Related papers: Rough differential equations containing path-depen…
In the present paper, we consider in detail the aspects of the Heisenberg's equations of motion, related to their transformation to the representation dependent of external sources. We provide with a closed solution as to the…
The construction of stochastic solutions for nonlinear partial differential equations is a powerful method to obtain new exact results and to develop efficient numerical algorithms, in particular when domain decomposition techniques are…
While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial…
The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of…
Rough stochastic differential equations (RSDEs) are common generalisations of Ito SDEs and Lyons RDEs and have emerged as new tool in several areas of applied probability, including non-linear stochastic filtering, pathwise stochastic…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
We consider a random process as a solution of stochastic differential equations with dependence of the coefficients on small parameter $\varepsilon$ and we suppose that the drift coefficients of these equations are unbounded on the…
We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is H\"older-continuous,…
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting…
We investigate the dependence on parameters for the discrete boundary value problem connected with the Emden-Fowler equation. A variational method is used in order to obtain a general scheme allowing for investigation the dependence on…
The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Necessary optimality conditions in the literature identify potentially optimal paths, but do not show how to perturb paths to…
Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However,…
We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and…
This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually…
We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor…
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on…
In this work we mainly prove the existence and pathwise uniqueness of solutions to general backward doubly stochastic differential equations with jumps appearing in both forward and backward integral parts. Several comparison theorems under…
We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse…
We consider a d-dimensional stochastic differential equation with additive noise and a drift coefficient which is assumed only to be a bounded Borel function. We show that, for almost all choices of the driving Brownian path, the equation…
We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite…