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

   
@article{EJP604,
	author = {Mohammud Foondun},
	title = {Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {14},
	year = {2009},
	keywords = {Integro-differential operators. Harnack inequality. Heat kernel, Holder continuity},
	abstract = {We consider the Dirichlet form given by  $$ {\cal E}(f,f) = \frac{1}{2}\int_{R^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx$$ $$ + \int_{R^d \times R^d} (f(y)-f(x))^2J(x,y)dxdy.$$ Under the assumption that the ${a_{ij}}$ 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 $\cal E$.},
	pages = {no. 11, 314-340},
	issn = {1083-6489},
	doi = {10.1214/EJP.v14-604},    
        url = {http://ejp.ejpecp.org/article/view/604}}