A computable Gaussian approximation for model uncertainty
\[
\frac{\mathrm{d}z_t}{\mathrm{d}t} = u\left(z_t, t\right), \quad z_0 = x.
\]
\[
\frac{\mathrm{d}F_0^t(x)}{\mathrm{d}t} = u\left(F_0^t(x), t\right), \qquad F_0^t(x) = x.
\]
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\[
\frac{\mathrm{d}y_t}{\mathrm{d}t} = u\left(y_t, t\right) + \varepsilon \sigma\left(y_t, t\right){\,}^{``}\xi_t{\!}^{"}, \quad y_0 = x.
\]
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\[
\frac{\mathrm{d}y_t}{\mathrm{d}t} = u\left(y_t, t\right) + \varepsilon \sigma\left(y_t, t\right){\,}^{``}\xi_t{\!}^{"}, \quad y_0 = x.
\]
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\[
y_t \sim \,???\,
\]
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\[
y_t \sim \,???\,
\]
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\[
y_t \,\dot{\sim}\, \mathcal{N}\left(F_0^t(x), \varepsilon^2\Sigma\left(x,t\right)\right) \\
\]
\[
\Sigma\left(x,t\right) = \int_0^t{L\left(t, \tau\right)L\left(t, \tau\right)^{T}\mathrm{d}\tau}
\] \[
L(t, \tau) = \left[\nabla F_0^t(x)\right]^{-1} \nabla F_0^\tau(x)\,\sigma\left(F_0^\tau(x), \tau\right)
\]
\[
y_t \,\dot{\sim}\, \mathcal{N}\left(F_0^t(x), \varepsilon^2\Sigma\left(x,t\right)\right) \\
\]
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We are given the SDE \[
\mathrm{d}y_t = u\left(y_t, t\right)\mathrm{d}t + \varepsilon\sigma\left(y_t, t\right)\mathrm{d}W_t
\] subject to \(y_0 = x\). Define \[
z_t^{(\varepsilon)} \coloneqq \frac{y_t - F_0^t(x)}{\varepsilon},
\] and consider the Gaussian process \[
z_t \sim \mathcal{N}\left(0, \Sigma(x,t)\right).
\] Then, for any \(r \geq 1\) there exists a constant \(D_r(x,t)\) constructed from the flow map data and \(\sigma\) such that \[
\mathbb{E}\left[\left|\!\left| z_t^{(\varepsilon)} - z_t\right|\!\right|^r\right] \leq D_r(x,t)\varepsilon^r.
\] Taking \(\epsilon \to 0\) gives us that \(z_t^{(\varepsilon)}\) converges in \(r\)th mean to \(z_t\). Then, for small \(\varepsilon\) \[
y_t = \varepsilon z_t^{(\varepsilon)} + F_0^t(x) \approx \varepsilon z_t + F_0^t(x).
\]
