Related papers: A regularization approach to functional It\^o calc…
We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large…
In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second…
Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise…
We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…
A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is…
We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and {\O}ksendal in [7] for scalar quasi-linear It\^o stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a…
Motivated by a problematic coming from mathematical finance, this paper is devoted to existing and additional results of continuity and differentiability of the It\^o map associated to rough differential equations. These regularity results…
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…
In this paper we propose a new type of viscosity solutions for fully nonlinear path dependent PDEs. By restricting to certain pseudo Markovian structure, we remove the uniform non- degeneracy condition imposed in our earlier works [9, 10].…
This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when $N\geq 2$ in…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in…
We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions…
Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE…
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDEs given as a sup--envelope function. This is done through a dynamic programming principle derived from Denis, Hu, Peng (2010). The…
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on…
We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential…
In this paper, we address the problem of approximating solutions of ill-posed problems using mollification. We quickly review existing mollification regularization methods and provide two new approximate solutions to a general ill-posed…