Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths

Abstract

In this work, we generalise the stochastic local time space integration introduced in [8] to the case of Brownian sheet. This allows us to prove a generalised two-parameter Itô formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.

Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients

“…The next three lemmas whose proofs are given in Section 5 provide an estimate for nkk (x, y) and nkk (0, x) for every dyadic numbers x, y ∈ Q. Lemmas 3.6 and 3.7 are counterparts of Lemmas 3.1 and 3.2 in [13] for the Brownian sheet. The proof of Lemma 3.6 uses the local time-space integration formula for the Brownian sheet as given in [5]. Lemma 3.8 follows from Lemma 3.7 using the fact that the set of dyadic numbers is dense in R.…”

“…In order to prove the auxiliary lemmas provided in the previous section, we need some preliminary results that have been obtained by applying a local time-space integration formula for Brownian sheets (see [5] for related results). Let us first recall the notion of local time in the plane of the Brownian sheet.…”