On Comparison Results for Neutral Stochastic Differential Equations of Reaction-Diffusion Type in L_{2}(ℝ^{d})

In the present paper, we establish a comparison result for solutions to the Cauchy problems for two stochastic integro-differential equations of reaction-diffusion type with delay. On this subject number of authors have obtained their comparison results. We deal with the Cauchy problems for two stochastic integro-differential equations of reaction-diffusion type with delay. Except drift and diffusion coefficients, our equations include also one integro-differential term. Basic difference of our case from the case of all earlier investigated problems is presence of this term. Presence of this term turns this equation into a nonlocal neutral stochastic equation of reaction-diffusion type. Nonlocal deterministic equations of this type are well known in literature and have wide range of applications. Such equations arise, for instance, in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics, and population dynamics. These equations are used in modeling of phytoplankton growth, distant interactions in epidemic models and nonlocal consumption of resources. We introduce a concept of mild solutions to our problems and state and prove a comparison theorem for them. According to our result, under certain assumptions on coefficients of equations under consideration, their solutions depend on the transient coefficients in a monotone way

Uncertainty analysis using Bayesian Model Averaging: a case study of input variables to energy models and inference to associated uncertainties of energy scenarios

Background Energy models are used to illustrate, calculate and evaluate energy futures under given assumptions. The results of energy models are energy scenarios representing uncertain energy futures. Methods The discussed approach for uncertainty quantification and evaluation is based on Bayesian Model Averaging for input variables to quantitative energy models. If the premise is accepted that the energy model results cannot be less uncertain than the input to energy models, the proposed approach provides a lower bound of associated uncertainty. The evaluation of model-based energy scenario uncertainty in terms of input variable uncertainty departing from a probabilistic assessment is discussed. Results The result is an explicit uncertainty quantification for input variables of energy models based on well-established measure and probability theory. The quantification of uncertainty helps assessing the predictive potential of energy scenarios used and allows an evaluation of possible consequences as promoted by energy scenarios in a highly uncertain economic, environmental, political and social target system. Conclusions If societal decisions are vested in computed model results, it is meaningful to accompany these with an uncertainty assessment. Bayesian Model Averaging (BMA) for input variables of energy models could add to the currently limited tools for uncertainty assessment of model-based energy scenarios