Naslov
Heat kernels of non-symmetric jump processes: beyond the stable case

Autori
Kim, Panki ; Song, Renming ; Vondraček, Zoran

Izvornik
Potential analysis (0926-2601) 49 (2018), 1; 37-90

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
heat kernel estimates, subordinate Brownian motion, symmetric L\'evy process, non-symmetric operator, non-symmetric Markov process

Sažetak
Let $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{; ; ; ; ; R}; ; ; ; ; ^d$ with its L\'evy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non- local operator $$ \mathcal{; ; ; ; ; L}; ; ; ; ; ^{; ; ; ; ; \kappa}; ; ; ; ; f(x):= \lim_{; ; ; ; ; \eps \downarrow 0}; ; ; ; ; \int_{; ; ; ; ; \{; ; ; ; ; z \in \R^d: |z|>\eps\}; ; ; ; ; }; ; ; ; ; (f(x+z)- f(x))\kappa(x, z)J(z)\, dz\, , $$ where $\kappa(x, z)$ is a Borel function on $\mathbb{; ; ; ; ; R}; ; ; ; ; ^d\times \\mathbb{; ; ; ; ; R}; ; ; ; ; ^d$ satisfying $0<\kappa_0\le \kappa(x, z)\le \kappa_1$, $\kappa(x, z)=\kappa(x, - z)$ and $|\kappa(x, z)-\kappa(y, z)|\le \kappa_2|x-y|^{; ; ; ; ; \beta}; ; ; ; ; $ for some $\beta\in (0, 1]$. We construct the heat kernel $p^\kappa(t, x, y)$ of $\LL^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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