We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient. arXiv:1404.2330v4 [math-ph] 24 Oct 2014 II. SMOLUCHOWSKI-KRAMERS APPROXIMATION For the main theorem, we assume x m t , x t ∈ U ⊂ R d , an open set, and v m t ∈ R d for all 0 ≤ t ≤ T . For an arbitrary vector a ∈ R d , |a| is the Euclidean norm and, for a d × d matrix A ∈ R d×d , |A| is the induced operator norm. We now state the assumptions and main theorem.

Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system's state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Itô calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equationconvention pair must be determined from the available experimental data before being able to predict the system's behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Itô calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics.

A linear stochastic model is presented for the dynamics of water vapor and tropical convection. Despite its linear formulation, the model reproduces a wide variety of observational statistics from disparate perspectives, including (i) a cloud cluster area distribution with an approximate power law; (ii) a power spectrum of spatiotemporal red noise, as in the “background spectrum” of tropical convection; and (iii) a suite of statistics that resemble the statistical physics concepts of critical phenomena and phase transitions. The physical processes of the model are precipitation, evaporation, and turbulent advection–diffusion of water vapor, and they are represented in idealized form as eddy diffusion, damping, and stochastic forcing. Consequently, the form of the model is a damped version of the two-dimensional stochastic heat equation. Exact analytical solutions are available for many statistics, and numerical realizations can be generated for minimal computational cost and for any desired time step. Given the simple form of the model, the results suggest that tropical convection may behave in a relatively simple, random way. Finally, relationships are also drawn with the Ising model, the Edwards–Wilkinson model, the Gaussian free field, and the Schramm–Loewner evolution and its possible connection with cloud cluster statistics. Potential applications of the model include several situations where realistic cloud fields must be generated for minimal cost, such as cloud parameterizations for climate models or radiative transfer models.

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations. © 2012 Springer Science+Business Media, LLC

Brownian motion of microscopic particles is driven by collisions with surrounding fluid molecules. The resulting noise is not white, but coloured, due, e.g., to the presence of hydrodynamic memory. The noise characteristic time-scale is typically of the same order of magnitude as the inertial time-scale over which the particle's kinetic energy is lost due to friction. We demonstrate theoretically that, in the presence of a temperature gradient, the interplay between these two characteristic time-scales can have measurable consequences on the particle's long-time behaviour. Using homogenization theory, we analyse the infinitesimal generator of the stochastic differential equation describing the system in the limit where the two time-scales are taken to zero keeping their ratio constant and derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Studying the long-term stationary particle distribution, we show that particles accumulate towards the colder (positive thermophoresis) or the hotter (negative thermophoresis) regions depending on their physical parameters.

Abstract. We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as m → 0, its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents the Brownian motion on the manifold as a limit of inertial systems.Mathematics Subject Classification (2010). 58J65, 60H10, 82C31.

This paper investigates stochastic models whose dynamics switch depending on the state/regime of the system. Such models have been called "hybrid switching diffusions" and exhibit "sliding dynamics" with noise. Here the aim is an application to models of rainfall, convection, and water vapor, where two states/regimes are considered: precipitation and non-precipitation. Regime changes are modeled with a "trigger function," and four trigger models are considered: deterministic triggers (i.e. Heaviside function) or stochastic triggers (finite-state Markov jump process), with either a single threshold for regime transitions or two distinct thresholds (allowing for hysteresis). These triggers are idealizations of those used in convective parameterizations of global climate models, and they are investigated here in a model for a single atmospheric column. Two types of results are presented here. First, exact statistics are presented for all four models, and a comparison indicates how the trigger choice influences rainfall statistics. For example, it is shown that the average rainfall is identical for all four triggers, whereas extreme rainfall events are more likely with the stochastic trigger. Second, the stochastic triggers are shown to converge to the deterministic triggers in the limit of fast transition rates. The convergence is shown using formal asymptotics on the Master-Fokker-Planck equations, where the limit is an interesting Fokker-Planck system with Dirac delta coupling terms. Furthermore, the convergence is proved in the mean-square sense for pathwise solutions.

Clouds are repeatedly identified as a leading source of uncertainty in future climate predictions. Of particular importance are stratocumulus clouds, which can appear as either (i) closed cells that reflect solar radiation back to space or (ii) open cells that allow solar radiation to reach the Earth's surface. Here we show that these clouds regimes—open versus closed cells—fit the paradigm of a phase transition. In addition, this paradigm characterizes pockets of open cells as the interface between the open‐ and closed‐cell regimes, and it identifies shallow cumulus clouds as a regime of higher variability. This behavior can be understood using an idealized model for the dynamics of atmospheric water as a stochastic diffusion process. With this new conceptual viewpoint, ideas from statistical mechanics could potentially be used for understanding uncertainties related to clouds in the climate system and climate predictions.