scholarly journals Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1917
Author(s):  
Junmei Wang ◽  
James Hoult ◽  
Yubin Yan

Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.

2018 ◽  
Vol 18 (05) ◽  
pp. 1850039 ◽  
Author(s):  
Ahmadou Bamba Sow ◽  
Yaya Sagna

In this paper, we deal with a backward doubly stochastic differential equations with jumps. Under stochastic Lipschitz conditions on the coefficients, we prove the existence and uniqueness of solution and provide a comparison theorem. Using this comparison theorem, we show the existence of a minimal solution when the drift satisfy a stochastic growth condition.


Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.


2011 ◽  
Vol 8 (6) ◽  
pp. 12247-12283
Author(s):  
P. Sabatier ◽  
J.-L. Reyss ◽  
J. M. Hall-Spencer ◽  
C. Colin ◽  
N. Frank ◽  
...  

Abstract. Here we show the use of the 210Pb-226Ra excess method to determine the growth rate of corals from one of the world's largest known cold-water coral reef, the Røst Reef off Norway. Two large branching framework-forming cold-water coral specimens, one Lophelia pertusa and one Madrepora oculata were collected alive at 350 m water depth from the Røst Reef at ~67° N and ~9° E. Pb and Ra isotopes were measured along the major growth axis of both specimens using low level alpha and gamma spectrometry and the corals trace element compositions were studied using ICP-QMS. Due to the different chemical behaviors of Pb and Ra in the marine environment, 210Pb and 226Ra were not incorporated the same way into the aragonite skeleton of those two cold-water corals. Thus to assess of the growth rates of both specimens we have here taken in consideration the exponential decrease of initially incorporated 210Pb as well as the ingrowth of 210Pb from the decay of 226Ra. Moreover a~post-depositional 210Pb incorporation is found in relation to the Mn-Fe coatings that could not be entirely removed from the oldest parts of the skeletons. The 226Ra activities in both corals were fairly constant, then assuming constant uptake of 210Pb through time the 210Pb-226Ra chronology can be applied to calculate linear growth rate. The 45.5 cm long branch of M. oculata reveals an age of 31 yr and a~linear growth rate of 14.4 ± 1.1 mm yr−1, i.e. 2.6 polyps per year. However, a correction regarding a remaining post-depositional Mn-Fe oxide coating is needed for the base of the specimen. The corrected age tend to confirm the radiocarbon derived basal age of 40 yr (using 14C bomb peak) with a mean growth rate of 2 polyps yr−1. This rate is similar to the one obtained in Aquaria experiments under optimal growth conditions. For the 80 cm-long specimen of L. pertusa a remaining contamination of metal-oxides is observed for the middle and basal part of the coral skeleton, inhibiting similar accurate age and growth rate estimates. However, the youngest branch was free of Mn enrichment and this 15 cm section reveals a growth rate of 8 mm yr−1 (~1 polyp every two to three years). However, the 210Pb growth rate estimate is within the lowermost ranges of previous growth rate estimates and may thus reflect that the coral was not developing at optimal growth conditions. Overall, 210Pb-226Ra dating can be successfully applied to determine the age and growth rate of framework-forming cold-water corals, however, removal of post-depositional Mn-Fe oxide deposits is a prerequisite. If successful, large branching M. oculata and L. pertusa coral skeletons provide unique oceanographic archive for studies of intermediate water environmentals with an up to annual time resolution and spanning over many decades.


Author(s):  
Annie Millet ◽  
Svetlana Roudenko ◽  
Kai Yang

Abstract We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.


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