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Register a new userBrownian sheet and time inversion -- From $G$-orbit to $L(G)$-orbit
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Authors
Manon Defosseux
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We have proved in a previous paper that a space-time Brownian motion conditioned to remain in a Weyl chamber associated to an affine Kac-Moody Lie algebra is distributed as the radial part process of a Brownian sheet on the compact real form of the underlying finite dimensional Lie algebra, the radial part being defined considering the coadjoint action of a loop group on the dual of a centrally extended loop algebra. We present here a very brief proof of this result based on a time inversion argument and on elementary stochastic differential calculus.
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In this paper, we obtain L{e}vys martingale characterization of $G$-Brownian motion without the nondegenerate condition. Base on this characterization, we prove the reflection principle of $G$-Brownian motion. Furthermore, we use Krylovs estimate to get the reflection principle of $tilde{G}$-Brownian motion.
We introduce $n$-parameter $Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each time parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of $n$ linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) connection to fourth order linear PDEs. Our BTBS does not belong to the classical theory of random fields; and to prove our new PDEs connections, we generalize our BTP approach in cite{Abtp1,Abtp2} and we mix it with the Brownian sheet connection to a linear PDE system, which we also give along with its corresponding nonlinear second order PDE and $2n$-th order linear PDE. In addition, we introduce the $n$-parameter $d$-dimensional linear Kuramoto-Sivashinsky (KS) sheet kernel (or transition density); and we link it to an intimately connected system of new linear Kuramoto-Sivashinsky-variant interacting PDEs, generalizing our earlier one parameter imaginary-Brownian-time-Brownian-angle kernel and its connection to the KS PDE. The interactions here mean that our PDEs systems are to be solved for a family of functions, a feature shared with well known fluids dynamics models. The interacting PDEs connections established here open up another new fundamental front in the rapidly growing field of iterated-type processes and their connections to both new and important higher order PDEs and to some equivalent fractional Cauchy problems. We connect the BTBS fourth order interacting PDEs system given here with an interacting fractional PDE system and further study it in another article.
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