Limit laws for local times of the Brownian sheet (Q1823550)

Let B(s,t), \(s,t>0\) be a Brownian sheet. Then for all \(s>0\), the process \(B_ s(t):=B(s,t),\) \(t>0\) is a (scaled) Brownian motion which admits a local time \(L_ s(x;t)\), which is jointly continuous in \(x\in {\mathbb{R}}\) and s, \(t>0\). \(s^{1/2}L_ s\) is a standard Brownian local time. We prove that \[ \lim_{\lambda \to +\infty}-\lambda^{-2}\log P(\sup_{1<s<2}\sup_{x}s^{1/2}L\quad_ s(x;1)>\lambda)=1/2. \] This result has several corollaries, both for \(L_ s\) and the local time of B(s,t), most of them new. The new ingredient in the proof has applications to other questions concerning local times. In particular, we give a new proof of the well known large deviations result for Brownian maximum local time, \[ \lim_{\lambda \to +\infty}-\lambda^{-2}\log P(\sup_{x}L_ 1(x;1)>\lambda)=1/2. \] Previous proofs of this, unlike the present one, have relied on the techniques specific to Brownian motion.