## Analysis & PDE

### Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations

Seiichiro Kusuoka

#### Abstract

We consider nondegenerate second-order parabolic partial differential equations in nondivergence form with bounded measurable coefficients (not necessary continuous). Under certain assumptions weaker than the Hölder continuity of the coefficients, we obtain Gaussian bounds and Hölder continuity of the fundamental solution with respect to the initial point. Our proofs employ pinned diffusion processes for the probabilistic representation of fundamental solutions and the coupling method.

#### Article information

Source
Anal. PDE, Volume 8, Number 1 (2015), 1-32.

Dates
Received: 16 October 2013
Revised: 6 November 2014
Accepted: 21 December 2014
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1511895957

Digital Object Identifier
doi:10.2140/apde.2015.8.1

Mathematical Reviews number (MathSciNet)
MR3336920

Zentralblatt MATH identifier
1316.35064

#### Citation

Kusuoka, Seiichiro. Hölder continuity and bounds for fundamental solutions to nondivergence form parabolic equations. Anal. PDE 8 (2015), no. 1, 1--32. doi:10.2140/apde.2015.8.1. https://projecteuclid.org/euclid.apde/1511895957

#### References

• D. G. Aronson, “Bounds for the fundamental solution of a parabolic equation”, Bull. Amer. Math. Soc. 73 (1967), 890–896.
• V. I. Bogachev, M. R öckner, and S. V. Shaposhnikov, “\cyr Global'naya regulyarnost' i otsenki resheniĭ parabolicheskikh uravneniĭ”, Teor. Veroyatn. Primen. 50:4 (2005), 652–674. Translated as “Global regularity and bounds for solutions of parabolic equations for probability measures” in Theory Probab. Appl. 50:4 (2006), 561–581.
• V. I. Bogachev, N. V. Krylov, and M. R öckner, “\cyr E1llipticheskie i parabolicheskie uravneniya dlya mer”, Uspekhi Mat. Nauk 64:6(390) (2009), 5–116. Translated as “Elliptic and parabolic equations for measures” in Russian Math. Surveys 64:6 (2009), 973–1078.
• Z.-Q. Chen and T. Kumagai, “Heat kernel estimates for stable-like processes on $d$-sets”, Stochastic Process. Appl. 108:1 (2003), 27–62.
• M. Cranston, “Gradient estimates on manifolds using coupling”, J. Funct. Anal. 99:1 (1991), 110–124.
• L. Escauriaza, “Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form”, Comm. Partial Differential Equations 25:5-6 (2000), 821–845.
• E. B. Fabes and C. E. Kenig, “Examples of singular parabolic measures and singular transition probability densities”, Duke Math. J. 48:4 (1981), 845–856.
• A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
• E. De Giorgi, “Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. $(3)$ 3 (1957), 25–43. Reprinted, pp. 167–184 in Selected papers, edited by G. Dal Maso et al., Springer, New York, 2006; translated as “On the differentiability and the analyticity of extremals of regular multiple integrals”, ibid., 149–166.
• N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library 24, North-Holland, Amsterdam, 1989.
• I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics 113, Springer, New York, 1991.
• S. Karrmann, “Gaussian estimates for second-order operators with unbounded coefficients”, J. Math. Anal. Appl. 258:1 (2001), 320–348.
• N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics 12, American Mathematical Society, Providence, RI, 1996.
• S. Kusuoka and D. W. Stroock, “Applications of the Malliavin calculus, II”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32:1 (1985), 1–76.
• O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, \cyr \em Lineĭnye i kvazilineĭnye uravneniya parabolicheskogo tipa, Nauka, Moscow, 1967. Translated as Linear and quasilinear equations of parabolic type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, RI, 1968.
• P. D. Lax and A. N. Milgram, “Parabolic equations”, pp. 167–190 in Contributions to the theory of partial differential equations, edited by L. Bers et al., Annals of Mathematics Studies 33, Princeton University Press, 1954.
• T. Lindvall and L. C. G. Rogers, “Coupling of multidimensional diffusions by reflection”, Ann. Probab. 14:3 (1986), 860–872.
• G. Metafune, D. Pallara, and A. Rhandi, “Global properties of transition probabilities of singular diffusions”, Teor. Veroyatn. Primen. 54:1 (2009), 116–148.
• J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math. 80 (1958), 931–954.
• F. O. Porper and S. D. Eidelman, “\cyr Dvustoronnie otsenki fundamental'nykh resheniĭ parabolicheskikh uravneniĭ vtorogo poryadka i nekotorye ikh prilozheniya”, Uspekhi Mat. Nauk 39:3(237) (1984), 107–156. Translated as “Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications” in Russian Math. Surveys 39:3 (1984), 119–178.
• F. O. Porper and S. D. Eidelman, “\cyr Svoĭstva resheniĭ parabolicheskikh uravneniĭ vtorogo poryadka s mladshimi chlenami”, Trudy Moskov. Mat. Obshch. 54 (1992), 118–159. Translated as “Properties of solutions of second-order parabolic equations with lower-order terms” in Trans. Moscow Math. Soc. 54 (1993), 101–137.
• D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften 293, Springer, Berlin, 1999.
• D. W. Stroock, “Diffusion semigroups corresponding to uniformly elliptic divergence form operators”, pp. 316–347 in Séminaire de Probabilités XXII, edited by J. Azéma et al., Lecture Notes in Math. 1321, Springer, Berlin, 1988.
• D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften 233, Springer, Berlin, 1979.