a large number of independent variables. It is shown that the reduced equations cannot be solved by traditional methods, since they do not possess the required algebraic symmetry.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 55–60, November, 1993.     

Probabilistic models for stochastic elliptic partial differential equations

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically.     

On the relations between Markovian master equations and stochastic differential equations

We have investigated the relationship between Markovian master equations (m.e.) and the corresponding stochastic differential equations (s.d.e.) for closed systems, i.e., systems not subjected to external pumping. We show that the form of the fluctuations in the s.d.e., i.e., additive or multiplicative, depends upon the properties of the kernel of the m.e. and the range of the state space of the stochastic variable(s), i.e., bounded or unbounded. The knowledge of these two properties of the m.e. permits the determination of the way in which the fluctuations enter into the s.d.e. (i.e., additive or multiplicative) and the calculation of their statistics. Several examples are presented to illustrate the general theory.     

On the linearization of nonlinear langevin-type stochastic differential equations

We show that a logical extension of the piecewise optimal linearization procedure leads to the Gaussian decoupling scheme, where no iteration is required. The scheme is equivalent to solving a few coupled equations. The method is applied to models which represent (a) a single steady state, (b) passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state, and (c) a bivariate case of passage from an unstable state to a final stable state. The results are shown to be in very good agreement with the Monte Carlo calculations carried out for these cases. The method should be of much value in multidimensional cases.