I am new in the field of stochastic differential equations and I came across the following problem. We are given a particle system with
$$ X_t^i = X_0 + B_t^i $$
Now I have to calculate the expectation value of
$ \mathbb{E}[\phi(x + B_0(t)] $ for $\phi$ bounded continous. The author states that this equals $ e^{(\frac{1}{2}\Delta t)}\phi)(x)$. I am an absolute beginner so I don´t even know where to look for. Good references are also very welcome. What I found is, that for the Ornstein-Uhlenbeck process, given by
$$ dX_t = - \lambda X_t dt + \sigma dB_t \quad \quad X_0=x_0 $$
We have $\mathbb{E}[X_t] = e^{-\lambda t}x_0 $. If I apply this result to my case i.e. $\lambda = 0$ I get
$ \mathbb{E}[X_0+ B_0] = x_0 $ but that is obviously wrong. Any thoughts?