### Abstract

We consider the Dirichlet form given by E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. Under the assumption that the aij are symmetric and uniformly elliptic and with suitable conditions on J, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E.

Original language | English |
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Pages (from-to) | 314-340 |

Number of pages | 27 |

Journal | Electronic Journal of Probability |

Volume | 14 |

DOIs | |

Publication status | Published - 7 Jan 2009 |

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### Keywords

- Harnack inequality
- heat kernel
- Hölder continuity
- integro-differential operators

### Cite this

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**Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part.** / Foondun, Mohammud.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

AU - Foondun, Mohammud

PY - 2009/1/7

Y1 - 2009/1/7

N2 - We consider the Dirichlet form given by E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. Under the assumption that the aij are symmetric and uniformly elliptic and with suitable conditions on J, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E.

AB - We consider the Dirichlet form given by E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. Under the assumption that the aij are symmetric and uniformly elliptic and with suitable conditions on J, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E.

KW - Harnack inequality

KW - heat kernel

KW - Hölder continuity

KW - integro-differential operators

UR - http://www.scopus.com/inward/record.url?scp=62649112453&partnerID=8YFLogxK

UR - https://projecteuclid.org/all/euclid.ejp

U2 - 10.1214/EJP.v14-604

DO - 10.1214/EJP.v14-604

M3 - Article

VL - 14

SP - 314

EP - 340

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -