It is BM, indeed. Maybe it is better to try to describe the problem (already discussed "ODE asymptotic .. thread (Aluffi, Gemam et al))

Recall

[$]d \vec{x}_t = -\nabla f(\vec{x_t}) dt + \sqrt{(2T)} d \vec{W}_t[$] (1)

When there is no random fluctuation I can solve constrained optimisation as a gradient system. Now I want to do hlll-climbing (basin hopping) by solving (1).

My issues are

A) Solving (1) as a SDE is not ideal (?) we lose out on ODE solver

B) What I want is to use Strang splitting into deterministiic and random legs, and which form the latter leg should be ie really the trigger for my question. So, what would be the 'best' approach for the stochastic part?

Here is idea for stochastic Burgers

https://arxiv.org/pdf/1907.12747.pdf