There is no formal gerund in Russian, but English expressions can often be translated closely to original mood by using different parts of speech, e.g.:
running away (as in 'while running away, he didn't look back') - убегая (деепричастие)
running away (as in 'his running away was unexpected') - бегство (существительное)
running - бег (существительное)
to keep on running - продолжать бежать/бег
Verb-to-noun and adv. participle forms mainly exist with some prefixed versions of the verb (убегая/убегание, избегая/избежание/избегание, забегая/забегание, etc.). The form you asked of doesn't exist (if there were such a word, it would sound like "бежа"), but still it can be translated as "на бегу".
P.S. There's also a related imperfective verb with no prefix: бегать, which can mean to run around, to run continuously or from time to time. This one gives the corresponding adverbial participle бегая with the sense '(while) running around'.