Related papers: Sharp pathwise nonuniqueness for additive SDEs

We consider the transport equation driven by the fractional Brownian motion. We study the existence and the uniqueness of the weak solution and, by using the tools of the Malliavin calculus, we prove the existence of the density of the…

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\"older continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} \Delta X_t(x)…

We consider non-degenerate SDEs with a $\beta$-Holder continuous and bounded drift term and driven by a Levy noise $L$ which is of $\alpha$-stable type. If $\alpha \in [1,2)$ and $\beta \in (1 - \frac{\alpha}{2},1) $ we show pathwise…

In the recent article [A. Jentzen, B. Kuckuck, T. M\"uller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a…

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the…

We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional…

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By…

In this paper, we establish large deviation principle for the strong solution of a doubly nonlinear PDE driven by small multiplicative Brownian noise. Motononicity arguments and the weak convergence approach have been exploited in the…

McKean-Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker-Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and…

We prove the existence of probabilistically strong solutions for large classes of possibly degenerate stochastic differential equations with locally Sobolev-regular coefficients, using the restricted Yamada-Watanabe theorem. Our approach…

A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic skew product flows, Nonlinearity 24 (2011) 1361-1367, gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential…

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic…

We consider a well-known family of SDEs with irregular drifts and the correspondent zero noise limits. Using (mollified) local times, we show which trajectories are selected. The approach is completely probabilistic and relies on elementary…

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is…

We first establish the unique ergodicity of the stochastic theta method (STM) with $\theta \in [1/2, 1]$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient…

We study differential equations with a linear, path dependent drift and discrete delay in the diffusion term driven by a $\gamma$-H\"older rough path for $\gamma > \frac{1}{3}$. We prove well-posedness of these systems and establish a…

In a recent paper by the first two named authors, existence of martingale solutions to a stochastic nonlinear Schr\"odinger equation driven by a L\'evy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and…

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand,…

In this paper, we prove the existence and uniqueness of maximally defined strong solutions to SDEs driven by multiplicative noise on general space-time domains $Q\subset\mathbb{R}_+\times\mathbb{R}^d$, which have continuous paths on the…

The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's…