I am trying to find a reference of a proof of a continuous time version of a result of Dvoretzky and Erdos from their paper "Some problems on random walk in space" that says the probability $\gamma_2(n)$ that a two dimensional random walk does not return to the origin after first leaving it up until time n is asymptotically $$\gamma_2(n)=\frac{\pi}{\log n}+O\left(\frac{\log\log n}{\log^2 n}\right).$$

Explicitly, I am looking to prove a result of the following kind. Let $\tilde{S}_t$ be a continuous time random walk on the two dimensional integer lattice then $$\mathbb{P}[\tilde{S}_s\neq 0 \text{ for }0<s\leq t]\approx\frac{\pi}{\log t}\text{ as }t\to\infty.$$ I feel the proof should be an easy adaption with integrals in place of sums etc. but it is alluding.

Thanks in advance for any help.

exactly$0$. It will return arbitrarily close to $0$ arbitrarily often, but you have to add an allowable distance to $0$ in the mix to phrase your problem correctly. $\endgroup$5more comments