Asymptotic behaviour of stochastic inertial dynamics incorporating a Tikhonov regularization term

In a separable Hilbert space, we study the minimization problem of a convex smooth function with Lipschitz continuous gradient whose evaluations are corrupted by random noise. To this end, we associate a stochastic inertial system that incorporates Tikhonov regularization with the optimization problem. We establish existence and uniqueness of a solution trajectory for this system. Then, we derive an upper bound on the expected value of an appropriate associated energy function given square-integrability of the diffusion $σ_X$ before focusing on the particular case where the parameter function multiplied by the Tikhonov term is given by $\frac{1}{t^r}$ for $0<r<2$. For this setting, we show a.s. convergence rates as well as convergence rates in expectation for the function values along the trajectory to an infimal value, the trajectory process to an optimal solution and its time derivative to zero under a stronger integrability condition on $σ_X$.