New Feynman-Kac type formula

Variation of iterated Brownian motion

“…Suppose for a moment that u k i −1 and u k i are such that A 2 holds, for example, the intervals corresponding to i = 1, 2 are in (a, ∞) and the other two are contained in the other half-line. The same argument that leads to (8) gives in the present case…”

“…If the event A 2 occurs then the interval with endpoints Y (t i−1 ) and Y (t i ) is disjoint from the interval with endpoints Y (t j−1 ) and Y (t j ). Hence we may integrate over suitable u i−1 , u i , u j−1 and u j in (8) to obtain…”

Some Path Properties of Iterated Brownian Motion

“…Funaki (1979) proved that a similar process is related to "squared Laplacian." Krylov (1960) and Hochberg (1978) considered finitely additive signed measures on the path space corresponding to squared Laplacian (there exists a genuine probabilistic approach, see, e.g., Madrecki and Rybaczuk (1992)). A paper of Vervaat (1985) contains a section on the composition of self-similar processes.…”

Itô formula for an asymptotically $4$-stable process

“…The research presented in Madrecki and Rybaczuk (1993) was motivated by several concrete examples. Here is a review of two of those examples.…”

“…The relativistic kinetic energy E is given by See Madrecki and Rybaczuk (1993) for more details and bibliography. This note is a part of a larger project which aims at developing a usable stochastic calculus which may be successfully applied to equations involving the squared Laplacian in the same way the classical stochastic calculus is applied to equations involving classical Laplacian.…”