SDEs are only diffusion and drift [efr-0020]

Naively, we would think that a stochastic differential equation on a manifold X should be a stochastic section of the tangent bundle, f: X \to \Delta (TX) (with the understanding that such a thing may not always have a solution).

Indeed, we will see that such a thing does lead to a differential equation. But unlike the deterministic case, there is not a 1-1 correspondence---different sections may determine the same differential equation (in the sense of having the same solutions).

The issue is that, as we have discussed, by necessity solutions of stochastic differential equations are defined by integration, in a certain sense. The solution concept we have in mind is a Markov process, where each derivative is sampled independently from its distribution. But by the central limit theorem, integrating a family of independent random variables will always give a normal distribution, no matter what those variables are.

In particular, the only aspect of the assigned distribution of derivatives f(x) \in \Delta (T_xX) is its mean Ef(x) \in T_xX and its covariance matrix, which is the expectation of v \otimes v inside the tensor product T_xX \otimes T_xX when v \sim f(x).

(Another way of viewing the covariance is as an operator on the tensor product of the covector space T_x^*X \otimes T_x^*X \to \mathbb {R}, which carries a simple tensor of two covectors \phi \otimes \psi to the expected value of \phi (v)\psi (v) when v is sampled according to the given distribution).