Conditional probabilities in multiplicative noise processes

Effects of Multiplicative Noise in Bistable Dynamical Systems

This study explores the escape dynamics of bistable systems influenced by multiplicative noise, extending the classical Kramers rate formula to scenarios involving state-dependent diffusion in asymmetric potentials. Using a generalized stochastic calculus framework, we derive an analytical expression for the escape rate and corroborate it with numerical simulations. The results highlight the critical role of the equilibrium potential Ueq(x), which incorporates noise intensity, stochastic prescription, and diffusion properties. We show how asymmetries and stochastic calculus prescriptions influence transition rates and equilibrium configurations. Using path integral techniques and weak noise approximations, we analyze the interplay between noise and potential asymmetry, uncovering phenomena such as barrier suppression and metastable state decay. The agreement between numerical and analytical results underscores the robustness of the proposed framework. This work provides a comprehensive foundation for studying noise-induced transitions in stochastic systems, offering insights into a broad range of applications in physics, chemistry, and biology.

Statistics of Stochastic Entropy for Recorded Transitions between ENSO States

We analyze the transitions between established phases of the El Niño Southern Oscillation (ENSO) by surveying the daily data of the southern oscillation index from an entropic viewpoint using the framework of stochastic statistical physics. We evaluate the variation of entropy produced due to each recorded path of that index during each transition as well as taking only into consideration the beginning and the end of the change between phases and verified both integral fluctuation relations. The statistical results show that these entropy variations have not been extreme entropic events; only the transition between the strong 1999-2000 La Niña to the moderate 2002-2003 El Niño is at the edge of being so. With that, the present work opens a long and winding avenue of research over the application of stochastic statistical physics to climate dynamics.

Renormalization of stochastic differential equations with multiplicative noise using effective potential methods

We present a method to renormalize stochastic differential equations subjected to multiplicative noise. The method is based on the widely used concept of effective potential in high-energy physics and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise. We derive a general formula for the one-loop effective potential of a single ordinary stochastic differential equation (with arbitrary interaction terms) subjected to multiplicative Gaussian noise (provided the noise satisfies a certain normalization condition). To illustrate the usefulness (and limitations) of the method, we use the effective potential to renormalize a toy chemical model based on a simplified Gray-Scott reaction. In particular, we use it to compute the scale dependence of the toy model's parameters (in perturbation theory) when subjected to a Gaussian power-law noise with short time correlations.

State-dependent diffusion in a bistable potential: Conditional probabilities and escape rates

We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similar to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state-dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.