Characterization of memory states of the Preisach operator with stochastic inputs

The Preisach operator with inputs defined by a Markov process x ^t is considered. The question we address is: what is the distribution of the random memory state of the Preisach operator at a given time moment t ₀ in the limit r→∞ of infinitely long input history x ^t, ^t0-r≤t≤ ^t0? In order to answer this question, we introduce a Markov chain (called the memory state Markov chain) where the states are pairs ( ^mk, ^Mk) of elements from the monotone sequences of the local minimum input values m _k and the local maximum input values M _k recorded in the memory state and the index k of the elements plays the role of time. We express the transition probabilities of this Markov chain in terms of the transition probabilities of the input stochastic process and show that the memory state Markov chain and the input process generate the same distribution of the memory states. These results are illustrated by several examples of stochastic inputs such as the Wiener and Bernoulli processes and their mixture (we first discuss a discrete version of these processes and then the continuous time and state setting). The memory state Markov chain is then used to find the distribution of the random number of elements in the memory state sequence. We show that this number has the Poisson distribution for the Wiener and Bernoulli processes inputs. In particular, in the discrete setting, the mean value of the number of elements in the memory state scales as lnN, where N is the number of the input states, while the mean time it takes the input to generate this memory state scales as N ² for the Wiener process and as N for the Bernoulli process. A similar relationship between the dimension of the memory state vector and the number of iterations in the numerical realization of the input is shown for the mixture of the Wiener and Bernoulli processes, thus confirming that the memory state Markov chain is an efficient tool for generating the distribution of the Preisach operator memory states resulting from the effect of a stochastic input.