Related papers: About semilinear low dimension Bessel PDEs

The Bessel process in low dimension (0 $\le$ $\delta$ $\le$ 1) is not an It{\^o} process and it is a semimartingale only in the cases $\delta$ = 1 and $\delta$ = 0. In this paper we first characterize it as the unique solution of an SDE…

In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all $\delta \geq 0$ and $T>0$, we compute the derivative of…

We consider diffusion operator $-\Delta + b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible…

We study the differentiability of Bessel flow $\rho : x \to \rho ^x_t$, where $(\rho ^x_t)_{t\geq 0}$ is BES $^x(\delta $) process of dimension $\delta >1$ starting from $x$. For $\delta \geq 2$ we prove the existence of bicontinuous…

Let $(L_t)_{t \geq 0}$ be a $k$-dimensional L\'evy process and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the L\'evy-driven stochastic differential equation (SDE) $$dX_t = \sigma(X_{t-}) \, dL_t,…

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized…

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}…

We consider the Schr\"odinger operator on the halfline with the potential $(m^2-\frac14)\frac1{x^2}$, often called the Bessel operator. We assume that $m$ is complex. We study the domains of various closed homogeneous realizations of the…

In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption…

We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a…

This paper aims at solving a one-dimensional backward stochastic differential equation (BSDE for short) with only integrable parameters. We first establish the existence of a minimal $L^1$ solution for the BSDE when the generator $g$ is…

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the…

In this paper, we introduce a simple Bessel $\delta$-method to the theory of exponential sums for $\rm GL_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level…

We study the existence and uniqueness of minimal supersolutions of backward stochastic differential equations with generators that are jointly lower semicontinuous, bounded below by an affine function of the control variable and satisfy a…

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb…

We study the existence of minimal supersolutions of BSDEs under a family of mutually singular probability measures. We consider generators that are jointly lower semicontinuous, positive, and either convex in the control variable and…

In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we investigate the shape of solutions whose derivative in $\theta$ vanishes at the…

We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…

Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its local time at $y$. The stochastic differential equation (SDE) whose solution corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given $a\geq 0$ is…

We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes…