Related papers: Symmetry of stochastic equations
Symmetry relationships for optical observations of matter generally fall into several common scattering geometries. The 'planetary' configuration is preferred among a group of observers of extraterrestrial planets, 'laboratory' observations…
We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in…
Symmetry is one of the most central concepts in physics, and it is no surprise that it has also been widely adopted as an inductive bias for machine-learning models applied to the physical sciences. This is especially true for models…
This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be specifically given on the Moment-SOS hierarchy in polynomial optimization, where results from…
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a…
Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the…
The aim of this work is to use systematically the symmetries of the (one dimensional) bacward heat equation with potentiel in order to solve certain one dimensional It\^o's stochastic differential equations. The special form of the drift…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders…
This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if Fokker--Planck--Kolmogorov equation has non-trivial Lie symmetry, then boundary crossing identity…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the…
We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $…
For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered.…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are…
The note provides the relation between symmetries and first integrals of It\^{o} stochastic differential equations and symmetries of the associated Kolmogorov backward equation. Relation between the symmetries of the Kolmogorov backward…
In this paper, we consider the composition of two independent processes : one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler…