Harmonic extension technique: probabilistic and analytic perspectives
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908476131770368 |
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| author | Kwaśnicki, Mateusz |
| author_facet | Kwaśnicki, Mateusz |
| contents | Consider a path of the reflected Brownian motion in the half-plane $\{y \ge 0\}$, and erase its part contained in the interior $\{y > 0\}$. What is left is, in an appropriate sense, a path of a jump-type stochastic process on the line $\{y = 0\}$ -- the boundary trace of the reflected Brownian motion. It is well known that this process is in fact the 1-stable Lévy process, also known as the Cauchy process.
The PDE interpretation of the above fact is the following. Consider a bounded harmonic function $u$ in the half-plane $\{y > 0\}$, with sufficiently smooth boundary values $f$. Let $g$ denote the normal derivative of $u$ at the boundary. The mapping $f \mapsto g$ is known as the Dirichlet-to-Neumann operator, and it is again well known that this operator coincides with the square root of the 1-D Laplace operator $-Δ$. Thus, the Dirichlet-to-Neumann operator coincides with the generator of the boundary trace process.
Molchanov and Ostrovskii proved that isotropic stable Lévy processes are boundary traces of appropriate diffusions in half-spaces. Caffarelli and Silvestre gave a PDE counterpart of this result: the fractional Laplace operator is the Dirichlet-to-Neumann operator for an appropriate second-order elliptic equation in the half-space. Again, the Dirichlet-to-Neumann operator turns out to be the generator of the boundary trace process.
During my talk I will discuss boundary trace processes and Dirichlet-to-Neumann operators in a more general context. My main goal will be to explain the connections between probabilistic and analytical results. Along the way, I will introduce the necessary machinery: Brownian local times and additive functionals, Krein's spectral theory of strings, and Fourier transform methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic extension technique: probabilistic and analytic perspectives Kwaśnicki, Mateusz Probability Analysis of PDEs Consider a path of the reflected Brownian motion in the half-plane $\{y \ge 0\}$, and erase its part contained in the interior $\{y > 0\}$. What is left is, in an appropriate sense, a path of a jump-type stochastic process on the line $\{y = 0\}$ -- the boundary trace of the reflected Brownian motion. It is well known that this process is in fact the 1-stable Lévy process, also known as the Cauchy process. The PDE interpretation of the above fact is the following. Consider a bounded harmonic function $u$ in the half-plane $\{y > 0\}$, with sufficiently smooth boundary values $f$. Let $g$ denote the normal derivative of $u$ at the boundary. The mapping $f \mapsto g$ is known as the Dirichlet-to-Neumann operator, and it is again well known that this operator coincides with the square root of the 1-D Laplace operator $-Δ$. Thus, the Dirichlet-to-Neumann operator coincides with the generator of the boundary trace process. Molchanov and Ostrovskii proved that isotropic stable Lévy processes are boundary traces of appropriate diffusions in half-spaces. Caffarelli and Silvestre gave a PDE counterpart of this result: the fractional Laplace operator is the Dirichlet-to-Neumann operator for an appropriate second-order elliptic equation in the half-space. Again, the Dirichlet-to-Neumann operator turns out to be the generator of the boundary trace process. During my talk I will discuss boundary trace processes and Dirichlet-to-Neumann operators in a more general context. My main goal will be to explain the connections between probabilistic and analytical results. Along the way, I will introduce the necessary machinery: Brownian local times and additive functionals, Krein's spectral theory of strings, and Fourier transform methods. |
| title | Harmonic extension technique: probabilistic and analytic perspectives |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2409.19118 |