Perhaps Robert Browning read Dante Rossetti?
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rpo.library.utoronto.ca]
Nah, likely just coincidence.
Did 'chase' use to be 'chace'? Not sure, could be a typo, but the same meaning so no matter. Sounds to me like the speaker (not necessarily Browning) is writing one of those all-too-frequent angst poems to a lover (not nec. Elizabeth) who has spurned him. Stalking her, you say? Yeah, could be that too.
Note, for example, the lines:
Me the loving and you the loth
While the one eludes, must the other pursue.
Loth is similar to loathe, but not exactly the same. I am loth to eat grits is to say I am normally unwilling. If I loathe grits, I hate them.
The rhyme scheme is interesting, abba-type stanzas, with the first three and last three lines virtual mirror images.