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==== Stability and dynamics====
[[File:Figure WSDE.png|thumb|429x429px|'''''Fig. 5''''' Approximate mean of the SDE (8.2) computed via Monte Carlo with ''100'' simulations of various schemes with ''h=1/16'' and ''p=q=6''.]] By construction, the weak LL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear SDEs, but it is not the case of the weak LL schemes in general. WLL schemes, with <math>p\leq q\leq p+2,</math> preserve the [[Moment (mathematics)|first two moments]] of the linear SDEs, and inherits the mean-square stability or instability that such solution may have<ref name=":11" />. This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover, these WLL schemes preserve the [[ergodicity]] of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs<ref name=":6"> Hansen N.R. (2003) Geometric ergodicity of discre-time approximations to multivariate diffusion. Bernoulli.9 : 725-743 [[doi:10.3150/bj/1066223276]]</ref>. For nonlinear SDEs with small noise (i.e., (8.1) with <math>\mathbf{g}_{i}(t)\approx 0</math>), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme<ref name=":11" />. For instance, Fig. 5 shows the approximate mean of the SDE
<math>dx=-t^{2}x\text{ }dt+\frac{3}{2(t+1)}e^{-t^{3}/3}\text{ }dw_{t},\qquad \qquad x(0)=1, \qquad \quad(8.2)</math>
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