Abstract
We consider the stochastic differential equation on \(\mathbb {R}^{d}\) given by
where B is a Brownian motion and b is considered to be a distribution of regularity \( > -\frac 12\). We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat-kernel estimates for Γt with explicit dependence on t and the norm of b.
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Acknowledgements
This work was supported by the German Science Foundation (DFG) via the Forschergruppe FOR2402 “Rough paths, stochastic partial differential equations and related topics”. WvZ was supported by the DFG through SPP1590 “Probabilistic Structures in Evolution”. NP thanks the DFG for financial support through the Heisenberg program. The main part of the work was done while NP was employed at Humboldt-Universität zu Berlin and Max-Planck-Institute for Mathematics in the Sciences, Leipzig. The authors are also grateful to the anonymous referees for their valuable feedback, suggestions and careful reading.
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A Appendix
A Appendix
Theorem A.1
Suppose α < 0 and β > 0 are such that α + β > 0. Let \(p,p_{1},p_{2}, q_{1},q_{2} \in [1,\infty ]\) be such that
For all r ≥ q1
Proof
For the proof see also [18, Corollary 2.1.35]. By slightly adapting [2, Theorem 2.82] and by using the Hölder inequality and [2, Theorem 2.79] (for Eq. 4), we obtain implies the following two estimates.
As [2, Theorem 2.52] implies \( \| u v \|_{B_{p,q}^{\alpha +\beta } } \lesssim \|u\|_{B_{p_{1},q_{1}}^{\alpha }} \|v\|_{B_{p_{2},q_{2}}^{\beta }}\), combining the above inequalities proves Eq. 2. □
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Perkowski, N., van Zuijlen, W. Quantitative Heat-Kernel Estimates for Diffusions with Distributional Drift. Potential Anal 59, 731–752 (2023). https://doi.org/10.1007/s11118-021-09984-3
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DOI: https://doi.org/10.1007/s11118-021-09984-3