it is known that a stochastic dierential equation (sde) induces two probabilisticobjects, namely a diusion process and a stochastic ow. while the diusion process isdetermined by the innitesimal mean and variance given by the coecients of the sde,this is not the case for the stochastic ow induced by the sde. in order to characterize thestochastic ow uniquely the innitesimal covariance give...

Averaging neural network weights sampled by a backbone stochastic gradient descent (SGD) is simple-yet-effective approach to assist the SGD in finding better optima, terms of generalization. From statistical perspective, weight-averaging contributes variance reduction. Recently, well-established (SWA) method was proposed, which featured application cyclical or high-constant (CHC) learning-rate ...

We consider a 2D stochastic wave equation driven by Gaussian noise, which is temporally white and spatially colored described the Riesz kernel. Our first main result functional central limit theorem for spatial average of solution. And we also establish quantitative marginal rate convergence total-variation distance. A fundamental ingredient in our proofs pointwise Lp-estimate Malliavin derivat...

It is known that a stochastic differential equation (SDE) induces two probabilistic objects, namely a difusion process and a stochastic flow. While the diffusion process is determined by the innitesimal mean and variance given by the coefficients of the SDE, this is not the case for the stochastic flow induced by the SDE. In order to characterize the stochastic flow uniquely the innitesimal cov...

Due to the intrinsic link with (kinetic) nonlinear Fokker–Planck equations and many diverse applications, distribution dependent stochastic differential have been investigated intensively in recent years. The appearance of probability distributions (or laws) random variables solutions coefficients is a distinct feature equations. In this paper, under certain averaging conditions, we establish p...