Feller generators with singular drifts in the critical range
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arXiv
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| Format: | Preprint |
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2024
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| author | Kinzebulatov, D. Semenov, Yu. A. |
| author_facet | Kinzebulatov, D. Semenov, Yu. A. |
| contents | We consider diffusion operator $-Δ+ b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of $b$ (but excluding the borderline value), we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in $L^p$ for $p$ sufficiently large, hence the gain in the assumptions on singular drift.
For the critical borderline value of the magnitude of singularities of $b$, we construct a strongly continuous semigroup in a ``critical'' Orlicz space on $\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \leq p<\infty$ but is slightly weaker than that of $L^\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Feller generators with singular drifts in the critical range Kinzebulatov, D. Semenov, Yu. A. Probability Analysis of PDEs Functional Analysis We consider diffusion operator $-Δ+ b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of $b$ (but excluding the borderline value), we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in $L^p$ for $p$ sufficiently large, hence the gain in the assumptions on singular drift. For the critical borderline value of the magnitude of singularities of $b$, we construct a strongly continuous semigroup in a ``critical'' Orlicz space on $\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \leq p<\infty$ but is slightly weaker than that of $L^\infty$. |
| title | Feller generators with singular drifts in the critical range |
| topic | Probability Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2405.12332 |