Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

An Intelligent Calculation Method of Volterra Time-Domain Kernel Based on Time-Delay Artificial Neural Network

To solve the problems of high complexity and low accuracy in Volterra time-domain kernel calculation of a nonlinear system, this paper proposes an intelligent calculation method of Volterra time-domain kernel by time-delay artificial neural networks (TDANNs) and also designs a root mean square error (RMSE) index to choose the neuron number of the network input layer. Firstly, a three-layer TDANN is designed according to the characteristics of the Volterra model. Secondly, the relationship between parameters of TDANN and Volterra time-domain kernel is analyzed, and then three-order expressions of Volterra time-domain kernel are derived. The calculation of Volterra time-domain kernel is completed by network training. Finally, it is verified by a nonlinear system. Simulation results indicate that compared with traditional methods, the new method has higher accuracy, and it can realize the batch calculation of Volterra kernel, which not only improves the calculation efficiency but also provides accurate data for fault diagnosis based on Volterra kernel in further research work.

Parametric Estimation of Volterra Kernel for the Dynamic Response of an Offshore Structure Using Laguerre Polynomials

The behavior of a nonlinear dynamic system under arbitrary excitation can be represented by the Volterra series if the Volterra kernels of different orders are known. This study presents a methodology for a direct estimation of the Volterra kernel coefficients up to the second-order using prepared data obtained by running a time-domain analysis of the system of interest. To avoid potential problems during kernel estimation, the Volterra kernel is expanded into a polynomial series using the Laguerre polynomials, and the coefficients of the Laguerre polynomials are then estimated using a least-square method. A nonlinear oscillator with a quadratic stiffness term is introduced, and the methodology is applied to check the applicability and accuracy. The methodology is applied to a more realistic engineering problem of a simplified riser under irregular wave excitation.

Diagnosis of Inter-Turn Short Circuit of Synchronous Generator Rotor Winding Based on Volterra Kernel Identification

The inter-turn short circuit is a common fault in the synchronous generator. This fault is not easily detected at early stage. However, with the development of the fault, it will pose a threat to the safe operation of the generator. To detect the inter-turn short circuit of rotor winding, the feasibility of identifying the stator branch characteristics of synchronous generator during inter-turn short circuit was analyzed. In this paper, an on-line fault identification method based on Volterra kernel identification is presented. This method uses the stator branch voltage and stator unbalance branch current collected from the generator as input and output signals of the series model. Recursive batch least squares method is applied to calculate the three kernels of Volterra series. When the generator is in normal state or fault state, the Volterra kernel will change accordingly. Through the identification of the time-domain kernel of the nonlinear transfer model, the inter-turn short circuit fault of the synchronous generator is diagnosed. The correctness and effectiveness of this method is verified by using the data of fault experimental synchronous generator.