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 - https://www.scopus.com/pages/publications/62649112453
UR - https://projecteuclid.org/all/euclid.ejp
U2 - 10.1214/EJP.v14-604
DO - 10.1214/EJP.v14-604
M3 - Article
AN - SCOPUS:62649112453
SN - 1083-6489
VL - 14
SP - 314
EP - 340
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
ER -