I would like to have an Itô Diffusion $$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$ where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower regularity than the usual requirements that they be Lipschitz continuous.

**My question is: Can such a process be well-defined?**

Let me briefly explain how this came up: For the Brownian motion on a Riemannian manifold, locally the functions $U$ and $\sigma$ should locally be given by $$\sigma_{ij} = g^{ij}, \quad b^k = \sum_{ij}g^{ij}\Gamma_{ij}^k,$$ $g^{ij}$ being the coefficients of the (inverse) metric and $\Gamma_{ij}^k$ the Christoffel symbols of the Levi-Civita-connection, which contain derivatives of $g^{ij}$.

Now, for example, I would like consider Brownian motion on the suspension of a circle (http://upload.wikimedia.org/wikipedia/commons/c/c3/Suspension.svg) embedded in $\mathbb{R}^3$.

This is not a Riemannian manifold in the usual sense, as the metric is not smooth: In case of the suspension, if you give it the smooth structure obtained by projecting to the $2$-sphere, but take the metric induced from $\mathbb{R}^3$, this metric will not be smooth: at the fold in the middle, the coefficient funcions will be only Lipschitz, so the Christoffel symbols will not be continuous there (even though bounded). Near the cusps at the top and bottom, the metric is not even bounded.

What to do here?