Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure.

Write $T_{\neq 0}M \subset TM$ for the non-zero tangent bundle of $M$. For any $C^\infty$ vector field $b \colon M \to TM$ on $M$, define the vector field $b' \colon T_{\neq 0}M \to T(T_{\neq 0}M)$ on $T_{\neq 0}M$ by $$ \hspace{40mm} b'(v) \ = \ \kappa(\mathrm{d}b_x(v)) \hspace{8mm} \forall \, v \in T_xM \!\setminus\! \{0_{(x)}\} \ \ \ \forall \, x \in M$$ where $\mathrm{d}b_x \colon T_xM \to T_{b(x)}TM$ is the derivative of $b$ at $x$ and $\kappa \colon TTM \to TTM$ is the canonical flip.

SDE and associated stochastic flow:

Consider a Stratonovich stochastic differential equation on $M$ of the form $$\tag{1} \label{sde} dX_t \ = \ b(X_t) \, dt \; + \, \sum_{i=1}^n \sigma_i(X_t) \circ dW_t^i $$ where (for some $n \geq 1$) $b,\sigma_1,\ldots,\sigma_n$ are $C^\infty$ vector fields and $\mathbf{W}:=((W_t^1,\ldots,W_t^n))_{t \in [0,\infty)}$ is an $n$-dimensional standard Brownian motion. Define an associated stochastic flow $$ ( \, \phi_{\mathbf{W},t}(x) \, : \, t \geq 0, \, x \in M, \, \mathbf{W} \!\in C_0([0,\infty),\mathbb{R}^n)) $$ such that for all $x \in M$,

  • $\phi_{\mathbf{W},0}(x)=x$;
  • given an $n$-dimensional standard Brownian motion $\mathbf{W}$, the stochastic process $X_t:=(\phi_{\mathbf{W},t}(x))_{t \geq 0}$ is a solution of \eqref{sde}.

Note that for any $x \in M$ and $v \in T_xM \!\setminus\!\{0_{(x)}\}$, the stochastic process $V_t := (\mathrm{d}\phi_{\mathbf{W},t})_x(v)$ is a solution of the SDE $$ dV_t \ = \ b'(V_t) \, dt \; + \, \sum_{i=1}^n \sigma_i'(V_t) \circ dW_t^i $$ on $T_{\neq 0}M$.

Very noisy SDE:

Definition. We say that \eqref{sde} is very noisy if the following two statements hold:

  1. For all distinct $x,y \in M$ there exists $t > 0$ such that, taking $\mathbf{W}$ to be an $n$-dimensional standard Brownian motion, the event $$ \{ \, d(\phi_{\mathbf{W},t}(x),\phi_{\mathbf{W},t}(y)) < d(x,y) \, \} $$ has positive probability.
  2. Letting $L$ be the Lie algebra generated by the vector fields $\sigma_1',\ldots,\sigma_n'$, we have that for all $v \in T_{\neq 0}M$, $$ \{l(v):l \in L\} = T_vTM. $$

Vanishing maximal Lyapunov exponent:

Definition. We say that \eqref{sde} has vanishing maximal Lyapunov exponent if for $\lambda$-almost all $x \in M$, taking $\mathbf{W}$ to be an $n$-dimensional standard Brownian motion, we have $$ \frac{\log\|(\mathrm{d}\phi_{\mathbf{W},t})_x\|}{t} \,\to\, 0 \ \ \textrm{ as } t \to \infty $$ almost surely.

Asymptotic stability in probability:

Definition. We say that \eqref{sde} is asymptotically stable in probability if for all $\varepsilon>0$ there exists $T>0$ such that for every $t>T$, taking $\mathbf{W}$ to be an $n$-dimensional standard Brownian motion, the event $$ \{\exists \, A \in \mathcal{B}(M) \textrm{ with } \lambda(M \setminus A)<\varepsilon \textrm{ and } \mathrm{diam}(\phi_{\mathbf{W},t}(A)) < \varepsilon \} $$ has probability greater than $1-\varepsilon$.

Now one of the results in the paper [1] is a surprising result (with a very sophisticated proof) that can essentially be formulated as follows:

Theorem. Suppose \eqref{sde} is very noisy and has vanishing maximal Lyapunov exponent. Then \eqref{sde} is asymptotically stable in probability.

Question. Are there any known examples of a stochastic differential equation \eqref{sde} that is very noisy and has vanishing maximal Lyapunov exponent?

(I think it is worth noting that, should it turn out that being very noisy and having vanishing maximal Lyapunov exponent are mutually exclusive, that would not necessarily devalue Baxendale's result: it could be that Baxendale's result is the first step in proving this exclusivity, or that the main ideas in the proof of Baxendale's result can be used in proving this exclusivity.)

[1] Peter H. Baxendale. “Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms”. Spatial stochastic processes. Vol. 19. Progr. Probab. Birkhäuser Boston, Boston, MA, 1991, pp. 189–218


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