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I'm considering a second-order Hamiltonian ODE and am trying to understand different approaches for introducing noise (and corresponding numerical integration techniques).
Given that the deterministic part is Hamiltonian, I've been looking at Langevin-dynamics as a base model, that is, adding noise to the momentum variables.
Ultimately I want to understand the difference between adding noise to the momentum variables and adding noise to the position variables.

Typically, I've seen numerical simulations of Langevin dynamics:

begin{align}
m_i ddot{x}_i = -frac{partial V(x(t))}{partial x_i} - gamma dot{x} + sqrt{2 gamma_i k_B tau}, xi_i(t)
tag{1}
end{align}

split the second order equation (1) into two first order equations represented as:

begin{align}
begin{cases}
dot{x}_i(t) = m_i^{-1} p_i(t) \
dot{p}_i(t) = -frac{partial V(x(t))}{partial x_i} - gamma_i m_i^{-1} p_i(t) + sqrt{2 gamma_i k_B tau}, xi_i(t)
tag{2}
end{cases}
end{align}

for $tau$ temperature, $gamma_i$ damping, $x$ position, $p$ generalized momentum, and $xi_i sim N(0,1)$.
Then the Euler-Maruyama scheme for (2) would give
begin{align}
begin{cases}
x^{(k+1)}_i = left( m_i^{-1} p^{(k)}_i right) Delta t \
p^{(k+1)}_i = left( -frac{partial V(x^{(k)})}{partial x_i} - gamma_i m_i^{-1} p^{(k)}_i right) Delta t + left( sqrt{2 gamma_i k_B tau},xi_i^{(k)} right) , sqrt{Delta t}.
tag{2.5}
end{cases}
end{align}

A standard diffusion process can be represented as

begin{align}
mathrm{d}X_t = b(X_t) + sigma(X_t), mathrm{d} W_t
tag{3}
end{align}

for $Xin mathbb{R}^d$, drift $b:mathbb{R}^d to mathbb{R}^d$ the drift, and diffusion $sigma: mathbb{R}^d to mathbb{R}^{d times d}$, and we can fit (2) into the form of (3) by taking appropriate $b$ and $sigma$.

My question is what happens when we add noise to the position $x$ variables instead of the $p$ variables?
If we describe it as a general diffusion as in (3), I imagine that we can just change $sigma$ to reflect the new noise, but should this different noise affect the new second order SDE's stationary distribution?
Is there any way to add noise to the position variables that recovers the same stationary distribution as Langevin dynamics in (2)?

Thanks!