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 sys
tem 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.
Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivativ
es in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional derivatives in space, and power law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Levy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index $1<alpha<2$ to the density of the hitting time of a stable subordinator with index $1/alpha$, and thereby unify some recent results in the literature. These results also provide a concrete interpretation of Zolotarev duality in terms of the fractional diffusion model.
