Related papers: Sharp solvability for singular SDEs
We prove the unique weak solvability of stochastic differential equations with time-inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e.\,with integrability parameter $q>1$ close to $1$. The constructed weak…
The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique…
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by…
SDE driven by an $\alpha $-stable process, $\alpha \in \lbrack 1,2),$ with Lipshitz continuous coefficient and $\beta $-H\"older drift is considered. The existence and uniqueness of a strong solution is proved when $\beta >1-\alpha /2$ by…
We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a…
A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The…
The existence of an unbounded sequence of solutions to a conformally invariant elliptic equation having nonlocal critical-power nonlinearity is established. The primary obstacle to establishing existence of solutions is the failure of…
In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove…
We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative…
We prove unique continuation properties for linear variable coefficient Schr\"odinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any…
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…
In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{\sigma |X_s|^\alpha dW_s}, \;x_0>0,\;\sigma>0,\;…
Using analysis for 2-admissible functions in weighted Sobolev spaces and stochastic calculus for possibly degenerate symmetric elliptic forms, we construct weak solutions to a wide class of stochastic differential equations starting from an…
We prove weak uniqueness for admissible solutions of It\^o's equations with uniformly nondegenerate $a$ which is almost in VMO and $b$ in a Morrey class of functions with low integrability property. If $b\in L_{d}$ any solution is…
In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic…
A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
We establish the well-posedness of linear elliptic equations with critical-order drifts in $L^d$ and positive zero-order coefficients in $L^1$ or $L^{\frac{2d}{d+2}}$, where classical methods are often too restrictive. Our approach relies…
In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of $p$-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary…
We prove the existence and uniqueness of strong solutions for stochastic differential equations in which the drift coefficient is square integrable in time variable and H\"{o}lder continuous in space variable. Moreover, we prove that the…
We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $\alpha$-stable process under H\"older regularity conditions for the drift term. We partially recover the…