Related papers: Some properties of solutions of It\^o equations wi…
This paper is a natural continuation of \cite{Kr_20_2}, where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in…
This paper is a natural continuation of \cite{Kr_20_2} and \cite{Kr_21_1} where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in…
We prove the solvability of It\^o stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different.…
We consider It\^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. We prove the unique strong solvability for any starting point and prove that as…
We consider It\^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{2+\varepsilon,loc}$, and the drift in a Morrey class containing $L_{d}$. We prove the unique strong solvability in…
The aim of the book is to present some recent results in the theory of stochastic It\^o equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order…
We prove the existence of strong solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Strong uniqueness is also discussed.
For It\^o stochastic equations in $\mathbb{R}^{d}$ with drift in $L_{d}$ several results are discussed such as the existence of weak solutions, the existence of the corresponding Markov process, Aleksandrov type estimates of their Green's…
The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables,…
We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators $L=a^{ij}D_{ij}+b^{i}D_{i}$, acting on functions on $\mathbb{R}^{d}$, with measurable coefficients, bounded and uniformly elliptic $a$ and…
For It\^o stochastic processes in $\mathbb{R}^{d}$ with drift in $L_{d}$ Aleksandrov's type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding…
We prove strong existense of solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class type.
We prove the existence of weak solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are…
We prove It{\^o}'s formula for the flow of measures associated with an It{\^o} process having a bounded drift and a uniformly elliptic and bounded diffusion matrix, and for functions in an appropriate Sobolev-type space. This formula is the…
We study a class of R^d-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not…
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and…
K. It\^{o} characterised in \cite{ito} zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of It\^{o} in the case of Hilbert spaces: Gauss-Markov families that are…
We present a detailed analysis of non-degenerate time-homogeneous It\^o-stochastic differential equations with low local regularity assumptions on the coefficients. In particular the drift coefficient may only satisfy a local integrability…
Comparison results are given for time-inhomogeneous Markov processes with respect to function classes induced stochastic orderings. The main result states comparison of two processes, provided that the comparability of their infinitesimal…
We prove strong existence and uniqueness of solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class type. In a sense we are treating a "supercritical" case.