Below is a paragraph from an academic paper.
Polish the writing to meet the academic style, improve the spelling, grammar, clarity, concisio and overall readability. When necessary, rewrite the whole sentence. Furthermore, list all modifigstign and explain the reasons to do so in markdown table.
“In order to establish the existence of invariant measures of equation (2.1), we need to prove
that for any ε > 0, there exists R > 0, for any T ≥ 1 we haveP(ku(t,u0)k2,ˆ ρ> R) ≤ ε.
For this aim, we have the following results.”