Related papers: Uniqueness for continuity equations in Hilbert spa…

In this paper we prove a new strong uniqueness result and a weak existence result for possibly {\it degenerate} multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples…

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order…

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing…

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…

In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability…

In this paper, we establish the strong well-posedness of SDEs with merely integrable time-dependent drifts driven by fractional Brownian motions with Hurst parameter H<1/2. Our result holds over the entire subcritical regime and can be…

Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $\rho_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of…

Recently, A. Gruenrock and H. Pecher proved global well-posedness of the 2d Dirac-Klein-Gordon equations given initial data for the spinor and scalar fields in $H^s$ and $H^{s+1/2} \times H^{s-1/2}$, respectively, where $s\ge 0$, but…

We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to H$ is a negative definite self-adjoint…

Consider the stochastic differential equation $\mathrm dX_t = -A X_t \,\mathrm dt + f(t, X_t) \,\mathrm dt + \mathrm dB_t$ in a (possibly infinite-dimensional) separable Hilbert space, where $B$ is a cylindrical Brownian motion and $f$ is a…

In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$…

We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth;…

Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x…

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the…

We consider the stationary diffusion equation $-\mathrm{div} (\nabla u + bu )=f$ in $n$-dimensional torus $\mathbb{T}^n$, where $f\in H^{-1}$ is a given forcing and $b\in L^p$ is a divergence-free drift. Zhikov (Funkts. Anal. Prilozhen.,…

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $\alpha$-stable processes with $\alpha \in(1,2)$. The existence part is based on Euler's approximation as…

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…

The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying…

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite…

Let $U,H$ be two separable Hilbert spaces and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align}…