Limits of random differential equations on manifolds

Abstract

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form \(\sum _kY_k\alpha _k(z_t^\epsilon (\omega ))\) where \(Y_k\) are vector fields, \(\epsilon \) is a positive number, \(z_t^\epsilon \) is a \({1\over \epsilon } {\mathcal {L}}_0\) diffusion process taking values in possibly a different manifold, \(\alpha _k\) are annihilators of \(\mathrm{ker}({\mathcal {L}}_0^*)\). Under Hörmander type conditions on \({\mathcal {L}}_0\) we prove that, as \(\epsilon \) approaches zero, the stochastic processes \(y_{t\over \epsilon }^\epsilon \) converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.

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Appendix

Appendix

We began with the proof of Lemma 3.1, follow it with a discussion on conditional inequalities without assuming conditions on the \(\sigma \)-algebra concerned.

Proof of Lemma 3.1

Step 1. Denote \(\psi (t)=ae^{-\delta t}\). Firstly, if \(f \in {\mathcal {B}}_b(G;{\mathbb {R}})\) and \(z\in G\),

$$\begin{aligned} |Q_tf(z)-\pi f| \le \Vert f\Vert _W \cdot \psi (t)\cdot W(z). \end{aligned}$$

Next, by the Markov property of \((z_t)\) and the assumption that \(\int g d\pi =0\):

$$\begin{aligned}&\left| {\mathbf E}\{f(z_{s_2}) g(z_{s_1})|{\mathcal {F}}_{s}\} -\int _G fQ_{s_1-s_2}g d\pi \right| \\&\quad =\left| {\mathbf E}\left\{ (fQ_{s_1-s_2} g)(z_{s_2})\Big | {\mathcal {F}}_s\right\} -\int _G fQ_{s_1-s_2}g d\pi \right| \\&\quad \le \psi (s_2\!-\!s) \;\Vert fQ_{s_1-s_2}g\Vert _W \;W(z_s) \le \psi (s_2-s) \sup _{z\in G}\left( { |f(z)| |Q_{s_1-s_2}g(z) |\over W(z)}\right) W(z_s) \\&\quad \le \psi (s_2-s)\psi (s_1-s_2) |f|_\infty \, \Vert g\Vert _W W(z_s) \le a\psi (s_1-s) |f|_\infty \Vert g\Vert _W W(z_s) . \end{aligned}$$

From this we see that,

$$\begin{aligned}&\left| {1\over t-s} \int _s^{t} \int _s^{s_1} \left( {\mathbf E}\left\{ f(z_{s_2}) g(z_{s_1}) \Big | {\mathcal {F}}_s\right\} -\int _G fQ_{s_1-s_2}g d\pi \right) ds_2ds_1\right| \\&\quad \le a |f|_\infty \, \Vert g\Vert _W W(z_s) {1\over t-s} \int _s^{t} \int _s^{s_1} \psi \left( s_1-s\right) \; ds_2 \; ds_1\\&\quad \le {a^2\over \delta ^2 (t-s)}|f|_\infty \, \Vert g\Vert _W W(z_s)\int _0^{(t-s)\delta } re^{-r} \; dr \le {a^2\over \delta ^2 (t-s)}|f|_\infty \, \Vert g\Vert _W W(z_s). \end{aligned}$$

This concludes (1). Step 2. For (2), we compute the following:

$$\begin{aligned}&{1\over t-s} \int _s^{t} \int _s^{s_1}\int _G fQ_{s_1-s_2}g \; d\pi \; ds_2 \; ds_1 =\int _G {1\over t-s} \int _0^{t-s}fQ_rg(t-s-r) \; dr d \pi \\&\quad =\int _G \int _0^\infty \! \left( f Q_r g\right) \; dr\; d\pi -\int _G \int _{t-s}^\infty f Q_r g \; dr \; d\pi {-}{1\over t-s}\int _G \int _0^{t-s}\! r f Q_r g \; dr d \pi . \end{aligned}$$

We estimate the last two terms. Firstly,

$$\begin{aligned}&\left| \int _G \int _{t-s}^\infty f (z)Q_r g(z) \; dr \; d\pi (z)\right| \le |f|_\infty \left| \int _G \int _{t-s}^\infty |Q_rg(z)| \; dr \; d\pi (z)\right| _\infty \\&\quad \le |f|_\infty \Vert g\Vert _W \int _G W(z) \pi (dz) \int _{t-s}^\infty \psi (r)dr \le {1\over \delta } |f|_\infty \Vert g\Vert _W \bar{W}\int _{(t-s)\delta }^\infty ae^{-r}dr\\&\quad \le {a\over \delta } |f|_\infty \Vert g\Vert _W \bar{W}. \end{aligned}$$

It remains to calculate the following:

$$\begin{aligned} \left| {1\over t-s} \int _G \int _0^{t-s} r f Q_r g \; dr d \pi \right|\le & {} {1\over t-s} |f|_\infty \Vert g\Vert _W \bar{W} \int _0^{t-s} r\psi (r)\; dr\\\le & {} {a\over (t-s)\delta ^2} |f|_\infty \Vert g\Vert _W \bar{W}. \end{aligned}$$

Gathering the estimates together we obtain the bound:

$$\begin{aligned}&\left| {1\over t-s} \int _s^{t} \int _s^{s_1}\int _G fQ_{s_1-s_2}g \; d\pi \; ds_2 \; ds_1 -\int _G \int _0^\infty \left( f Q_r g\right) \; dr\; d\pi \right| \\&\quad \le {a\over \delta }|f|_\infty \Vert g\Vert _W \bar{W}+ {a\over (t-s)\delta ^2} |f|_\infty \Vert g\Vert _W \;\bar{W}. \end{aligned}$$

By adding this estimate to that in part (1), we conclude part (2):

$$\begin{aligned}&\left| {1\over t-s} \int _s^{t} \int _s^{s_1} {\mathbf E}\left\{ f(z_{s_2}) g(z_{s_1}) \Big | {\mathcal {F}}_s\right\} -\int _G \int _0^\infty \left( f Q_r g\right) \; dr\; d\pi \right| \nonumber \\&\quad \le {a\over \delta }|f|_\infty \Vert g\Vert _W \bar{W}+ {a\over (t-s)\delta ^2} |f|_\infty \Vert g\Vert _W \;\bar{W} +{a^2\over \delta ^2 (t-s)}|f|_\infty \Vert g\Vert _W W(z_s).\nonumber \\ \end{aligned}$$
(9.1)

We conclude part (2). Step 3. We first assume that \(\bar{g} =0\), then,

$$\begin{aligned}&\left| {\epsilon \over t-s} \int _{s\over \epsilon }^{t\over \epsilon } \int _{s\over \epsilon }^{s_1} {\mathbf E}\left\{ f(z^\epsilon _{s_2}) g(z^\epsilon _{s_1}) \Big | {\mathcal {F}}_{s\over \epsilon }\right\} \; ds_2 \; ds_1\right| \\&\quad \le \left| {\epsilon \over t-s} \int _{s\over \epsilon }^{t\over \epsilon } \int _{s\over \epsilon }^{s_1} {\mathbf E}\left\{ f(z^\epsilon _{s_2}) g(z^\epsilon _{s_1}) \Big | {\mathcal {F}}_{s\over \epsilon }\right\} \; ds_2 \; ds_1 -\int _G \int _0^\infty f Q_r^\epsilon g \; dr \; d\pi \right| \\&\qquad +\left| \int _G \int _0^\infty f Q_r^\epsilon g \; dr \; d\pi \right| . \end{aligned}$$

We note that for every \(x\in G\), \(\Vert Q_r^\epsilon (x, \cdot )-\pi \Vert _{TV,W}\le \psi ({ r\over \epsilon }) W(x)\). In line (9.1) we replace s, t, \(\delta \) by \({s\over \epsilon }\), \({t\over \epsilon }\), and \( {\delta \over \epsilon }\) respectively to see the first term on the right hand side is bounded by

$$\begin{aligned} {a\epsilon ^3\over \delta ^2 (t-s)}( a W(z^\epsilon _{s\over \epsilon })+ \bar{W}) |f|_\infty \Vert g\Vert _W + {a\epsilon \over \delta } |f|_\infty \Vert g\Vert _W \bar{W}. \end{aligned}$$

Next we observe that

$$\begin{aligned} \int _0^\infty f(z) Q_s^\epsilon g(z) \; ds= & {} \int _0^\infty f (z)Q_{s\over \epsilon }(z) \; ds =\epsilon \int _0^\infty f (z)Q_s g(z) \; ds\\ \left| \int _G \int _0^\infty f(z) Q_s^\epsilon g(z) \; ds \; d\pi (z)\right|\le & {} \epsilon \,|f|_\infty \Vert g\Vert _W \bar{W} \int _0^\infty \psi (s) \; ds= {a\epsilon \over \delta }|f|_\infty \Vert g\Vert _W \bar{W} . \end{aligned}$$

This gives the estimate for the case of \(\bar{g}=0\):

$$\begin{aligned} \left| {\epsilon \over t-s} \int _{s\over \epsilon }^{t\over \epsilon } \int _{s\over \epsilon }^{s_1} {\mathbf E}\left\{ f(z^\epsilon _{s_2}) g(z^\epsilon _{s_1}) \Big | {\mathcal {F}}_{s\over \epsilon }\right\} \; ds_2 \; ds_1\right| \le C_1(z_{s\over \epsilon }^\epsilon ) {\epsilon ^3 \over t-s}+C_2'(z_{s\over \epsilon }^\epsilon )\epsilon . \end{aligned}$$

where

$$\begin{aligned} C_1={a\over \delta ^2} (aW(\cdot )+\bar{W})|f|_\infty \Vert g\Vert _W, \quad C_2'={2a\over \delta }|f|_\infty \Vert g\Vert _W\bar{W}. \end{aligned}$$

If \(\int g \; d\pi \not =0\), we split \(g=g-\bar{g}+\bar{g}\) and estimate the remaining term. We use the fact that \(\pi f=0\),

$$\begin{aligned}&\left| {\epsilon \over t-s} \int _{s\over \epsilon }^{t\over \epsilon } \int _{s\over \epsilon }^{s_1} {\mathbf E}\left\{ f(z^\epsilon _{s_2}) \bar{g}\big | {\mathcal {F}}_{s\over \epsilon }\right\} \; ds_2 \; ds_1\right| \\&\quad \le |\bar{g}| \left| {\epsilon \over t-s} \int _0^{t-s\over \epsilon } \int _0^{s_1} \left| Q^\epsilon _{s_2} fSpa(z_{s\over \epsilon }) \right| \; ds_2 \; ds_1\right| \\&\quad \le |\bar{g}\Vert |f\Vert _W W(z_{s\over \epsilon }^\epsilon ) \sup _{s_1>0} \left\{ \left| \int _0^{s_1} \psi \left( {s_2\over \epsilon }\right) ds_2\right| \right\} \le |\bar{g}| \; \Vert |f\Vert _W W(z_{s\over \epsilon }^\epsilon )\epsilon \int _0^\infty \psi (r)dr\\&\quad ={a\epsilon \over \delta } |\bar{g}| \; \Vert f\Vert _W W(z_{s\over \epsilon }^\epsilon ). \end{aligned}$$

Finally we obtain the required estimate in part (3):

$$\begin{aligned}&\left| {\epsilon \over t-s} \int _{s\over \epsilon }^{t\over \epsilon } \int _{s\over \epsilon }^{s_1} {\mathbf E}\left\{ f(z^\epsilon _{s_2}) g(z^\epsilon _{s_1}) \Big | {\mathcal {F}}_{s\over \epsilon }\right\} \; ds_2 \; ds_1\right| \\&\quad \le C_1(z_{s\over \epsilon }^\epsilon ) \left( {\epsilon ^3\over t-s}\right) +C_2'(z_{s\over \epsilon }^\epsilon )\epsilon + \epsilon {a \over \delta }|\bar{g}| \; \Vert f\Vert _W W(z_{s\over \epsilon }^\epsilon ), \end{aligned}$$

thus concluding part (3).

The following conditional inequalities are elementary. We include a proof for a partial conditional Burkholder–Davis–Gundy inequality for completeness. We do not assume the existence of regular conditional probabilities.

Lemma 10.1

Let \((M_t)\) be a continuous \(L^2\) martingale vanishing at 0. Let \((H_t)\) be an adapted stochastic process with left continuous sample paths and right limits. If for stopping times \(s<t\), \({\mathbf E}\int _s^t(H_r)^2 d\langle M\rangle _r<\infty \). Then

$$\begin{aligned} {\mathbf E}\left\{ \left( \int _s^t H_r dM_r\right) ^2\Big |F_s\right\} ={\mathbf E}\left\{ \int _s^t (H_r)^2 d\langle M\rangle _r\Big |F_s\right\} . \end{aligned}$$

Lemma 10.2

Let \(p>1\) and \((M_t)\) is a right continuous \(({\mathcal {F}}_t)\) martingale or a right continuous positive sub-martingale index by an interval I of \({\mathbb {R}}_+\). Then,

$$\begin{aligned} {\mathbf E}\left\{ \sup _{s\in I} |M_t|^p \Big | \, {\mathcal {F}}_s\right\} \le \left( {p\over p-1}\right) ^p \sup _{s\in I}{\mathbf E}\left\{ |M_s|^p\Big | \, {\mathcal {F}}_s\right\} . \end{aligned}$$

If \((M_u, s\le u \le t)\) is a right continuous \(({\mathcal {F}}_t)\) martingale and \(p\ge 2\), there exists a constant \(c(p)>0\) s.t.

$$\begin{aligned} {\mathbf E}\left\{ \sup _{s\le u \le t} |M_u|^p\Big |F_s\right\} \le c_p {\mathbf E}\left\{ \langle M\rangle _t^{p\over 2} \Big |F_s\right\} . \end{aligned}$$

This proof is the same as the proof for \({\mathcal {F}}_s\) the trivial \(\sigma \)-algebra, c.f. Revuz and Yor [38].

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Li, XM. Limits of random differential equations on manifolds. Probab. Theory Relat. Fields 166, 659–712 (2016). https://doi.org/10.1007/s00440-015-0669-x

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  • 60D