Related papers: Weak regularization by degenerate L\'evy noise and…
In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla…
Regularization plays a pivotal role in ill-posed machine learning and inverse problems. However, the fundamental comparative analysis of various regularization norms remains open. We establish a small noise analysis framework to assess the…
In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged…
In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in $H_x^1\times H_x^1$. To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear…
We show that perturbing ill-posed differential equations with (potentially very) smooth random processes can restore well-posedness -- even if the perturbation is (potentially much) more regular than the drift component of the solution. The…
We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear…
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 $…
We establish heat-kernel bounds and regularity estimates for the transition densities of the diffusion associated with the martingale problem corresponding to the generator of a formal multidimensional Brownian SDE with singular drift. As a…
We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a…
We consider stochastic differential equations (SDEs) driven by small L\'evy noise with some unknown parameters, and propose a new type of least squares estimators based on discrete samples from the SDEs. To approximate the increments of a…
We address parameter estimation in second-order stochastic differential equations (SDEs), which are prevalent in physics, biology, and ecology. The second-order SDE is converted to a first-order system by introducing an auxiliary velocity…
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only…
We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of…
For a discrete-negative-time discrete-space SDE, which admits no strong solution in the classical sense, a weak solution is constructed that is a (necessarily nonmeasurable) non-anticipative function of the driving i.i.d. noise. The result…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the…
We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of…
Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an…
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical {\alpha}-stable L\'evy processes via modulation or amplitude equations. We study SPDEs with a cubic…
The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and H\"older continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise,…
We study a generalized 1d periodic SPDE of Burgers type: $$ \partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi $$ where $\theta > 1/2$, $-A$ is the 1d Laplacian, $\xi$ is a space-time white noise and the initial condition $u_0$…