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

2009/01/26 by Foondun, Mohammud · 6 citations
#60J35 #60J75 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0901.4127

Abstract

We consider the Dirichlet form given by \sE(f,f)&=&1/2∫_\bR^d∑_i,j=1^d a_ij(x)(∂ f(x))/(∂ x_i) (∂ f(x))/(∂ x_j) dx &+&∫_{\bR^d× \bR^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 \sE.

Cited by

• Interior C^1,α regularity of mixed local-nonlocal (p,q)-energy minimizers for p≤ sq
• Regularity theory for mixed local-nonlocal problem involving general stable operators
• Two alternative proofs of weak Harnack inequality for mixed local and nonlocal p-Laplace equations with a nonhomogeneity
• Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group
• Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

Related

• Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms
• Spectral bounds and heat kernel upper estimates for Dirichlet forms
• Weak parabolic Harnack inequality and Hölder regularity for non-local Dirichlet forms
• Robust estimates for elliptic nonlocal operators on doubling spaces
• Heat kernel estimates for general symmetric pure jump Dirichlet forms