# probability distributions – Fokker-Planck Equation of an Euler Scheme

Consider the overdamped Langevin equation

$$dX_t= -nabla V(X_t) dt + dW_t$$

with associated FPE for its density $$rho_cdot(cdot):mathbb{R}times mathbb{R}^dto mathbb{R}$$

$$partial_t rho_t = text{div}(nabla rho_t+rho_tnabla V).$$

Now consider the Euler Maruyama approximation with step $$tau$$

$$X^tau_{n+1}=X^tau_{n}-nabla V(X^tau_{n})big((n+1)tau-ntaubig) + W_{(n+1)tau }-W_{ntau}$$

and its interpolation

$$X^tau_t=X^tau_n~~text{for}~tin (ntau,(n+1)tau).$$

My question : is there a FPE associated to the above process?