Between resets the process is a Brownian motion with drift −c; at the epochs of an independent Poisson process of rate λ it is sent back to the reset point xR, so Xt = xR + (Wt − ct) − (Wτ(t) − cτ(t)) after the last reset τ(t); before the first reset it starts at 0 and follows Wt − ct. Three things to watch. Search: resetting makes the time τ to reach a level u integrable. For xR < u, E τ = (eαu − 1) e−αxR/λ with α = √(c²+2λ) + c, and often there is a finite best rate λ*. With drift towards the target the optimum may be no resetting; when xR ≥ u, faster resets are always better and E τ = (1 − e−αu)/λ. Steady state: Xt converges to an asymmetric Laplace law with its peak at xR, and for the stationary version the pair (Yt, Yt+δ) splits into a no-reset part that remembers Yt and a reset part that forgets it. High levels: the law of sup[0,T] Xt is a renewal series, and its tail depends on the sign of xR: for xR ≤ 0 the cheapest way up is never to reset, P ∼ 2e−λTΨ((u+cT)/√T), while for xR > 0 it pays to reset exactly once, immediately, P ∼ 4λT²e−λTu−2Ψ((u+cT−xR)/√T); for the stationary process with c = 0 the tail is exponential, eαxR[Φ(α√T) + λ∫0TΦ(α√s)ds] e−αu. From Dębicki, Hashorva & Michna, Distributional and extremal behaviour of Brownian motion with exponential resetting.