Associate professor at University of Jyväskylä.
Docent of Mathematics

Contact information

Research interests

• nonlinear partial differential equations (parabolic, in particular), stochastic game theory, calculus of variations, nonlinear potential theory.

Conferences and minicourses

• Jyväskylä summer school at University of Jyväskylä, Finland, on 5.-9. Aug 2024. Speakers Juha Kinnunen (Harmonic analysis methods for nonlinear parabolic PDEs), and Julio D. Rossi (Convexity, partial differential equations and game theory)
• Finnish mathematical days 2024, Geometric analysis and PDEs -session
• Minicourse on Tug-of-war games and p-Laplace equation, 10.1.2022-21.1.2022, at Beijing Normal University (Zoom).
• Workshop on stochastic analysis and nonlinear partial differential equations at the Department of Mathematics and Statistics, University of Jyväskylä, on 11.-12.6.2020.
• Jyväskylä summer school at University of Jyväskylä, Finland, on 6.-17.8.2018. Application DL for students outside JyU April 30.
• Nonlinear, nonlocal problems and stochastic methods at Aalto University, Finland, on 7.-9.12.2016.
• Nonlinear PDEs and stochastic methods at the Department of Mathematics and Statistics, University of Jyväskylä, on 10.-13.06.2014.
• Workshop on Stochastics, Harmonic Analysis and PDEs at the Department of Mathematics and Statistics, University of Jyväskylä, on 19.12.-20.12.2013.

Research team in 'Stochastic games and nonlinear partial differential equations'

• PI Mikko Parviainen
• Visiting professor Juan J. Manfredi
• Postdoctoral researcher Saara Sarsa
• Visiting postdoctoral researcher Jarkko Siltakoski
• Doctoral researcher Riku Anttila
• Doctoral researcher Yawen Feng
• Doctoral researcher Huajie Liu

  Finished PhDs and some former members:

• Visiting professor Julio D. Rossi
• Visiting professor Ángel Arroyo
• Postdoc Jarkko Siltakoski
• Postdoc Pablo Blanc
• Postdoc Jeongmin Han
• Postdoc Eero Ruosteenoja
• Academy postdoc Amal Attouchi
• Doctoral researcher Tapio Kurkinen (PhD 2024)
• Jeongmin Han (PhD 2021)
• Jarkko Siltakoski (PhD 2020)
• Joonas Heino (PhD2018)
• Eero Ruosteenoja (PhD2017)

• Part of the Nonlinear partial differential equations research group.
• The group organizes a research seminar on 'Stochastics and partial differential equations' in collaboration with the stochastics group.

Teaching, advising and service

• Editor for Math. Scand.
• Courses and seminars
• I am glad to discuss possible topics for bachelor, master and doctoral theses for example in the fields of partial differential equations, stochastic games and control theory.

  Please contact me by email or come to talk with me any time I am in my office MaD313.

Lecture notes

• Partial differential equations (classical solutions)
• Partial differential equations 2 (weak solutions)
• Viscosity theory (viscosity solutions)
• Notes on tug-of-war games and the p-Laplace equation (game theory and PDEs) SpringerBriefs on PDEs and Data Science, Springer, 2024. Preprint in arXiv.

Research papers and other publications

1. Krylov-Safonov theory for Pucci-type extremal inequalities on random data clouds (with Á. Arroyo and P. Blanc). Preprint in arXiv.
   

2. Second order Sobolev regularity results for the generalized p-parabolic equation (with Y. Feng and S. Sarsa). Preprint in arXiv.
   

3. Dynamic programming principle in cost-efficient sequential design: application to switching measurements (with J. Han and J. Karvanen). Preprint in arXiv.
   

4. A bridge between convexity and quasiconvexity (with P. Blanc and J.D. Rossi). Preprint in arXiv.
   

5. Notes on tug-of-war games and the p-Laplace equation (game theory and PDEs) SpringerBriefs on PDEs and Data Science, Springer, 2024. Preprint in arXiv.
   

6. Hölder estimate for a tug-of-war game with 1<p<2 from Krylov-Safonov regularity theory (with Á. Arroyo). Rev. Mat. Iberoam.40(3): 1023-1044, 2024. Preprint in arXiv.
   

7. Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian (with P. Blanc, J. Han, and E. Ruosteenoja). Potential Anal. 59: 1995-2015, 2023. Preprint in arXiv.
   

8. Hölder regularity for stochastic processes with bounded and measurable increments (with Á. Arroyo and P. Blanc). Ann. Inst. H. Poincaré Anal. Non Linéaire. 40: 215-258, 2023. Preprint in arXiv.
   

9. Asymptotic C^{1,\gamma}-regularity for value functions to uniformly elliptic dynamic programming principles (with P. Blanc and J.D. Rossi). Math. Ann. 387: 1249-1270, 2023. Preprint in arXiv.
   

10. A systematic approach on the second order regularity of solutions to the general parabolic p-Laplace equation (with Y. Feng and S. Sarsa). Calc. Var. Partial Differential Equations 62(204), 2023. Preprint in arXiv.
    

11. Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non-divergence form (with T. Kurkinen and J. Siltakoski). Bull. Lond. Math. Soc. 55(1): 470-489, 2023. Preprint in arXiv.
    

12. Local regularity estimates for general discrete dynamic programming equations (with Á. Arroyo and P. Blanc). J. Math. Pures Appl. 167: 225-256, 2022. Preprint in arXiv.
    

13. On the second order regularity of solutions to the parabolic p-Laplace equation (with Y. Feng and S. Sarsa). Journal of Evolution Equations 22(1): Paper No. 6, 17, 2022. Preprint in arXiv.
    

14. Convergence of dynamic programming principles for the p-Laplacian (with F. del Teso, J. J. Manfredi) Adv. Calc. Var. 15(2):191-212, 2022. Preprint in arXiv.
    

15. Gradient and Lipschitz estimates for tug-of-war type games (with A. Attouchi and H. Luiro) SIAM J. Math. Anal 53(2): 1295-1319, 2021. Preprint in arXiv.
    

16. Asymptotic Hölder Regularity for Ellipsoid Process (with Á. Arroyo) ESAIM Control Optim. Calc. Var. 26: Paper No. 112, pages 31, 2020. Preprint in arXiv.
    

17. Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications (with J. L. Vázquez) Ann. Sc. Norm. Super. Pisa Cl. Sci. 21:303-359, 2020. Preprint in arXiv.
    

18. Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities (with Á. Arroyo, H. Luiro, and E. Ruosteenoja) Potential Anal. 53(2):565-589, 2020. Preprint in arXiv.
    

19. A remark on infinite initial values for quasilinear parabolic equations (with P. Lindqvist) Nonlinear Anal. 194:111391, 2020. Preprint in arXiv.
    

20. The tusk condition and Petrovski criterion for the normalized p-parabolic equation (with A. and J. Björn) J. Lond. Math. Soc. 100 (2):623-643, 2019. Preprint in arXiv.
    

21. Supercaloric functions for the porous medium equation (with J. Kinnunen, P. Lehtelä, and P. Lindqvist) J. Evol. Equ. 19 (1):249-270, 2019. Preprint in arXiv.
    

22. Regularity for nonlinear stochastic games (with H. Luiro). Ann. Inst. H. Poincaré Anal. Non Linéaire. 35 (6):1435-1456, 2018. Preprint in arXiv.
    

23. Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian (with A. Attouchi) Commun. Contemp. Math. 20 (4):1750035, 2018. Preprint in arXiv.
    

24. Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian (with Á. Arroyo and J. Heino) Commun. Pure Appl. Anal. 16 (3):915-944, 2017. Preprint in arXiv.
    

25. Gradient walk and p-harmonic functions (with H. Luiro) Proc. Amer. Math. Soc. 145: 4313-4324, 2017. Preprint in arXiv.
    

26. C^{1,a} regularity for the normalized p-Poisson problem (with A. Attouchi and E. Ruosteenoja) J. Math. Pures Appl. 108 (4): 553-591, 2017. Preprint in arXiv.
    

27. Tug-of-war, market manipulation and option pricing (with K. Nyström). Math. Finance 27 (2), 279-312, 2017. Preprint in arXiv.
    

28. Local regularity for time-dependent tug-of-war games with varying probabilities (with E. Ruosteenoja) J. Differential Equations 261(2): 1357-1398, 2016. Preprint in arXiv.
    

29. Discontinuous gradient constraints and the infinity Laplacian (with P. Juutinen and J.D. Rossi). Int. Math. Res. Not. IMRN 8: 2451-2492, 2016. Preprint in arXiv.
    

30. Shadows of infinities (with T. Kuusi and P. Lindqvist). Ann. Mat. Pura Appl. 195(4): 1185-1206, 2016. Preprint in arXiv.
    

31. Stability of degenerate parabolic Cauchy problems (with T. Lukkari). Commun. Pure Appl. Anal. 14(1): 201-216, 2015. Preprint in arXiv.
    

32. Boundary regularity for degenerate and singular parabolic equations (with A. Björn, J. Björn, and U. Gianazza). Calc. Var. Partial Differential Equations. 52(3): 797-827, 2015. Preprint in arXiv.
    

33. Variational parabolic capacity (with B. Avelin and T. Kuusi). Discrete Contin. Dyn. Syst.35(12):5665-5688, 2015. Preprint pdf.
    

34. Global higher integrability for parabolic quasiminimizers in metric spaces (with M. Masson). Journal d'Analyse Mathematique 126(1):307-339, 2015. Preprint in arXiv.
    

35. On the existence and uniqueness of p-harmonious functions (with H. Luiro and E. Saksman). Differential and Integral Equations 27(3/4):201-216, 2014. Preprint in arXiv.
    

36. Harnack's inequality for p-harmonic functions via stochastic games (with H. Luiro and E. Saksman). Comm. Partial Differential Equations 38(12):1985-2003, 2013. Preprint in arXiv.
    

37. Local higher integrability for parabolic quasiminimizers in metric spaces (with M. Masson, M. Miranda Jr, and F. Paronetto). Ricerche Mat. 62(2):279-305, 2013. Preprint in arXiv.
    

38. Local approximation of superharmonic and superparabolic functions in nonlinear potential theory (with J. Kinnunen and T. Lukkari). J. Fixed Point Theory Appl. 13(1):291-307, 2013. Preprint pdf.
    

39. Nonlinear parabolic capacity and polar sets of superparabolic functions (with J. Kinnunen, R. Korte and T. Kuusi). Math. Ann. 355(4):1349-1381, 2013. Preprint pdf .
    

40. Solutions of nonlinear PDEs in the sense of averages (with B. Kawohl and J.J. Manfredi). J. Math. Pures Appl. 97(2):173-188, 2012. Preprint pdf .
    

41. Irregular time dependent obstacles (with P. Lindqvist). J. Funct. Anal. 263 (8):2458-2482, 2012. Preprint in arXiv.
    

42. Nonhomogeneous variational problems and quasi-minimizers on metric spaces (with J. Gong and J.J. Manfredi). Manuscripta Math. 137(1):247-271, 2012. Preprint in arXiv.
    

43. Dynamic programming principle for tug-of-war games with noise (with J.J. Manfredi and J.D. Rossi). ESAIM Control Optim. Calc. Var. 18:81-90, 2012. Preprint pdf.
    

44. On the definition and properties of p-harmonious functions (with J.J. Manfredi and J.D. Rossi). Ann. Sc. Norm. Super. Pisa Cl. Sci. 11(2):215-241, 2012. Preprint pdf.
    

45. Equivalence of viscosity and weak solutions for the p(x)-Laplacian (with P. Juutinen and T. Lukkari). Ann. Inst. H. Poincaré Anal. Non Linéaire. 27(6):1471-1487, 2010. Preprint pdf.
    

46. An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games (with J.J. Manfredi and J.D. Rossi). SIAM J. Math. Anal. 42(5):2058-2081, 2010. Preprint pdf.
    

47. An existence result for superparabolic functions (with J. Kinnunen and T. Lukkari), J. Funct. Anal. 258:713-728, 2010. Preprint pdf.
    

48. An asymptotic mean value characterization for p-harmonic functions (with J.J. Manfredi and J.D. Rossi). Proc. Amer. Math. Soc. 138(3):881-889, 2010. Preprint pdf.
    

49. Stability for degenerate parabolic equations (with J. Kinnunen), Adv. Calc. Var. 3(1):29-48, 2010. Preprint pdf.
    

50. Self-improving property of nonlinear higher order parabolic systems near the boundary (with V. Bögelein), NoDEA Nonlinear Differential Equations Appl. 17(1):21-54, 2010. Preprint pdf.
    

51. Lebesgue points and the fundamental convergence theorem for superharmonic functions (with A. Björn, and J. Björn), Rev. Mat. Iberoam. 26(1):147-174, 2010. Preprint pdf.
    

52. A connection between a general class of superparabolic functions and supersolutions (with R. Korte, and T. Kuusi), J. Evol. Equ. 10(1):1-20, 2010. Preprint pdf
    

53. Existence for a degenerate Cauchy problem (with T. Kuusi), Manuscripta Math. 128(2):213-249, 2009. Preprint pdf.
    

54. Nonlinear balayage on metric spaces (with A. Björn, J. Björn, and T. Mäkäläinen), Nonlinear Anal. 71(5-6):2153-2171, 2009. Preprint pdf.
    

55. Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations 246:512-540, 2009. Preprint pdf.
    

56. Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl. 188(2):333-358, 2009. Preprint pdf.
    

57. Harnack's inequality for quasiminimizers with non-standard growth conditions (with P. Harjulehto, T. Kuusi, T. Lukkari, and N. Marola), J. Math. Anal. Appl. 344(1): 504-520, 2008. Preprint pdf.
    

58. Global higher integrability for parabolic quasiminimizers in nonsmooth domains, Calc. Var. Partial Differential Equations 31(1):75-98, 2008.

Other:

1. Peliteoria on päätöksenteon tiedettä, jota sovelletaan niin biologiassa kuin ydinasestrategioissa, Tiedeykkönen, Yle, 2/2019.

2. Väitöstutkimuksia epälineaarisista parabolisista yhtälöistä ja systeemeistä (with T. Kuusi), Arkhimedes, 1/2008. pdf.

3. Global higher integrability for nonlinear parabolic partial differential equations in nonsmooth domains, Dissertation, Helsinki University of Technology, Institute of Mathematics, 2007. pdf.

Research groups

• Nonlinear PDE group at Aalto University and University of Jyväskylä.