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N. V. Krylov

Publications and source records attributed to N. V. Krylov.

At least 73 records · Page 4Linked to original sources
arXiv2016

On nonequivalence of regular boundary points for second-order elliptic operators

In this paper we present examples of nondivergence form second order elliptic operators with continuous coefficients such that $L$ has an irregular boundary point that is regular for the Laplacian. Also for any eigenvalue spread <1 of the matrix of the coefficients we provide an example of operator with discontinuous coefficients that has regular boundary points nonequivalent to Laplacian's (we give examples for each direction of nonequivalence). All examples are constructed for each dimension starting with 3.

math.AP↗
arXiv2016

Poisson stochastic process and basic Schauder and Sobolev estimates in the theory of parabolic equations

We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the {\em same\/} constants as in the case of the one-dimensional heat equation. The method is based on using the Poisson stochastic process. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results.

math.AP↗
arXiv2016

On the existence of $W^{2}_{p}$ solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

We establish the existence of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on $H$, apart from ellipticity, but of a "cut-off" version of the equation $H(v,Dv,D^{2}v,x)=0$.

math.AP↗
arXiv2015

On parabolic equations in one space dimension

Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of $(t,x)$ bounded away from zero. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations.

math.AP↗
arXiv2014

On regularity properties and approximations of value functions for stochastic differential games in domains

We prove that for any constant $K\geq1$, the value functions for time homogeneous stochastic differential games in the whole space can be approximated up to a constant over $K$ by value functions whose second-order derivatives are bounded by a constant times $K$. On the way of proving this result we prove that the value functions for stochastic differential games in domains and in the whole space admit estimates of their Lipschitz constants in a variety of settings.

math.OC↗
arXiv2014

To the theory of viscosity solutions for uniformly parabolic Isaacs equations

We show how a theorem about the solvability in $W^{1,2}_{\infty}$ of special parabolic Isaacs equations can be used to obtain the existence and uniqueness of viscosity solutions of general uniformly nondegenerate parabolic Isaacs equations. We apply it also to establish the $C^{1+χ}$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in $C^γ$ with respect to the spatial variables with $γ$ slightly less than $1/2$.

math.AP↗
arXiv2014

To the theory of viscosity solutions for uniformly elliptic Isaacs equations

We show how a theorem about solvability in $C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the $C^{1+χ}$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in $C^γ$ with $γ$ slightly less than $1/2$.

math.AP↗
arXiv2014

On the independence of the value function for stochastic differential games of the probability space

We show that the value function in a stochastic differential game does not change if we keep the same space $(Ω,\mathcal{F})$ but introduce probability measures by means of Girsanov's transformation {\em depending} on the policies of the players. We also show that the value function does not change if we allow the driving Wiener processes to depend on the policies of the players. Finally, we show that the value function does not change if we perform a random time change with the rate depending on the policies of the players.

math.PR↗
arXiv2014

Hörmander's theorem for stochastic partial differential equations

We prove Hörmander's type hypoellipticity theorem for stochastic partial differential equations when the coefficients are only measurable with respect to the time variable. The need for such kind of results comes from filtering theory of partially observable diffusion processes, when even if the initial system is autonomous, the observation process enters the coefficients of the filtering equation and makes them time-dependent with no good control on the smoothness of the coefficients with respect to the time variable.

math.PR↗
arXiv2013

On a result of Jessica Lin

In Theorem 1.5 of [5] an estimate from below is given for nonnegative supersolutions of parabolic equations in terms of the measure of the set where the right-hand side is less than $-1$. The given proof contains many interesting details which are certainly useful in many other situations. However, as long as the statement of Theorem 1.5 of \cite{JL} is concerned, it can be proved in a much shorter way, which we present here.

math.AP↗
arXiv2013

Hörmander's theorem for parabolic equations with coefficients measurable in the time variable

We are dealing with possibly degenerate second-order parabolic operators whose coefficients are infinitely differentiable with respect to space variables and only measurable with respect to the time variable. We impose the Hörmander condition on the diffusion coefficients and prove that the solutions of the corresponding equations with right-hand sides which are infinitely differentiable in the space variables in a space-time domain have also this property.

math.AP↗
arXiv2013

Hypoellipticity for filtering problems of partially observable diffusion processes

We prove that under Hörmander's type conditions on the coefficients of the unobservable component of a partially observable diffusion process the filtering density is infinitely differentiable and can be represented as the integral of an infinitely differentiable kernel against the prior initial distribution. These results are derived from more general results obtained for SPDEs. One the main novelty of the paper is the existence and smoothness of the kernel, another that we allow the coefficients of our partially observable process to be just measurable with respect to the time variable.

math.PR↗
arXiv2013

On singularity as a function of time of a conditional distribution of an exit time

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

math.PR↗
arXiv2013

An ersatz existence theorem for fully nonlinear parabolic equations without convexity assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a continuous solution with the first and the second spatial derivatives under control: bounded in the case of the whole space and locally bounded in case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the second spatial derivatives of the unknown function. This is different from a previous work of Hongjie Dong and the author where the modification was done for large values of the unknown function and its spatial derivatives.

math.AP↗
arXiv2012

On $C^{1+α}$ regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_{p}$-viscosity solutions, which are in $C^{1+α}$ for an $α\in(0,1)$. The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.

math.AP↗
arXiv2012

A relatively short proof of Itô's formula for SPDEs and its applications

We give a short proof of Itô's formula for stochastic Hilbert-space valued processes in the setting $V\subset H\subset V^{*}$ based on the possibility to lift the stochastic differentials, which are originally in $V^{*}$, into $H$. Using this result we also prove the maximum principle for second-order SPDEs in arbitrary domains.

math.PR↗