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On diffusion processes with drift in \(L_{d}\)

Abstract

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 solution, 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.

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Acknowledgements

The author’s sincere thanks are due to T. Yastrzhembskiy for pointing out several mistakes and misprints in the first draft of the paper. The author is also very grateful to the referees whose comments helped improve the presentation.

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Krylov, N.V. On diffusion processes with drift in \(L_{d}\). Probab. Theory Relat. Fields 179, 165–199 (2021). https://doi.org/10.1007/s00440-020-01007-3

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  • DOI: https://doi.org/10.1007/s00440-020-01007-3

Keywords

  • Itô equations
  • Markov processes
  • Diffusion processes

Mathematics Subject Classification

  • 60J60
  • 60J35