Persiguiendo means "chasing" or "pursuing." While "chasing" might seem common, its use here is particularly evocative and less literal than one might expect.
In the song, Diego Torres sings, "Los días pasan persiguiendo mi soledad" (The days pass chasing my loneliness). This personification of days actively pursuing loneliness creates a vivid and somewhat melancholic image, making the word stand out and inviting listeners to explore the deeper meaning of the lyrics.
The song keeps returning to one hour of the day. Cae la noche y yo vuelvo a estar (Night falls and I am once again), and then the same three words every time: Sólo otra vez (Alone once again). What comes back with the dark is a smell before it is a memory — Tu perfume en mi almohada (Your perfume on my pillow), still in the bed of someone who left long enough ago that Tanto tiempo ha pasado (So much time has passed).
Torres never asks her to come back. He describes what the waiting does to the daylight instead: Los días pasan / Persiguiendo mi soledad (The days go by / Chasing my loneliness), and a mind that circles without arriving — Dando vueltas como en sueños (Wandering in circles like in dreams). The one question he does put, Cuanto mas / Yo me pregunto (How much longer / I wonder), gets the flattest answer available: No hay respuestas para dar (There are no answers to give). The chorus adds no argument at all, just two gerunds left hanging — Esperándote (Waiting for you), Recordándote (Remembering you).