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This paper addresses the gradient flow -- the continuous-time representation of the gradient method -- with the smooth approximation of a non-differentiable objective function and presents convergence analysis framework. Similar to the…

This paper focuses on time-varying delayed stochastic differential systems with stochastically switching parameters formulated by a unified switching behavior combining a discrete adapted process and a Cox process. Unlike prior studies…

The spectral density function describes the second-order properties of a stationary stochastic process on $\mathbb{R}^d$. This paper considers the nonparametric estimation of the spectral density of a continuous-time stochastic process…

Under the uniform H\"{o}rmander's hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity…

In this paper we give log-convexity properties for solutions to discrete Schr\"odinger equations with different discrete versions of Gaussian decay at two different times. For free evolutions, we use complex analysis arguments to derive…

We present limit theorems for a sequence of Piecewise Deterministic Markov Processes (PDMPs) taking values in a separable Hilbert space. This class of processes provides a rigorous framework for stochastic spatial models in which discrete…

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process…

In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin's…

We introduce multiple hidden Markov models (MHMMs) where an observed multivariate categorical time series depends on an unobservable multivariate Mar- kov chain. MHMMs provide an elegant framework for specifying various independence…

We consider the approximation of weakly T-coercive operators. The main property to ensure the convergence thereof is the regularity of the approximation (in the vocabulary of discrete approximation schemes). In a previous work the existence…

We present a general scheme to calculate within the independent interval approximation generalized (level-dependent) persistence properties for processes having a finite density of zero-crossings. Our results are especially relevant for the…

In this paper, we address stochastic optimization problems involving a composition of a non-smooth outer function and a smooth inner function, a formulation frequently encountered in machine learning and operations research. To deal with…

In the paper we develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of first-order ordinary differential equations in spaces of smooth functions on a finite interval. This problems are set…

In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment…

We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the…

It is well known that the transition semigroup of an Ornstein Uhlenbeck process with delay is not strong Feller for small times, so it has no regularizing effects when acting on bounded and continuous functions. In this paper we study…

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and…

There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the traditional HMM. However, in many settings the HDP-HMM's strict Markovian constraints are…

We define the notion of a thick open set $\Omega$ in a Euclidean space and show that a local Hardy-Littlewood inequality holds in $L^p(\Omega)$, $p \in (1, \infty]$. We then establish pointwise and $L^p(\Omega)$ convergence for families of…

We introduce a methodology for online estimation of smoothing expectations for a class of additive functionals, in the context of a rich family of diffusion processes (that may include jumps) -- observed at discrete-time instances. We…