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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.
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We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, dots, B^{m})$ with Hurst parameter $H in (frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs. Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-frac12 -H}$. In the general case when $m>1$, the exact rate equals $n^{frac12 -2H}$. In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions.
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.
Let $B^{alpha_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${alpha_i}$ ($i=1,2$), and let $B^{alpha_1}$ and $B^{alpha_2}$ be independent. We prove that, if $frac{N_1}{alpha_1}+frac{N_2}{alpha_2}>d$, then the intersection local times of $B^{alpha_1}$ and $B^{alpha_2}$ exist, and have a continuous version. We also establish H{o}lder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre ({it J. Theor. Probab.} {bf 20} (2007)), where the existence of the intersection local times of two independent $(1,d)$-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.
We prove the existence of the intersection local time for two independent, d -dimensional fractional Brownian motions with the same Hurst parameter H. Assume d greater or equal to 2, then the intersection local time exists if and only if Hd<2.
Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.
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