Approssimazione numerica structure-preserving di problemi stocastici di evoluzione

In this Ph.D. thesis, we address our attention to the geometric numerical integration of stochastic differential equations and stochastic partial differential equations whose exact dynamics are characterized by invariant laws, (see, for example, [3,4, 8, 12, 27–29, 31, 32, 34, 41, 42, 45, 54, 63, 64, 70, 116, 117, 119, 145, 147–151, 167, 175, 188] and reference therein). The first part of this work is oriented to a nonlinear stability analysis of numerical discretizations to mean-square dissipative stochastic differential systems [116] with the aim of understanding if stochastic θ-Maruyama and Milstein methods [113,115] as well as stochastic Runge-Kutta methods [159–161] are capable of exhibiting the same exponential mean-square contractive behaviour visible along the exact dynamics. The investigation reveals that the contractivity of the continuous problem is translated into bounds to the stepsize of the time integration, in the case of stochastic θ-methods while, for stochastic Runge-Kutta methods, contractivity is retained through suitable algebraic constraints on the choice of the coefficients of the methods. The second part of the treatise is devoted to stochastic Ito and Stratonovich Hamiltonian systems [31,32,147–151]. It has been proven, indeed, that the Hamiltonian computed along the exact solutions to stochastic Ito Hamiltonian systems grows linearly in time according to the well-known trace equation [31, 32, 41], while in the Stratonovich case, the Hamiltonian is conserved along the exact flow [148, 150, 151]. Here, we aim to provide a backward error analysis in order to provide long-term estimates aiming to understand the structure-preserving character of the numerical schemes under investigation. We discover that, in the Ito setting, spurious quadratic terms (with respect to the stepsize) in the expected Hamiltonian error affect the preservation of the trace equation along the numerical dynamics while, for the Stratonovich case, an exponentially increasing parasitic term in the Hamiltonian deviation destroys the overall long-term behaviour. The influence of Monte Carlo estimates on the conservative behaviour of selected numerical methods has been also addressed. Moreover, the preservation properties of stochastic θ-methods applied to the stochastic Korteweg-de Vries equation [131,135] have been also investigated. The last part of this Ph.D. thesis regards the design and the study of stochastic multiscale models aimed to describe networks of individuals under the triadic closure principle, providing a rigorous analysis of a bistable regime that has been heuristically observed in [105]. Finally, several numerical experiments have been also provided to confirm the effectiveness of all theoretical results.