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Register a new userFrom Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky-variant interacting PDEs systems
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Hassan Allouba
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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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Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a feature exhibited in well known fluids dynamics models. On the other hand, the memory-preserving interaction between the PDE solution and the initial data is common to both the single and the multi parameter Brownian-time PDEs. Here, we introduce a new---even in the one parameter case---proof that judiciously combines stochastic analysis with analysis and fractional calculus to simultaneously link BTBS to a new system of temporally half-derivative interacting PDEs as well as to the fourth order system proved earlier and differently by Allouba. We then introduce a general class of random fields we call inverse-stable-Levy-time Brownian sheets (ISLTBSs), and we link them to $beta$-fractional-time-derivative systems of interacting PDEs for $0<beta<1$. When $beta=1/ u$, $ uinlbr2,3,...rbr$, our proof also connects an ISLTBS to a system of memory-preserving $ u$-Laplacian interacting PDEs. Memory is expressed via a sum of temporally-scaled $k$-Laplacians of the initial data, $k=1,..., u-1$. Using a Fourier-Laplace-transform-fractional-calculus approach, we give a conditional equivalence result that gives a necessary and sufficient condition for the equivalence between the fractional and the high order systems. In the one parameter case this condition automatically holds.
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.
We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, $KBtxy$, on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE, which we recently introduced. In sharp contrast to second order heat-based SPDEs--whose real-valued mild solutions are confined to $d=1$--we prove the existence of solutions to the BTBM SIE in $d=1,2,3$ with dimension-dependent and striking Holder regularity, under both less than Lipschitz and Lipschitz conditions. In space, we show an unprecedented nearly local Lipschitz regularity for $d=1,2$--roughly, the SIE is spatially twice as regular as the Brownian sheet in these dimensions--and nearly local Holder 1/2 regularity in d=3. In time, our solutions are locally Holder continuous with exponent $gammain(0,(4-d)/(8))$ for $1le dle3$. To investigate our SIE, we (a) introduce the Brownian-time random walk and we use it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and the mild forms of many SPDEs of different orders on the lattice. Here, solutions types to our SIE are both direct and limits of their lattice version. The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that it is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $KBtxy$ by the intimately connected kernel of our recently introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE becomes the mild form of a Kuramoto-Sivashinsky SPDE with linear PDE part. Ideas developed here are adapted in separate papers to give a new approach, via our explicit IBTBAP representation, to many KS-type SPDEs in multi spatial dimensions.
In this note we consider generalized diffusion equations in which the diffusivity coefficient is not necessarily constant in time, but instead it solves a nonlinear fractional differential equation involving fractional Riemann-Liouville time-derivative. Our main contribution is to highlight the link between these generalised equations and fractional Brownian motion (fBm). In particular, we investigate the governing equation of fBm and show that its diffusion coefficient must satisfy an additive evolutive fractional equation. We derive in a similar way the governing equation of the iterated fractional Brownian motion.
We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{alpha, beta}$ with Hurst parameter $(alpha, beta) in (0,1)^2$. When $0<alpha leq 1-frac{1}{2q}$ or $0<beta leq 1-frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $qgeq 2$, while for $1-frac{1}{2q}<alpha, beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
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