[Paper Review] Interacting time-fractional and $\Delta^{ u}$ PDEs systems via Brownian-time and Inverse-stable-L\'evy-time Brownian sheets

Hassan Allouba, Erkan Nane|arXiv (Cornell University)|May 3, 2011

Stochastic processes and financial applications17 references4 citations

TL;DR

This paper establishes a novel stochastic connection between Brownian-time Brownian sheets (BTBS) and inverse-stable-Lévy-time Brownian sheets (ISLTBS) with systems of interacting time-fractional and high-order PDEs. Using stochastic analysis, fractional calculus, and Fourier-Laplace transforms, it proves conditional equivalence between fractional and high-order PDE systems, showing memory-preserving solutions via spatial Laplacians of initial data. The key contribution is a unified framework linking multiparameter random processes to coupled PDEs with intrinsic interaction and memory effects.

ABSTRACT

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-L\'evy-time Brownian sheets (ISLTBSs), and we link them to $\beta$-fractional-time-derivative systems of interacting PDEs for $0<\beta<1$. When $\beta=1/ u$, $ u\in\lbr2,3,... br$, 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.

Motivation & Objective

• To establish a new stochastic connection between Brownian-time Brownian sheets (BTBS) and systems of temporally half-derivative and fourth-order interacting PDEs.
• To introduce inverse-stable-Lévy-time Brownian sheets (ISLTBS) and link them to β-fractional-time-derivative systems of interacting PDEs for 0 < β < 1.
• To prove conditional equivalence between high-order and fractional PDE systems using Fourier-Laplace transforms and fractional calculus.
• To extend results beyond bounded initial data by relaxing boundedness assumptions using integrability and growth conditions on f and its derivatives.
• To demonstrate that memory effects in solutions are encoded through temporally scaled spatial k-Laplacians of the initial data, k = 1, ..., ν−1, for ν-Laplacian systems.

Proposed method

• Uses a unified stochastic analytic proof combining Itô’s rule, properties of the Brownian sheet, and fractional calculus to simultaneously link BTBS to both half-derivative and fourth-order interacting PDEs.
• Introduces ISLTBS as a generalization of BTBS, where each time parameter is replaced by an inverse-stable-Lévy process.
• Applies a Fourier-Laplace-transform-fractional-calculus approach to derive a conditional equivalence between high-order and fractional PDE systems.
• Derives necessary and sufficient conditions for equivalence between fractional and high-order PDE systems, which hold automatically in the one-parameter case.
• Employs the Brownian-time Feynman-Kac formula to express solutions as expectations over Brownian paths with time changed by absolute Brownian motions or inverse-stable-Lévy processes.
• Relaxes boundedness assumptions on initial data f and its derivatives by introducing integrability and growth conditions (3.1) involving the BS density and its derivatives.

Experimental results

Research questions

• RQ1 How can Brownian-time Brownian sheets (BTBS) be used to generate systems of interacting time-fractional and fourth-order PDEs?
• RQ2 What is the conditional equivalence condition between high-order and fractional PDE systems linked to BTBS and ISLTBS?
• RQ3 How do memory effects manifest in solutions of these PDE systems, and how are they encoded via spatial Laplacians of the initial data?
• RQ4 Can the boundedness condition on f and its derivatives be relaxed while preserving the PDE connections?
• RQ5 What is the role of the inverse-stable-Lévy process in generating β-fractional-time-derivative systems of interacting PDEs?

Key findings

• The paper proves a new stochastic analytic link between BTBS and a system of temporally half-derivative interacting PDEs, which is novel in the one-parameter case.
• It establishes a conditional equivalence between the BTBS-based fourth-order and time-fractional interacting PDEs systems, with the condition automatically satisfied in the one-parameter setting.
• For ISLTBS with β = 1/ν, ν ∈ {2, 3, ...}, the method links the process to a system of ν-Laplacian interacting PDEs with memory preserved via a sum of temporally scaled k-Laplacians of the initial data, k = 1, ..., ν−1.
• The solution V(j)(t, x) to the fractional PDE satisfies ∂βtj V(j)(t, x) = 1/2 ∆x V(j)(t, x), confirming the fractional PDE connection.
• The boundedness assumption on f and its derivatives can be replaced by integrability and growth conditions (3.1), which are satisfied when f has polynomial growth and its derivatives are Hölder continuous.
• The results extend to the case ν = 1/2 (i.e., β = 1/2), recovering the BTBS fourth-order PDE system with a minor scaling adjustment in the density kernel.