The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and positive friction

A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to physically realizable examples where the coefficients defining the Langevin equation grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity.Comment: 11 pages, no figure

The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction

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.Comment: This paper has been corrected from a previous version. Author Austin McDaniel has been added. Lemma 2 has been rewritten, Lemma 3 added, previous version's Lemma 3 moved to Lemma 4. 20 pages, 1 figur

A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation ("the cell problem"), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.Comment: 32 page

Stratonovich-to-Ito transition in noisy systems with multiplicative feedback

Cataloged from PDF version of article.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, Ito calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation-convention 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 Ito 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

Thermophoresis of Brownian particles driven by coloured noise

The Brownian motion of microscopic particles is driven by the collisions with the molecules of the surrounding fluid. The noise associated with these collisions 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 as the time over which the particle's kinetic energy is lost due to friction (inertial time scale). 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 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 characteristic times are taken to zero; from this generator, we derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Furthermore, studying the long-term stationary particle distribution, we show that particles can accumulate towards the colder (positive thermophoresis) or the warmer (negative thermophoresis) regions depending on the dependence of their physical parameters and, in particular, their mobility on the temperature.Comment: 9 pages, 4 figure

Coastal Tropical Convection in a Stochastic Modeling Framework

Recent research has suggested that the overall dependence of convection near coasts on large-scale atmospheric conditions is weaker than over the open ocean or inland areas. This is due to the fact that in coastal regions convection is often supported by meso-scale land-sea interactions and the topography of coastal areas. As these effects are not resolved and not included in standard cumulus parametrization schemes, coastal convection is among the most poorly simulated phenomena in global models. To outline a possible parametrization framework for coastal convection we develop an idealized modeling approach and test its ability to capture the main characteristics of coastal convection. The new approach first develops a decision algorithm, or trigger function, for the existence of coastal convection. The function is then applied in a stochastic cloud model to increase the occurrence probability of deep convection when land-sea interactions are diagnosed to be important. The results suggest that the combination of the trigger function with a stochastic model is able to capture the occurrence of deep convection in atmospheric conditions often found for coastal convection. When coastal effects are deemed to be present the spatial and temporal organization of clouds that has been documented form observations is well captured by the model. The presented modeling approach has therefore potential to improve the representation of clouds and convection in global numerical weather forecasting and climate models.Comment: Manuscript submitted for publication in Journal of Advances in Modeling Earth System