A variational model for innovation diffusion under fuzzy uncertainty

We propose a variational approach to study the market penetration of new technologies under conditions of spatial heterogeneity in the economic factors influencing the process and imprecise knowledge about their intensities. Differently from other methodologies that describe the adoption process in terms of partial differential equations, we formulate the model as a minimization problem of an appropriate variational functional with fuzzy coefficients. This approach permits to consider in the analysis both the attractive and diffusive forces that drive the process and the subjective opinions of policy makers about the actual influence exerted by these determinants on the adoption decision. Interestingly, our results show that different degrees of uncertainty lead to significantly different predictions about the diffusion process and, therefore, our methodology could be applied to support strategic decisions concerning innovation diffusion plans. An application to the digital transition in agriculture is also provided to study the effectiveness of government policies.