Escape Probability for Stochastic Dynamical Systems with Jumps

A Markov jump process approximation of the stochastic Burgers equation

We consider the stochastic Burgers equation $ \dnachd{t} ψ(t,r) = Δψ(t,r) + \nabla ψ^2(t,r)+\sqrt{γψ(t,r)} η(t,r) $ with periodic boundary conditions, where $t \ge 0,$ $r \in [0,1],$ and $η$ is some space-time white noise. A certain Markov jump process is constructed to approximate a solution of this equation.}

Exponential Stability of Nonlinear Fractional Stochastic System with Poisson Jumps

In this paper exponential stability of nonlinear fractional order stochastic system with Poisson jumps is studied in finite dimensional space. Existence and uniqueness of solution, stability and exponential stability results are established by using boundedness properties of Mittag-Leffler matrix function, fixed point route and local assumptions on nonlinear terms. A numerical example is given to illustrate the efficiency of the obtained results. Finally, conclusion is drawn.

On jumps stochastic slowly diffusion equations with fast oscillation coefficients

We present a large deviation principle for some stochastic evolution equations with jumps which depend on two small parameters, when the viscosity parameter ε tends to zero more quickly than the homogenization's one δε (written as a function of ε). In particular, we highlighted a large deviation principle in path-space using some classical techniques and a uniform upper bound for the characteristic function of a Feller process.

Stochastic resetting in the Kramers problem: A Monte Carlo approach

The theory of stochastic resetting asserts that restarting a search process at certain times may accelerate the finding of a target. In the case of a classical diffusing particle trapped in a potential well, stochastic resetting may decrease the escape times due to thermal fluctuations. Here, we numerically explore the Kramers problem for a cubic potential, which is the simplest potential with a escape. Both deterministic and Poisson resetting times are analyzed. We use a Monte Carlo approach, which is necessary for generic complex potentials, and we show that the optimal rate is related to th…

Minimum Action Path theory reveals the details of stochastic biochemical transitions out of oscillatory cellular states

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Tran- sitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending non-locally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

通过最优传输实现随机跳跃线性系统的均方稳定性

Mean Square Stability for Stochastic Jump Linear Systems via Optimal Transport

本文通过最优传输理论，提出了一种统一框架用于分析随机跳跃线性系统的均方稳定性，无需假设跳跃过程特性，证明了该方法的有效性。

Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

Consider a manifold $M$ endowed locally with a pair of complementary distributions $Δ^H \oplus Δ^V=TM$ and let $\text{Diff}(Δ^H, M)$ and $\text{Diff}(Δ^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $φ_t = ξ_t \circ ψ_t$, where $ξ_t \in \text{Diff}(Δ^H, M)$ and $ψ_t \in \text{Diff}(Δ^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present…

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a dev…

Stochastic delay differential equations with jumps in differentiable manifolds

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an S…