Lévy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Lévy walks move with a finite speed. Here, we present an extension of the Lévy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Lévy walk processes and will be of use in a variety of systems, for which the particles are externally confined.

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

Stationary states for a particle moving in a single-well, steeper than parabolic, potential driven by Lévy noise can be bi-modal. Here, we explore in details conditions that are required in order to induce multimodal stationary states having more than two modal values. Phenomenological arguments determining necessary conditions for emergence of stationary states of higher multimodality are provided. Basing on these arguments, appropriate symmetric single-well potentials are constructed. Finally, using numerical methods it is verified that stationary states have anticipated multimodality.

A Lévy noise is an efficient description of out-of-equilibrium systems. The presence of Lévy flights results in a plenitude of noise-induced phenomena. Among others, Lévy flights can produce stationary states with more than one modal value in single-well potentials. Here, we explore stationary states in special double-well potentials demonstrating that a sufficiently high potential barrier separating potential wells can produce bimodal stationary states in each potential well. Furthermore, we explore how the decrease in the barrier height affects the multimodality of stationary states. Finally, we explore a role of the multimodality of stationary states on the noise induced escape over the static potential barrier.

Stochastic evolution of various reaction networks is commonly described in terms of noise assisted escape of an overdamped particle from a potential well, as devised by the paradigmatic Langevin equation. When implemented for systems close to equilibrium, the approach correctly explains emergence of Boltzmann distribution for the ensemble of trajectories generated by Langevin equation and relates intensity of the noise strength to the mobility. This scenario can be further generalized to include effects of non-thermal, external burst-like forcing modeled by Lévy noise. In the paper forward and reverse kinetics of Langevin equations with Lévy noise are analyzed for simple model of potential wells pointing to the most probable escape which is executed via a single long jump. Heavy tails of Lévy noise distributions not only facilitate escape kinetics, but more importantly, change the escape protocol by altering final stationary state to a non-Boltzmann, non-equilibrium form. As a result, contrary to the kinetics induced by a Gaussian white noise, escape rates in environments with Lévy noise are determined not by the barrier height, but instead, by the barrier width. We discuss consequences of simultaneous action of thermal and Lévy noises on statistics of passage times and population of reactants in double-well potentials.

The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity—full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between partial resetting scenarios which restart velocity or position, as in the limit of very frequent resets only the position resetting (regardless of initial velocity) can trap the particle in the finite domain of motion. The velocity resetting or the simultaneous position and velocity restarting provide a possibility of facilitating the undamped escape kinetics.

We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of stochastic resonant activation (SRA) in the presence of a statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise-induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories, while SRA is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, we show that stochastic resetting cannot be easily used to further optimize the SRA in symmetric setups.

The noise driven motion in a bistable potential acts as the archetypal model of various physical phenomena. Here, we contrast the overdamped dynamics with the full (underdamped) dynamics. For the overdamped particle driven by a non-equilibrium, α-stable noise the ratio of forward and backward transition rates depends only on the width of a potential barrier separating both minima. Using analytical and numerical methods, we show that in the regime of full dynamics, contrary to the overdamped case, the ratio of transition rates depends both on widths and heights of the potential barrier. The analytical formula for the ratio of transition rates is corroborated by extensive numerical simulations.