相关论文: An improved uniqueness result for a system of stoc…

We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\] and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initial conditions. Here, $L$ and $A$ are certain partial differential operators involving…

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove…

We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation…

For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise…

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…

This paper proves existence and stability results of solitary-wave solutions to coupled nonlinear Schr\"{o}dinger equations with power-type nonlinearities arising in several models of modern physics. The existence of solitary waves is…

An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed,…

We study a one-dimensional stochastic differential equation driven by a stable L\'evy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation.…

Spectral singularities such as exceptional points invoke specific physical effects. The present paper focuses upon the time dependent solutions of the Schr\"odinger equation. In a simple model it is demonstrated that - depending on initial…

We obtain uniqueness and existence of a solution $u$ to the following second-order stochastic partial differential equation (SPDE) : \begin{align} \label{abs eqn} du= \left( \bar a^{ij}(\omega,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t…

We put forward a new method for proving weak uniqueness of stochastic equations with singular drifts driven by a non-Markov or infinite-dimensional noise. We apply our method to study stochastic heat equation (SHE) driven by Gaussian…

In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.

We prove the existence and uniqueness of a strong solution for an SDE on a semi-axis with singularities at the point 0. The result obtained yields, for example, the strong uniqueness of non-negative solutions to SDEs governing Bessel…

We show weak existence and uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 1$, with prescribed sub-invariant measure $\widehat{\mu}$. The dispersion and drift coefficients of the…

We study existence and uniqueness of solutions for second order ordinary stochastic differential equations with Dirichlet boundary conditions on a given interval. In the first part of the paper we provide sufficient conditions to ensure…

We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $\alpha/x$, $\alpha>0$. We establish the exponential stability of the semigroup for all…

We consider in this article the weakly coupled system of wave equations in the \textit{scale-invariant case} and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the…

We survey and refine recent results on weak and strong well-posedness of stochastic differential equations with singular drift satisfying some minimal assumptions.

The differential equation (DE) with proportional delay is a particular case of the time-dependent delay differential equation (DDE). In this paper, we solve non-linear DEs with proportional delay using the successive approximation method…

For $\alpha\in (0,1)$, we consider stochastic differential equations driven by one-sided stable processes of order $\alpha$: \[dX_t= \phi(X_{t-})\ dZ_t.\] We prove that pathwise uniqueness holds for this equation under the assumptions that…