The Annals of Probability

Convergence and regularity of probability laws by using an interpolation method

Vlad Bally and Lucia Caramellino

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Fournier and Printems [Bernoulli 16 (2010) 343–360] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Hölder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. By employing some Besov space techniques, Debussche and Romito [Probab. Theory Related Fields 158 (2014) 575–596] have substantially improved the result of Fournier and Printems. In our paper, we show that this kind of problem naturally fits in the framework of interpolation spaces: we prove an interpolation inequality (see Proposition 2.5) which allows to state (and even to slightly improve) the above absolute continuity result. Moreover, it turns out that the above interpolation inequality has applications in a completely different framework: we use it in order to estimate the error in total variance distance in some convergence theorems.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 1110-1159.

Received: June 2015
Revised: October 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B70: Interpolation between normed linear spaces [See also 46M35]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Regularity of probability laws Orlicz spaces Hermite polynomials interpolation spaces Malliavin calculus integration by parts formulas


Bally, Vlad; Caramellino, Lucia. Convergence and regularity of probability laws by using an interpolation method. Ann. Probab. 45 (2017), no. 2, 1110--1159. doi:10.1214/15-AOP1082.

Export citation


  • [1] Bally, V. and Caramellino, L. (2011). Riesz transform and integration by parts formulas for random variables. Stochastic Process. Appl. 121 1332–1355.
  • [2] Bally, V. and Caramellino, L. (2013). Regularity of Wiener functionals under a Hörmander type condition of order one. Preprint. Available at arXiv:1307.3942.
  • [3] Bally, V. and Caramellino, L. (2014). On the distances between probability density functions. Electron. J. Probab. 19 no. 110, 33.
  • [4] Bally, V. and Clément, E. (2011). Integration by parts formula and applications to equations with jumps. Probab. Theory Related Fields 151 613–657.
  • [5] Bally, V. and Clément, E. (2011). Integration by parts formula with respect to jump times for stochastic differential equations. In Stochastic Analysis 2010 7–29. Springer, Heidelberg.
  • [6] Bally, V. and Fournier, N. (2011). Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probab. Theory Related Fields 151 659–704.
  • [7] Bally, V. and Pardoux, E. (1998). Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9 27–64.
  • [8] Bennett, C. and Sharpley, R. (1988). Interpolation of Operators. Pure and Applied Mathematics 129. Academic Press, Boston, MA.
  • [9] Debussche, A. and Fournier, N. (2013). Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. J. Funct. Anal. 264 1757–1778.
  • [10] Debussche, A. and Romito, M. (2014). Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise. Probab. Theory Related Fields 158 575–596.
  • [11] De Marco, S. (2011). Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 1282–1321.
  • [12] Dziubański, J. (1997). Triebel–Lizorkin spaces associated with Laguerre and Hermite expansions. Proc. Amer. Math. Soc. 125 3547–3554.
  • [13] Epperson, J. (1997). Hermite and Laguerre wave packet expansions. Studia Math. 126 199–217.
  • [14] Fournier, N. (2002). Jumping SDEs: Absolute continuity using monotonicity. Stochastic Process. Appl. 98 317–330.
  • [15] Fournier, N. (2015). Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 860–897.
  • [16] Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimensional processes. Bernoulli 16 343–360.
  • [17] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [18] Kebaier, A. and Kohatsu-Higa, A. (2008). An optimal control variance reduction method for density estimation. Stochastic Process. Appl. 118 2143–2180.
  • [19] Kosmol, P. and Müller-Wichards, D. (2011). Optimization in Function Spaces: With Stability Considerations in Orlicz Spaces. De Gruyter Series in Nonlinear Analysis and Applications 13. de Gruyter, Berlin.
  • [20] Löcherbach, E., Loukianova, D. and Loukianov, O. (2011). Polynomial bounds in the ergodic theorem for one-dimensional diffusions and integrability of hitting times. Ann. Inst. Henri Poincaré Probab. Stat. 47 425–449.
  • [21] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin.
  • [22] Ninomiya, S. and Victoir, N. (2008). Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15 107–121.
  • [23] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
  • [24] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [25] Pardoux, É. and Zhang, T. S. (1993). Absolute continuity of the law of the solution of a parabolic SPDE. J. Funct. Anal. 112 447–458.
  • [26] Petrushev, P. and Xu, Y. (2008). Decomposition of spaces of distributions induced by Hermite expansions. J. Fourier Anal. Appl. 14 372–414.
  • [27] Sanz-Solé, M. (2005). Malliavin Calculus with Applications to Stochastic Partial Differential Equations. EPFL Press, Lausanne.
  • [28] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509 (1991).
  • [29] Triebel, H. (1995). Interpolation Theory, Function Spaces, Differential Operators, 2nd ed. Johann Ambrosius Barth, Heidelberg.
  • [30] Triebel, H. (2006). Theory of Function Spaces. III. Monographs in Mathematics 100. Birkhäuser, Basel.
  • [31] Veretennikov, A. Yu. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115–127.
  • [32] Veretennikov, A. Yu. and Klokov, S. A. (2004). On the subexponential rate of mixing for Markov processes. Teor. Veroyatn. Primen. 49 21–35.
  • [33] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.