Skip to main content

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations


In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial- and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman–Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.

This is a preview of subscription content, access via your institution.


  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure Appl. Math., vol. 140. Academic Press, New York (2003)

    MATH  Google Scholar 

  2. Bernstein, S.: Sur les liaisons entre les grandeurs aléatoires. In: Verhandlungen des Internationalen Mathematikerkongress, vol. 1, pp. 288–309 (1932)

    Google Scholar 

  3. Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae. Probability and Its Applications Series. Birkhäuser, Basel (2000)

    Google Scholar 

  5. Carlen, E.A.: Conservative diffusions. Commun. Math. Phys. 94, 293–315 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chueshov, I.D., Vuillermot, P.-A.: Long-time behavior of solutions to a class of quasilinear parabolic equations with random coefficients. Ann. Inst. Henri Poincaré 15, 191–232 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Cruzeiro, A.B., Zambrini, J.C.: Malliavin calculus and Euclidean quantum mechanics, I. Functional calculus. J. Funct. Anal. 96, 62–95 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cruzeiro, A.B., Wu, L., Zambrini, J.C.: Bernstein processes associated with Markov processes. In: Rebolledo, R. (ed.) Stochastic Analysis and Mathematical Physics. Birkhäuser, Basel (2000)

    Google Scholar 

  9. Dobrushin, R.L., Sukhov, Y.M., Fritz, J.: A.N. Kolmogorov—the founder of the theory of reversible Markov processes. Russ. Math. Surv. 43(II), 157–182 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dynkin, E.B.: Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society Colloquium Publications, vol. 50. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  11. Eidelman, S.D., Ivasis̆en, S.D.: Investigation of the Green matrix for a homogeneous parabolic boundary value problem. Trans. Mosc. Math. Soc. 23, 179–242 (1970)

    MathSciNet  Google Scholar 

  12. Eidelman, S.D., Zhitarashu, N.V.: Parabolic Boundary Value Problems, Operator Theory. Advances and Applications, vol. 101. Birkhäuser, Basel (1998)

    Book  Google Scholar 

  13. Freidlin, M.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, vol. 109. Princeton University Press, Princeton (1985)

    MATH  Google Scholar 

  14. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  15. Gihman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes, III. Classics in Mathematics Series. Springer, New York (2007)

    Google Scholar 

  16. Gulisashvili, A., van Casteren, J.A.: Non-Autonomous Kato Classes and Feynman–Kac Propagators. World Scientific, Singapore (2006)

    Book  MATH  Google Scholar 

  17. Hsu, P.: Probabilistic approach to the Neumann problem. Commun. Pure Appl. Math. 38, 445–472 (1985)

    Article  MATH  Google Scholar 

  18. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  19. Jamison, B.: Reciprocal processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 30, 65–86 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113. Springer, New York (1987)

    Google Scholar 

  21. Kolmogorov, A.N.: On the reversibility of the statistical laws of nature. In: Shiryayev, A.N. (ed.) Selected Works of A.N. Kolmogorov. Mathematics and its Applications (Soviet Series), vol. 26. Kluwer, Boston (1992)

    Google Scholar 

  22. Ladyz̆enskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS Translations of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence (1968)

    Google Scholar 

  23. Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  25. Privault, N., Zambrini, J.C.: Markovian bridges and reversible diffusion processes with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 40, 599–633 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sato, K., Ueno, T.: Multi-dimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4, 529–605 (1964)

    MathSciNet  Google Scholar 

  27. Schrödinger, E.: Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. Henri Poincaré 2, 269–310 (1932)

    Google Scholar 

  28. Schwartz, L.: Théorie des Noyaux. In: Proceedings of the International Congress of Mathematicians, vol. 1, pp. 220–230 (1950)

    Google Scholar 

  29. Solonnikov, V.A.: On boundary-value problems for linear parabolic systems of differential equations of general form. In: Proceedings of the Steklov Institute of Mathematics, vol. 83 (1965)

    Google Scholar 

  30. Tanabe, H.: Equations of Evolution. Monographs and Studies in Mathematics, vol. 6. Pitman, London (1979)

    MATH  Google Scholar 

  31. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163–177 (1979)

    MATH  MathSciNet  Google Scholar 

  32. Van Casteren, J.A.: Markov Processes, Feller Semigroups and Evolution Equations. World Scientific, Singapore (2010)

    Google Scholar 

  33. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  34. Weinberger, H.F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover, New York (1995)

    Google Scholar 

  35. Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. 41, 327–340 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. Math. Phys. 27, 2307–2330 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references


The research of both authors was supported by the FCT of the Portuguese government under Grant PTDC/MAT/69635/2006, and by the Mathematical Physics Group of the University of Lisbon under Grant ISFL/1/208. The first author is also indebted to Madalina Deaconu and Elton Hsu for stimulating discussions and correspondence on the theme of reflected diffusions. Last but not least, he wishes to thank the Complexo Interdisciplinar da Universidade de Lisboa and the ETH-Forschungsinstitut für Mathematik in Zurich where parts of this work were completed for their financial support and warm hospitality.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pierre A. Vuillermot.

Appendix: A Variational Construction of Weak Solutions in L 2(D)

Appendix: A Variational Construction of Weak Solutions in L 2(D)

The solutions u φ and v ψ we used throughout this article are generated by two evolution systems, U A (t,s)0≤stT and \(U_{A}^{\ast}(t,s)_{0\leq s\leq t\leq T}\) on L 2(D). We show here how to construct these evolution systems by applying the standard methods of [30], under the following hypotheses regarding the coefficients k, l and V in (1) and (2):


The function \(k:D\times[ 0,T ] \mapsto \mathbb{R}^{d^{2}}\) is matrix-valued and for every i,j∈{1,…,d} we have k i,j =k j,i L (D×(0,T)); moreover, there exists a finite constant \(\underline{k}>0\) such that the inequality

$$ \bigl( k(x,t)q,q \bigr)_{\mathbb{R}^{d}}\geq\underline{k}| q |^{2} $$

holds uniformly in (x,t)∈D×[0,T] for all q∈ℝd. Finally, there exist finite constants c >0, \(\beta\in( \frac{1}{2},1 ] \) such that the Hölder continuity estimate

$$\max_{i,j\in\{ 1,\ldots,d \} }\bigl| k_{i,j}(x,t)-k_{i,j}(x,s)\bigr| \leq c_{\ast}| t-s|^{\beta}$$

is valid for every xD and all s,t∈[0,T].

As for the lower-order differential operators we assume that the following hypotheses are valid, where we assume without restricting the generality that the constants c and β are the same as in hypothesis (K′):


Each component of the vector-field l:D×[0,T]↦ℝd satisfies l i L (D×(0,T)). Moreover, the Hölder continuity estimate

$$\max_{i\in\{ 1,\ldots,d \} }\bigl| l_{i}(x,t)-l_{i} (x,s)\bigr| \leq c_{\ast}| t-s|^{\beta}$$

holds for every xD and all s,t∈[0,T].


The function V:D×(0,T)↦ℝ is such that VL (D×(0,T)) and satisfies

$$\bigl| V(x,t)-V(x,s)\bigr| \leq c_{\ast}| t-s |^{\beta}$$

for every xD and all s,t∈[0,T].

Moreover, both the initial condition φ and the final condition ψ are real-valued and the following hypothesis holds:


We have φ,ψL 2(D).


In the variational theory we are reviewing here we observe that the Hölder continuity requirement relative to the time variable in hypotheses (K′), (L′) and (V′) is stronger than that of hypotheses (K), (L) and (V), since \(\beta\in( \frac{1}{2},1 ] \) whereas \(\frac{\alpha}{2}\in( 0,\frac{1}{2} ) \). However, it is easy to show by uniqueness arguments that the evolution operators U A (t,s)0≤stT and \(U_{A}^{\ast }(t,s)_{0\leq s\leq t\leq T}\) introduced in Sect. 2 are identical to those constructed below. The reason why \(\beta\in( \frac{1}{2},1 ] \) is required here is intimately tied up with the variational structure of the problem, and is thoroughly discussed in [30].

Under the preceding three conditions, it is easily verified that the quadratic form a:[0,TH 1(DH 1(D)↦ℂ defined by

satisfies the estimates


for all s,t∈[0,T] and all f,hH 1(D), where ∥.∥2 and ∥.∥1,2 stand for the usual norms in L 2(D) and H 1(D), respectively, and where \(( .\,,. )_{\mathbb{C}^{d}}\) denotes the Hermitian inner product in ℂd. Consequently, the formal elliptic operator

$$A(t):=-\frac{1}{2}\operatorname{div} \bigl( k(.\,,t)\nabla\bigr) + \bigl( l(.,t),\nabla\bigr)_{\mathbb{C}^{d}}+V(.\,,t) $$

corresponding to the right-hand side of (1) can be realized as a regularly accretive operator defined on some time-dependent and dense domain \(\mathcal{D(}A(t))\subset L^{2}(D)\), and as such generates an evolution system U A (t,s)0≤stT in L 2(D) given by

$$ U_{A}(t,s)f(x)= \begin{cases} f(x)&\mbox{if}\ t=s,\\ \int_{D}dy\,g_{A}(x,t;y,s)f(y)&\mbox{if}\ t>s \end{cases} $$

for every fL 2(D), where g A denotes the parabolic Green function associated with (1). Indeed, all these assertions follow directly from estimates (135)–(137) and the general theory developed in Sect. 5.4 of [30], together with Schwartz’s kernel theorem which guarantees the existence of g A (see [28] for a summary of the many possible applications of that theorem).

In a similar way, the Hermitian conjugate form

$$a^{\ast} ( t,f,h ) :=\overline{a ( t,h,f ) }$$

is associated with the linear operator A (t) adjoint to A(t), which in turn generates the adjoint evolution system

$$ U_{A}^{\ast}(t,s)f(x)= \begin{cases} f(x)&\mbox{if}\ t=s,\\ \int_{D}dy\,g_{A}^{\ast}(x,s;y,t)f(y)&\mbox{if}\ t>s, \end{cases} $$

where \(G_{A}^{\ast}\) is the parabolic Green function associated with (2) that satisfies the relation

$$g_{A}^{\ast}(x,s;y,t)=g_{A}(y,t;x,s) $$

for all s,t∈[0,T] with t>s.

The important features of (138) and (139) are that they provide the real-valued functions defined by

$$ u_{\varphi}(x,t):=U_{A}(t,0)\varphi(x)=\int_{D}dy\,g_{A}(x,t;y,0) \varphi(y),\quad t\in( 0,T ] $$


$$ v_{\psi}(x,t):=U_{A}^{\ast}(T,t)\psi(x)=\int _{D}dy\,g_{A}^{\ast}(x,t;y,T)\psi(y),\quad t\in[ 0,T ) , $$

which satisfy

$$\biggl( \frac{\partial}{\partial t}u_{\varphi}(.\,,t),h \biggr)_{2} +a\bigl(t,u_{\varphi}(.\,,t),h\bigr)=0,\quad t\in( 0,T ] $$


$$- \biggl( \frac{\partial}{\partial t}v_{\psi}(.\,,t),h \biggr )_{2}+a^{\ast } \bigl(t,v_{\psi}(.\,,t),h\bigr)=0,\quad t\in[ 0,T ) $$

for every hH 1(D), respectively, where (. ,.)2 stands for the usual inner product in L 2(D). Moreover, we have u φ ,v ψ L 2(D×(0,T)), so that (140) and (141) provide weak solutions to (1) and (2), respectively (see e.g. Sect. 5.5 in [30]).

These solutions are those which ultimately possess the properties listed in Lemma 1 of Sect. 2, according to the above remark regarding the Hölder regularity in time.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vuillermot, P.A., Zambrini, J.C. Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations. J Theor Probab 27, 449–492 (2014).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


  • Diffusion processes
  • Parabolic partial differential equations

Mathematics Subject Classification

  • 35K20
  • 60H30
  • 60K99