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Register a new userTime-fractional and memoryful $Delta^{2^{k}}$ SIEs on $RptimesRd$: how far can we push white noise?
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Hassan Allouba
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High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs---through their fundamental solutions---on stochastic integral equations (SIEs) driven by space-time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase of the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in (Allouba 2013, Allouba 2006), where we introduced two different stochast
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How far can we use multi-wavelength cross-identifications to deconvolve far-infrared images? In this short research note I explore a test case of CLEAN deconvolutions of simulated confused 850 micron SCUBA-2 data, and explore the possible scientific applications of combining this data with ostensibly deeper TolTEC Large Scale Structure (LSS) survey 1.1mm-2mm data. I show that the SCUBA-2 can be reconstructed to the 1.1mm LMT resolution and achieve an 850 micron deconvolved sensitivity of 0.7 mJy RMS, an improvement of at least ~1:5x over naive point source filtered images. The TolTEC/SCUBA-2 combination can constrain cold (<10K) observed-frame colour temperatures, where TolTEC alone cannot.
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 consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp (Phi)_{2}$-quantum field model or H{o}egh-Krohns model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
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.
The present paper is a continuation of our previous work on the stochastic quantization of the $exp(Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L^{1}$-regime $vertalphavert<sqrt{8pi}$ of the charge parameter $alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
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