Related papers: Some remarks on Davie's uniqueness theorem
We investigate properties of the (conditional) law of the solution to SDEs driven by fractional Brownian noise with a singular, possibly distributional, drift. Our results on the law are twofold: i) we quantify the spatial regularity of the…
For classical finite time horizon stopping problems driven by a Brownian motion \[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem…
We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyans theorem.
Combining fractional calculus and the Rough Path Theory we study the existence and uniqueness of mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral…
We study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a…
In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $Ax-B|x|=b$ with $A, B\in \mathbb{R}^{n\times n}$ from the optimization field are first presented,…
The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems…
We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ stands…
We add a theorem to [J. Differential Equations 257 (2014), no. 3, 720--758] by F. Achleitner, C.M. Cuesta and S. Hittmeir. In that paper we studied travelling wave solutions of a Korteweg-de Vries-Burgers type equation with a non-local…
A celebrated result in bifurcation theory is that global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem when the operators involved…
We improve on the strong uniqueness results of [GLM+17], which deal with the following system of SDE. \begin{align*} dX_t&=Y_tdt \\ dY_t&=|X_{t}|^{\alpha}dB_t \end{align*} and $X_0=x_0,Y_0=y_0$. For $(x_0,y_0)\ne(0,0)$, we show that…
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.
One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\,dt+\lambda X\,dt=X\,dW,$ $t\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\partial\varphi(t)$, $t\in(0,T)$, is a maximal…
We consider a mixed stochastic differential equation $d{X_t}=a(t,X_t)d{t}+b(t,X_t) d{W_t}+c(t,X_t)d{B^H_t}$ driven by independent multidimensional Wiener process and fractional Brownian motion. Under Hormander type conditions we show that…
We prove the existence, uniqueness and non negativity of solutions for a nonlinear stationary Doi-Edwards equation. The existence is proved by a perturbation argument. We get the uniqueness and the non negativity by showing the convergence…
We prove that the uniqueness results obtained in \cite{urrea} for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With…
We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the…
We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $P=\{(t_i,x_i),\; i=1,\dots,N\}$, find an ODE $x^\prime(t) = v (x)$ that admits a solution $x(t)$ such that $x_i \approx…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an $m$-dimensional Brownian motion and a $d$-dimensional canonical process with uniform Lipschitzian coefficients. Such…