Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise (original) (raw)
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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q)xi, where e(q)xi is Gaussian white noise whose amplitude e(q) depends on q itself. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities are uniquely resolved if the system has a known equilibrium distribution exp[-V(q)/T] and if, at some more fundamental level, the physics of the system is reversible. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation &quml;+e(2)(q)&qdot;=-nablaV(q)+e(-1)(q)xi with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Itoinsertion mark.
The Hopf bifurcation with bounded noise
2011
We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.