arXiv subjects
Publications and source records attributed to N. V. Krylov.
We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b\in L_{d}$ and $c\in L_{q}$, $c\geq0$, $d>q\geq d/2$. We prove the solvability of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, $1 0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.
We consider Itô 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 a function of the starting point the solutions are Hölder continuous with any exponent $<1$. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.
We give a brief overview of the history of the Sobolev mixed norm theory of linear elliptic and parabolic equations and the recent development in this theory based on the Rubio de Francia extrapolation theorem. A self contained proof of this theorem along with other relevant tools of Real Analysis are also presented as well as an application to mixed norm estimates for fully nonlinear equations.
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 $b\in L_{d}(\mathbb{R}^{d})$. We show that each of them is strong Markov with strong Feller transition semigroup $T_{t}$, which is also a continuous bounded semigroup in $L_{d_{0}}(\mathbb{R}^{d})$ for some $d_{0}\in (d/2, d)$. We show that $T_{t}$, $t>0$, has a kernel $p_{t}(x,y)$ which is summable in $y$ to the power of $d_{0}/(d_{0}-1)$. This leads to the parabolic Aleksandrov estimate with power of summability $d_{0}$ instead of the usual $d+1$. For the probabilistic solutions, associated with such a process, of the problem $Lu=f$ in a bounded domain $D\subset\mathbb{R}^{d}$ with boundary condition $u=g$, where $f\in L_{d_{0}}(D)$ and $g$ is bounded, we show that it is Hölder continuous. Parabolic version of this problem is treated as well. We also prove Harnack's inequality for harmonic and caloric functions associated with such a process. Finally, we show that the probabilistic solutions are $L_{d_{0}}$-viscosity solutions.
In subdomains of $\mathbb{R}^{d}$ we consider uniformly elliptic equations $H\big(v( x),D v( x),D^{2}v( x), x\big)=0$ with the growth of $H$ with respect to $|Dv|$ controlled by the product of a function from $L_{d}$ times $|Dv|$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_{0}\in(d/2,d)$ such that for any $p\in(d_{0},d)$ the equation with prescribed continuous boundary data has a solution in class $W^{2}_{p,\text{loc}}$. Our results are new even if $H$ is linear.
For Itô 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 elliptic and parabolic operators in $L_{p}$ and $L_{p+1}$, respectively, where $p\geq d$.
We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $\varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $\varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev spaces of not the original Isaacs equation but of its appropriate modification. We also give an example of a uniformly nondegenerate game where our assumptions are not satisfied and where we conjecture that there are no not only optimal Markov but even $\varepsilon$-optimal adjoint (time-homogeneous) Markov strategies for one of the players.
We give a simple proof that for a continuous local martingale $M_{t}$ $$ \liminf_{\varepsilon\downarrow0}\varepsilon \log Ee^{(1-\varepsilon) \langle M\rangle_{\infty}/2}<\infty \Longrightarrow E\exp(M_{\infty}-\langle M\rangle_{\infty}/2)=1. $$
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, and be almost in VMO (vanishing mean oscillation) with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic Itô equations in the class of "good" solutions (which is nonempty). This also implies uniqueness of a Green's function in the class of "good" ones (which is always nonempty).
We give an appropriate version of the result in the paper by Dipierro, Savin, and Valdinoci for different, not necessarily fractional, powers of the Laplacian.
We prove several pointwise estimates for solutions of linear elliptic (parabolic) equations with measurable coefficients in smooth domains (cylinders) through the weighted $L_{d}$ ($L_{d+1}$)-norm of the free term. The weights allow the free term to blow up near the (latteral) boundary. We also present weighted estimates for occupation times of diffusion processes.
We give error estimates in Peng's central limit theorem for not necessarily nondegenerate case. The exposition uses the language of the classical probability theory instead of the language of the theory of sublinear expectations. We only consider the one-dimensional case. The higher dimensional extension is straightforward.
We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.
In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.
We present an example of smooth quasi-convex functions in the positive octant of $\mathbb{R}^{3}$ which cannot be obtained as the images of convex smooth functions under a monotone smooth mappings of $\mathbb{R}$.
We investigate the $C^{1+α}$-regularity of solutions of parabolic equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity solutions which are in $C^{1+α}_{loc}$. We allow $H$ to be just measurable in $t$ and for its principal part to have sufficiently small discontinuities as a function of~$x$. No Lipschitz continuity of $H$ with respect to $v,Dv$ is required.
We establish the existence of solutions of fully nonlinear parabolic second-order equations like $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$ in smooth cylinders 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 fixed 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 $t,x$ 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 $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$.
We give an elementary proof of a basic "energy" equality in the theory of evolution equations in the setting, which usually starts with introducing $V\subset H=H^{*}\subset V^{*}$.