On diffusion processes with drift in $$L_{d}$$ L d
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On diffusion processes with drift in Ld N. V. Krylov1 Received: 1 February 2020 / Revised: 5 September 2020 / Accepted: 22 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators L = a i j Di j + bi Di , acting on functions on Rd , with measurable coefficients, bounded and uniformly elliptic a and b ∈ L d (Rd ). We show that each of them is strong Markov with strong Feller transition semigroup Tt , which is also a continuous bounded semigroup in L d0 (Rd ) for some d0 ∈ (d/2, d). We show that Tt , t > 0, has a kernel pt (x, y) which is summable in y to the power of d0 /(d0 − 1). This leads to the parabolic Aleksandrov estimate with power of summability d0 instead of the usual d + 1. For the probabilistic solution, associated with such a process, of the problem Lu = f in a bounded domain D ⊂ Rd with boundary condition u = g, where f ∈ L d0 (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 d0 -viscosity solutions. Keywords Itô equations · Markov processes · Diffusion processes Mathematics Subject Classification 60J60 · 60J35
1 Introduction Let Rd be a Euclidean space of points x = (x 1 , ..., x d ). For a fixed throughout the article δ ∈ (0, 1) define Sδ as the set of d × d symmetric matrices whose eigenvalues are between δ and δ −1 . Fix a constant b ∈ (0, ∞). In this article we consider and discuss only uniformly nondegenerate processes with bounded diffusion coefficient.
B 1
N. V. Krylov [email protected] University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA
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N. V. Krylov
Assumption 1.1 We are given a Borel measurable Sδ -valued function a = a(x) and a Borel measurable Rd -valued function b = b(x) such that b L d (Rd ) ≤ b. Define Di =
∂ , ∂ xi
Di j = Di D j ,
L = (1/2)a i j (x)Di j + bi (x)Di .
(1.1)
The goal of this article is to investigate (time-homogeneous Markov) quasi-diffusion processes corresponding to L. In the more modern terminology from [18] these are called diffusion processes, but at this point and later on we will follow the terminology from [5] in which the notion of diffusion processes is defined differently from [18]. The definition of time-homogeneous diffusion processes first appeared in the book by Dynkin in 1963, [5], where he also constructs diffusion processes corresponding to elliptic operators as in (1.1) with bounded and Hölder continuous coefficients, such that the matrix (ai j (x)) is uniformly strictly positive. If xt (x), t ≥ 0, is a family of continuous processes on Rd , parametrized by x ∈ Rd , and the family is a diffusion process corresponding to the above L in Dynkin’s sense, then, for any bounded domain D ⊂ Rd and smooth function u, u(x) = E x u(xτ D ) −
τD
Lu(xt ) dt ,
0
where τ D = τ D (
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