Rewrite of SDE tutorial

[ArnoStrouwen]

Rewrite of MTK documentation continues with SDE modelling.

What this tutorial is still lacking, is an exact mapping of the MTK @brownian concept to the LaTeX equation.

I'm not at all an expert in SDEs, but @brownian does not seem to map directly to what we call Brownian motion but more something on the level of dB/dt.

[ChrisRackauckas]

I'm not at all an expert in SDEs, but @brownian does not seem to map directly to what we call Brownian motion but more something on the level of dB/dt.

They are Brownian increments, yes, so variables which when integrated are Brownian motions. Not referred to as dB/dt, instead just dB since it's only defined in the integral sense.

[ArnoStrouwen]

They are Brownian increments, yes, so variables which when integrated are Brownian motions. Not referred to as dB/dt, instead just dB since it's only defined in the integral sense.

I understand that this is only mathematically rigorously defined in the integral sense.
However, in the code, we do have: D = Differential(t); D(x), which in the Symbolics docs is defined as differential w.r.t time.

I need to figure out a way to succinctly express that even though we work in the integral sense, the dt present in the deterministic terms is omitted (when no @brownian is present), while the dW is implied from @brownian

[ChrisRackauckas]

The formulation here is effectively as a Langevin equation https://en.wikipedia.org/wiki/Langevin_equation

[ArnoStrouwen]

Can be merged when tests pass.

[ArnoStrouwen]

Errors seem to be in another tutorial:
https://github.com/SciML/ModelingToolkit.jl/actions/runs/11306893338/job/31448000464?pr=3084#step:6:2450