Journal of Evolution Equations

, Volume 19, Issue 4, pp 997–1040 | Cite as

Smooth measures and capacities associated with nonlocal parabolic operators

  • Tomasz KlimsiakEmail author
  • Andrzej Rozkosz
Open Access


We consider a family \(\{L_t,\, t\in [0,T]\}\) of closed operators generated by a family of regular (non-symmetric) Dirichlet forms \(\{(B^{(t)},V),t\in [0,T]\}\) on \(L^2(E;m)\). We show that a bounded (signed) measure \(\mu \) on \((0,T)\times E\) is smooth, i.e. charges no set of zero parabolic capacity associated with \(\frac{\partial }{\partial t}+L_t\), if and only if \(\mu \) is of the form \(\mu =f\cdot m_1+g_1+\partial _tg_2\) with \(f\in L^1((0,T)\times E;\mathrm{d}t\otimes m)\), \(g_1\in L^2(0,T;V')\), \(g_2\in L^2(0,T;V)\). We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator \(\frac{\partial }{\partial t}+L_t\) and a functional from the dual \({{\mathcal {W}}}'\) of the space \({{\mathcal {W}}}=\{u\in L^2(0,T;V):\partial _t u\in L^2(0,T;V')\}\) on the right-hand side of the equation.


Dirichlet form Parabolic capacity Smooth measure Hunt process Additive functional 

Mathematics Subject Classification

Primary 31C25 Secondary 35K58 31C15 60J45 



This work was supported by Polish National Science Centre (Grant No. 2016/23/B/ST1/01543).


  1. 1.
    Beznea, L., Cîmpean, I.: Quasimartingales associated to Markov processes. Trans. Amer. Math. Soc. 370 (2018) 7761–7787.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York and London 1968.zbMATHGoogle Scholar
  3. 3.
    Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 539–551.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, New York, 2011.zbMATHGoogle Scholar
  5. 5.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Publishing Co., Amsterdam, 1978.zbMATHGoogle Scholar
  6. 6.
    Droniou, J., Porretta, A., Prignet, A.: Parabolic Capacity and Soft Measures for Nonlinear Equations. Potential Anal. 19 (2003) 99–161.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fukushima, M.: On semi-martingale characterizations of functionals of symmetric Markov processes. Electron. J. Probab. 4 (1999), no. 18, 32 pp.Google Scholar
  8. 8.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Second revised and extended edition. Walter de Gruyter, Berlin 2011.zbMATHGoogle Scholar
  9. 9.
    Fukushima, M., Sato, K., Taniguchi, S.: On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28 (1991) 517–535.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Getoor R.K., Sharpe, M.J.: Naturality, standardness, and weak duality for Markov processes. Z. Wahrsch. Verw. Gebiete 67 (1984) 1–62.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Klimsiak, T.: Reflected BSDEs on filtered probability spaces. Stochastic Process. Appl. 125 (2015) 4204–4241.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klimsiak, T.: Semi-Dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations. J. Funct. Anal. 268 (2015) 1205–1240.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klimsiak, T.: Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form. J. Evol. Equ. 18 (2018) 681–713.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klimsiak, T., Rozkosz, A.: Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form. NoDEA Nonlinear Differential Equations Appl. 22 (2015) 1911–1934.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Klimsiak, T., Rozkosz, A.: On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form. Bull. Polish Acad. Sci. Math. 65 (2017) 45–56.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Klimsiak, T., Rozkosz, A.: On the structure of diffuse measures for parabolic capacities. arXiv:1808.06422.
  17. 17.
    Lions, J.-L.: Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires. Dunod, Gauthier Villars, Paris 1969.zbMATHGoogle Scholar
  18. 18.
    Meyer, P.-A.: Fonctionnelles multiplicatives et additives de Markov. Ann. Inst. Fourier, 12 (1962) 125–230.CrossRefGoogle Scholar
  19. 19.
    Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non–Symmetric) Dirichlet Forms. Springer–Verlag, Berlin 1992.CrossRefGoogle Scholar
  20. 20.
    Oshima, Y.: On a construction of Markov processes associated with time dependent Dirichlet spaces. Forum Math. 4 (1992) 395–415.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Oshima, Y.: Some properties of Markov processes associated with time dependent Dirichlet forms. Osaka J. Math. 29 (1992) 103–127.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Oshima, Y.: Time-dependent Dirichlet forms and related stochastic calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004) 281–316.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Oshima, Y.: Semi-Dirichlet Forms and Markov Processes. Walter de Gruyter, Berlin 2013.CrossRefGoogle Scholar
  24. 24.
    Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura Appl. 187 (2008) 563–604.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Petitta, F., Ponce, A.C., Porretta, A.: Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11 (2011) 861–905.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pierre, M.: Problèmes d’évolution avec contraintes unilatérales et potentiel paraboliques. Comm. Partial Differential Equations 4 (1979) 1149–1197.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pierre, M.: Représentant précis d’un potentiel parabolique. Séminaire de Théorie du Potentiel, Paris, No. 5, Lecture Notes in Math. 814 (1980) 186–228.Google Scholar
  28. 28.
    Pierre, M.: Parabolic capacity and Sobolev spaces. SIAM J. Math. Anal. 14 (1983) 522–533.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Revuz, D.: Mesures associees aux fonctionnelles additives de Markov I. Trans. Amer. Math. Soc. 148 (1970) 501–531.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Stannat, W.: Dirichlet forms and Markov processes: a generalized framework including both elliptic and parabolic cases. Potential Anal. 8 (1998) 21–60.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stannat, W.: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics. Mem. Amer. Math. Soc. 142 (1999), no. 678.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Trutnau, G.: Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions. Osaka J. Math. 37 (2000) 315–343.MathSciNetzbMATHGoogle Scholar
  33. 33.
    Trutnau, G.: Analytic properties of smooth measures in the non-sectorial case, (Stochastic Analysis of Jump processes and Related Topics), Kyoto University, Departamental Bulletin Paper (2010), 1672: pp. 45–62,

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Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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