# Oscillatory Survival Probability: Analytical and Numerical Study of a Non-Poissonian Exit Time

@article{Duc2016OscillatorySP, title={Oscillatory Survival Probability: Analytical and Numerical Study of a Non-Poissonian Exit Time}, author={Khanh Dao Duc and Zeev Schuss and David Holcman}, journal={Multiscale Model. Simul.}, year={2016}, volume={14}, pages={772-798} }

We consider the escape of a planar diffusion process from the domain of attraction $\Omega$ of a stable focus of the drift in the limit of small diffusion. The boundary $\partial\Omega$ of $\Omega$ is an unstable limit cycle of the drift, and the focus is close to the limit cycle. A new phenomenon of oscillatory decay of the peaks of the survival probability of the process in $\Omega$ emerges for a specific distance of the focus to the boundary which depends on the amplitude of the diffusion…

## 7 Citations

### Exit versus escape in a stochastic dynamical system of neuronal networks explains heterogenous bursting intervals.

- Physics
- 2020

Neuronal networks can generate burst events. It remains unclear how to analyse interburst periods and their statistics. We study here the phase-space of a mean-field model, based on synaptic…

### Exit Versus Escape for Stochastic Dynamical Systems and Application to the Computation of the Bursting Time Duration in Neuronal Networks

- PhysicsJournal of Nonlinear Science
- 2022

We study the exit time of two-dimensional dynamical systems perturbed by a small noise that exhibits two peculiar behaviors: (1) The maximum of the probability density function of trajectories is not…

### Escape from an attractor generated by recurrent exit

- MathematicsPhysical Review Research
- 2021

It is reported that for some systems, crossing the boundary is not enough, because stochastic trajectories return inside the basin with a high probability a certain number of times before escaping far away, due to a shallow potential.

### Dynamic Looping of a Free-Draining Polymer

- MathematicsSIAM J. Appl. Math.
- 2018

This work revisits the celebrated Wilemski--Fixman treatment for the looping time of a free-draining polymer and shows that under the condition of a small dimensionless $\epsilon$, the ratio of capture radius to the Kuhn length, it is able to systematically produce all known analytical and asymptotic results obtained by other methods.

### Stochastic Dynamics: Markov Chains, Random Transformations and Applications

- Mathematics
- 2018

The theory of noise-induced synchronization is introduced together with a more intuitive version of the multiplicative ergodic theory, and then is applied to hidden Markov models for developing an efficient algorithm of parameter inference.

### Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration

- BiologybioRxiv
- 2017

A novel passage time statistics method allows extracting transient dynamic at scales varying from one to few hundreds of nanometers, it predicts the local changes in the number of binding molecules following DSB and can be used to characterize the local dynamic of the chromatin.

### Exit versus escape a basin of attraction in noisy dynamical systems

- Physics, Mathematics
- 2020

We study a class of two-dimensional dynamical systems perturbed by small noise that exhibits a peculiar shift of the maximum associated to the probability density function from the point attractor.…

## References

SHOWING 1-10 OF 30 REFERENCES

### Oscillatory decay of the survival probability of activated diffusion across a limit cycle.

- Mathematics, PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2014

It is shown that in this limit the period of the oscillations is the winding number of the activated stochastic process, and that these peak probability oscillations should be detectable in planar dynamical systems with the topology described here.

### Diffusion Across Characteristic Boundaries

- Mathematics
- 1982

We consider the motion of a particle acted on by the deterministic force vector b${\bf b}({\bf x}( t ))$ and perturbed by random forces of white noise type. Such a particle will leave any bounded…

### Summing Logarithmic Expansions for Singularly Perturbed Eigenvalue Problems

- MathematicsSIAM J. Appl. Math.
- 1993

In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole.

### Mathematical modeling and numerical computation of narrow escape problems.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

A comprehensive study involving comparison with full numerical simulations shows that the full numerical and asymptotic results for the MFPT are within 1% accuracy even when total trap size is only moderately small, and for traps that may be rather close together.

### The Narrow Escape Problem

- MathematicsSIAM Rev.
- 2014

Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.

### Diffusion across characteristic boundaries with critical points

- Mathematics
- 1983

We consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with (i) an asymptotically stable equilibrium point or limit cycle and (ii)…

### Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle.

- Physics, MathematicsPhysical review letters
- 1996

The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle, and the implications for the weak-noise limit of stochastic resonance models are pointed out.

### The narrow escape problem for diffusion in cellular microdomains

- BiologyProceedings of the National Academy of Sciences
- 2007

Asymptotic formulas for the mean escape time are presented for a model of a Brownian particle confined to a bounded domain by a reflecting boundary by a small window through which it can escape, and several applications in cellular biology are presented.

### Theory and Applications of Stochastic Processes: An Analytical Approach

- Mathematics
- 2009

The Physical Brownian Motion: Diffusion And Noise.- The Probability Space of Brownian Motion.- It#x00F4 Integration and Calculus.- Stochastic Differential Equations.- The Discrete Approach and…

### Universality of first-passage- and residence-time distributions in non-adiabatic stochastic resonance

- Physics
- 2004

We present mathematically rigorous expressions for the first-passage-time and residence-time distributions of a periodically forced Brownian particle in a bistable potential. For a broad range of…