The Lyapunov spectrum for conditioned random dynamical systems

Abstract

We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations.

Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes

“…From time t k on they both evolve according to the SDE, but with different initial conditions at time t k , namely X δ t k and X δ,k−1 t k , respectively. In view of the observation (5), this is where SES, condition (7a), comes into play, with the exponential decay ensuring summability. However, the constant C in (5) depends on the distance between the initial conditions x and y, which are X δ t k and X δ,k−1 t k in the case at hand.…”

“…In view of the observation (5), this is where SES, condition (7a), comes into play, with the exponential decay ensuring summability. However, the constant C in (5) depends on the distance between the initial conditions x and y, which are X δ t k and X δ,k−1 t k in the case at hand. By definition of the two processes on the time interval [t k−1 , t k ] the (expected value of the) difference between X δ t k and X δ,k−1 t k is precisely the local weak error, i.e.…”

“…, for any m ≤ n. The first term is related to the local weak error, whereas the second is essentially a term of the form (5) and is bounded by using the SES condition. Choosing m = 1, by (7b) and then (7a), we have…”

“…We note in passing that this is related to the concept of pullback attractor, see e.g. [14], and to the definition of Lyapunov exponent, see [5] and references therein. The second and third authors of this paper have pushed a programme to show that SES is conceptually key and technically a flexible tool to produce uniform in time results, whether the approximation is produced numerically or not, and indeed SES has been successfully used as a key ingredient in [8], where the approximation is produced via averaging methods; in [1], where the approximation is given by a multi-scale particle system; and in [7] for other numerical approximations.…”