Super-Brownian Local Time: a Representation and Two Applications

Tanaka formula and local time for a class of interacting branching measure-valued diffusions

“…Nevertheless, Theorem 2.4 of Section 2 (our main result) establishes the existence of the local time Λ x t for SDSM directly through the characterization provided by (2.15), an explicit Tanaka formula expressed through a Green function with a singularity at the origin, in the spirit of the approach proposed in López-Mimbela and Villa [25] in their Theorem 3.1, of which our Theorem 2.4 is an extension. However, in order to make sense of it, we have to approximate this singular Green function and its derivatives by smooth functions to ensure that the various stochastic integrals in (2.15) are well-defined.…”

“…This was the method applied by Adler and Lewin [1] in their proof of the Tanaka formula for the local time of Super-Brownian motion and super stable processes. This also occurs when trying the approach proposed in López-Mimbela and Villa [25] for Super-Brownian motion, where an alternative representation of the local time simplifies the proof of its joint continuity by taking advantage of sharp estimates for the Green function of Brownian motion and its associated Tanaka formula. However, the higher order singularity of the Green function and its derivative in our case, raises some new technical difficulties in the moment estimation of the interacting term, as well as in the handling of a stochastic convolution integral term appearing in the corresponding Tanaka formula.…”

Joint Hölder continuity of local time for a class of interacting branching measure valued diffusions

“…The existence and the joint space-time continuity of paths for the local time of Super-Brownian motion, when d ≤ 3, were first obtained by Iscoe [14] and Sugitani [27]. Their results were variously sharpened and generalized, first by Adler and Lewin [1] for super stable processes and Krone [16] for superdiffusions; then by many others, most notably Ethier and Krone [9] for some related Fleming-Viot processes with diffusive mutations; López-Mimbela and Villa [21], who streamlined and unified the various definitions of the local time and clarified their interrelations in the above cases; Li and Xiong [19], who offered an alternative (trajectorial) definition of the local time when the superprocess is degenerate, that is, a purely atomic measure valued process, and proved its joint Hölder continuity, as well as scaling limit theorems, using a representation in terms of stochastic integrals with respect to the excursions of an underlying Poisson random measure.…”

Local time and Tanaka formula for a multitype Dawson–Watanabe superprocess

“…Following Dynkin's definition we prove existence of local time for dimensions d < 2α, where α ∈ (0, 2] denotes the smallest of the stable motion exponents. We also derive a representation of the local time which is a multitype analogue of Tanaka formula-like representations obtained in [1] and [6] for super-Brownian local time. This representation is used to show that Dynkin's notion of superprocess local time, when it makes sense, coincides with the occupation density of the multitype superprocess.…”

“…Dynkin's approach to local time is based on representations of functionals of a continuous superprocess by multiple stochastic integrals. In the case of super-Brownian motion, both notions L t and L t are equivalent [6].…”