## Revista Matemática Iberoamericana

### A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

#### Abstract

In this paper, we consider the following type of non-local (pseudo-differential) operators $\mathcal{L}$ on $\mathbb{R}^d$: \begin{align*} \mathcal{L} u(x) = \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\+ \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x)) J(x, y) dy, \end{align*} where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\mathbb{R}^d$ that is uniformly elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\mathbb{R}^d \times \mathbb{R}^d$ satisfying certain conditions. Corresponding to $\mathcal{L}$ is a symmetric strong Markov process $X$ on $\mathbb{R}^d$ that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of $\mathcal{L}$ and parabolic Harnack principle for positive parabolic functions of $\mathcal{L}$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\mathcal{L}$ and jump-diffusion $X$. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on $\mathbb{R}^d$. To establish these results, we employ methods from both probability theory and analysis.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 551-589.

Dates
First available in Project Euclid: 4 June 2010

https://projecteuclid.org/euclid.rmi/1275671311

Mathematical Reviews number (MathSciNet)
MR2677007

Zentralblatt MATH identifier
1200.60065

#### Citation

Chen, Zhen-Qing; Kumagai, Takashi. A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoamericana 26 (2010), no. 2, 551--589. https://projecteuclid.org/euclid.rmi/1275671311

#### References

• Barlow, M.T.: Diffusions on fractals. In Lectures on probability theory and statistics (Saint-Flour, 1995), 1-21. Lectures Notes in Math 1690. Springer, Berlin, 1998.
• Barlow, M.T., Bass, R.F., Chen, Z.Q. and Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963-1999.
• Barlow, M.T., Bass, R.F. and Kumagai, T.: Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261 (2009), no. 2, 297-320.
• Barlow, M.T., Grigor'yan, A. and Kumagai, T.: Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626 (2009), 135-157.
• Bass, R.F., Kumagai, T. and Uemura, T.: Convergence of symmetric Markov chains on $\mathbbZ^d$. To appear in Probab. Theory Relat. Fields.
• Carlen, E.A., Kusuoka, S. and Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245-287.
• Chen, Z.Q., Kim, P. and Kumagai, T.: Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math. Ann. 342 (2008), no. 4, 833-883.
• Chen, Z.Q. and Kumagai, T.: Heat kernel estimates for stable-like processes on $d$-sets. Stochastic Process. Appl. 108 (2003), no. 1, 27-62.
• Chen, Z.Q. and Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008), no. 1-2, 277-317.
• Chung, K.L.: Greenian bounds for Markov processes. Potential Anal. 1 (1992), no. 1, 83-92.
• Fabes, E.B. and Stroock, D.W.: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 (1986), no. 4, 327-338.
• Foondun, M.: Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part. Electron. J. Probab. 14 (2009), no. 11, 314-340.
• Saloff-Coste, L.: Aspects of Sobolev-type inequalities. London Mathematical Society Lecture Note Series 289. Cambridge University Press, Cambridge, 2002.
• Song, R. and Vondraček, Z.: Parabolic Harnack inequality for the mixture of Brownian motion and stable process. Tohoku Math. J. (2) 59 (2007), no. 1, 1-19.
• Stroock, D.W.: Diffusion semigroup corresponding to uniformly elliptic divergence form operator. In Séminaire de Probabilités, XXII, 316-347. Lecture Notes in Math. 1321. Springer, Berlin, 1988.
• Stós, A.: Symmetric $\alpha$-stable processes on $d$-sets. Bull. Polish Acad. Sci. Math. 48 (2000), no. 3, 237-245.