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  "body": "\n## Summary\n\nA random ordinary differential equation is an [[ordinary differential equation|0NT64JV96A]] whose right side depends on a [[stochastic process|0QH5ENKZM1]] with regular sample paths.\nFor each fixed outcome of the randomness it is an ordinary, non-autonomous differential equation, and it is solved path by path with classical calculus.\nSuch equations generate random dynamical systems, and many [[stochastic differential equations|B219G9SHJA]] can be turned into them by a change of variables.\n\n## Explanations\n\n### L1 human\n\nThink of a boat whose speed depends on a wind that changes in a random but smooth way.\nOnce the weather of a given day is known, the motion of the boat is an ordinary differential equation.\nA random ordinary differential equation collects all these equations, one for each possible weather.\nFor example, $x' = -x + \\eta_t$ with a random continuous signal $\\eta$ has, for each path of $\\eta$, the solution $x(t) = e^{-t} x(0) + \\int_0^t e^{-(t - s)} \\eta_s \\, ds$.\n",
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