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In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= \sigma_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate…
The solutions of SDEs with multiplicative noise are not Markovian. On a coarse-grained time scale they still are, but only in the "anti-Ito" case. This allows a simple computation of the most likely path. Any density peak moves along such a…
Dynamical properties of a generalized max-plus model for ultradiscrete limit cycles are investigated.This model includes both the negative feedback model and the Sel'kov model. It exhibits the Neimark-Sacker bifurcation, and possesses…
The term "hybrid system" refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical…
An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is…
We analyze the dynamics of streaming stochastic gradient descent (SGD) in the high-dimensional limit when applied to generalized linear models and multi-index models (e.g. logistic regression, phase retrieval) with general data-covariance.…
We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary…
We extend slow manifolds near a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the most flexible degrees of freedom. This Slow-Fast Decomposition (SFD) enslaves exponentially…
Stochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units,…
The purpose of this paper is to implement a random death process into a persistent random walk model which produces subballistic superdiffusion (L\'{e}vy walk). We develop a Markovian model of cell motility with the extra residence variable…
In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such…
We leverage commutative hypercomplex analysis to find closed-form solutions of some systems of stochastic differential equations. Specifically, we obtain necessary and sufficient conditions under which a system of stochastic differential…
Solving Markov Decision Processes (MDPs) remains a central challenge in sequential decision-making, especially when dealing with large state spaces and long-term optimization criteria. A key step in Bellman dynamic programming algorithms is…
We study nonlinear time-inhomogeneous Markov processes in the sense of McKean's seminal work [32]. These are given as families of laws $\mathbb{P}_{s,\zeta}$, $s\geq 0$, on path space, where $\zeta$ runs through a set of admissible initial…
We propose a differential difference equation in ${\mathcal R}^1\times {\mathcal Z}^2$ and study it by Hirota's bilinear method. This equation has a singular continuum limit into a system which admits the reduction to the Davey-Stewartson…
We present two fully probabilistic Euler schemes, one explicit and one implicit, for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth and random initial condition. We provide a…
Improving the efficiency of discrete time scale invariant (DSI) processes, we consider some flexible sampling of a continuous time DSI process ${X(t), t\in{R^+}}$ with scale $l>1$, which is in correspondence to some multi-dimensional…