Question: What is the most appropriate way to combine two diffusion processes in a way that "averages" their behavior?
Details
A diffusion process can be parameterized by two functions $(\mu,\sigma)$. This corresponds to an Ito process: $$ dX_t = \mu(X_t)dt + \sigma(X_t)dW_t $$ as well as a Fokker-Planck equation: $$ \partial_t p(x,t) = -\sum_i\partial_i[\mu(x) p(x,t)] +\frac{1}{2}\sum_i\sum_j \partial_{ij}[D_{ij}(x) p(x,t)]$$ where the diffusion tensor is $D(x)=\sigma(x)\,\sigma(x)^T$, which also appears in the equation for the infinitesimal generator. Furthermore, it is common to consider $D$ as the contravariant (inverse) metric tensor of a Riemannian manifold, i.e. $D=g^{-1}$.
Ignoring $\mu$, notice there are three reasonable ways to define the variance of the process: $\sigma$, $D$, or $g$.
My question is this: I have two diffusion processes, one with $\sigma$, $D$, and $g$, and another with $\hat{\sigma}$, $\hat{D}$, and $\hat{g}$. I want to create a new process with behaviour similar to the average between these two processes. So then my question is which variance measure should I combine?
I.e. which should I take: (1) $\tilde{\sigma}=\sigma+\hat{\sigma}$, (2) $ \tilde{g}=g+\hat{g}$, or (3) $\tilde{D}=D+\hat{D}$?
The application is not a physical one (in the experimental sense), but I would be happy with a physical reason. Indeed, I think any of the three would produce some kind of combined/average process, but it's not clear which one is the most sensible or natural.
My initial thoughts: my most immediate thought is (1), because it is the most "direct" to the process. However, if we consider two metrics, it is sensible to add them (i.e. the sum of two metric tensors is still a metric), so perhaps (2) is reasonable as well.
Diffusion process is linear, so as an option one can consider the arithmetic mean of random variables: $$ \tilde{X_t}=\frac{X_t+\hat{X_t}}2. $$ For independent processes, $n=1$ and constant diffusion coefficients it corresponds to $$\tilde{D}=\frac{D+\hat{D}}4.$$