Tail behaviour of stationary densities for one-dimensional random diffeomorphisms

Abstract

We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisation of the extremal diffeomorphism at the boundary point (in case the corresponding fixed point is hyperbolic), or the leading nonlinear term of the extremal diffeomorphism (in case the corresponding fixed point is not hyperbolic).

Figures (4)

• Figure 1: In (a), a plot of the extremal maps $T(x)\pm\sigma$ of the system \ref{['eq:toy_example']} for increasing values of $\sigma$ are shown, where the deterministic map is depicted in red (i.e. $\sigma=0$). Here $b=5$ was taken, and the blue, green, and orange plots correspond to noise strength values $\sigma=\sigma^*/4, \sigma^*$, and $2\sigma^*$, respectively, where $\sigma^*$ is given in \ref{['eq:sigma*']}. As a reference, the identity line is portrayed as black dotted line. In (b), the change of the minimal invariant sets is presented as $\sigma$ increases, where the coexistence of two disjoint minimal invariant sets is portrayed in light blue colour, while in light orange the case when there is a unique one. The system exhibits a topological bifurcation at $\sigma=\sigma^*$.
• Figure 2: In (a), the minimal invariant set contained in $\mathbb{R}_-:=\{x\leq 0\}$ from Figure \ref{['fig:bifMIS']} is depicted. Here $b=3$ was considered. The extremal maps are plotted for noise strengths $\sigma^*/2$ and $\sigma^*$, where $\sigma^*$ is the bifurcation value \ref{['eq:sigma*']}. In (b), their corresponding stationary densities $\phi(x)$ are presented. The stationary density becomes flatter at the right boundary as $\sigma\rightarrow\sigma^*$.
• Figure 3: A numerical validation of the main results for the system \ref{['eq:toy_example']}. In (a), the value $\sigma=\sigma^*/2$ was taken, and thus the boundary $x_+$ is hyperbolic. The stationary density is plotted (in blue) in a double logarithmic scale against $x_+-x$, and compared with $\ln\vert c_1\vert+2\ln\ln\left(\frac{1}{x_+-x} \right)$ (in red). In (b), the noise strength taken was $\sigma=\sigma^*$, so that $x_+$ is nonhyperbolic with $r=2$. The function $\ln\ln\left( \frac{1}{\phi(x)}\right)$ is plotted against $\ln\left( \frac{1}{x_+-x}\right)$ (in green), and compared to the function $\ln c_2 +u + \ln u$ (in red) in the corresponding logarithmic scale.
• Figure 4: In (a) and (b), a graphical summary of the integral estimation of the sums \ref{['eq:lower_1']} and \ref{['eq:lower_3']} are respectively sketched.

Theorems & Definitions (23)

• Theorem A: Asymptotic scaling of $\phi$ near a hyperbolic boundary
• Theorem B: Asymptotic scaling of $\phi$ near a nonhyperbolic boundary
• Proposition 2.1
• proof
• Lemma 3.1
• Proposition 3.2
• proof
• proof : Proof of Lemma \ref{['LEMMAcrucial']}
• Lemma 4.1: Scaling law for $n_{x_0}^x$ near a hyperbolic point
• proof