On divergence form SPDEs with VMO coefficients
We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space.
math.PR↗
arXiv subjects
N. V. Krylov
Publications and source records attributed to N. V. Krylov.
Elliptic and parabolic second-order PDEs with growing coefficients
We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
math.AP↗
Itô's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes
We prove Itô's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
math.PR↗
On factorizations of smooth nonnegative matrix-values functions and on smooth functions with values in polyhedra
We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron.
math.OC↗
On time-inhomogeneous controlled diffusion processes in domains
Time-inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered. Bellman's principle and its applications to proving the continuity of value functions are investigated.
math.PR↗
A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasilinear operators
A priori estimates for finite-difference approximations for the first and second order derivatives are obtained for solutions of parabolic equations described in the title.
math.NA↗
Parabolic equations with VMO coefficients in spaces with mixed norms
An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on $t$. The method of proving simplifies the methods previously used in the case $p=q$.
math.AP↗
Maximum principle for SPDEs and its applications
The maximum principle for SPDEs is established in multidimensional $C^{1}$ domains. An application is given to proving the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs.
math.PR↗
Parabolic equations with measurable coefficients
We investigate the unique solvability of second order parabolic equations in non-divergence form in $W_p^{1,2}((0,T) \times \bR^d)$, $p \ge 2$. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables.
math.AP↗
Elliptic differential equations with measurable coefficients
We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations.
math.AP↗
Parabolic and elliptic equations with VMO coefficients
An $L_{p}$-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all functions independent of $x$. Weak uniqueness of the martingale problem associated with such equations is obtained.
math.AP↗
On the rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients
We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order $h^{1/2}$ for certain types of finite-difference schemes are obtained.
math.OC↗
On diffusion approximation with discontinuous coefficients
Convergence of stochastic processes with jumps to diffusion processes is investigated in the case when the limit process has discontinuous coefficients. An example is given in which the diffusion approximation of a queueing model yields a diffusion process with discontinuous diffusion and drift coefficients.
math.PR↗