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We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian…
In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic…
Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $…
This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness…
We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly,…
We develop the rough path counterpart of It\^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It\^o / forward) stochastic differential equations treatable…
Let $u(t,x)$ be the solution to a stochastic heat equation $$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$ with initial condition…
In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise…
We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic…
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a…
For scalar conservation laws driven by a rough path $z(t)$, in the sense of Lions, Perthame and Souganidis in arXiv:1309.1931, we show that it is possible to replace $z(t)$ by a piecewise linear path, and still obtain the same solution at a…
High dimensional covariance estimation and graphical models is a contemporary topic in statistics and machine learning having widespread applications. An important line of research in this regard is to shrink the extreme spectrum of the…
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional…
In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven by an…
In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin…
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
In this paper we consider a system of non-linear stochastic heat equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and has a homogeneous spatial covariance. Under some suitable regularity and non degeneracy…
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the…
This paper calculates the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise.