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On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale. (English. Russian original) Zbl 0625.35041
Math. USSR, Sb. 58, 207-221 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 2, 207-221 (1986).
Estimates of the maximum of the solution of a nondivergent parabolic (elliptic) equation of the second order via $$L_ p$$-norm of the right side are established. The coefficients of the equation may be irregular (locally unbounded). In terms of semimartingales these estimates give new $$L_ p$$-estimates of solutions of stochastic differential equations (SDEs) with irregular coefficients. The solvability of SDEs may be studied and the theory of control of such processes may be built with the help of these estimates.
Reviewer: A.Veretennikov

##### MSC:
 35K20 Initial-boundary value problems for second-order parabolic equations 60G48 Generalizations of martingales 35B50 Maximum principles in context of PDEs 60E15 Inequalities; stochastic orderings
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