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Feynman-Kac formula under a finite entropy condition

Abstract

Motivated by entropic optimal transport, we are interested in the Feynman-Kac formula associated to the parabolic equation \( ( {\mathsf {L}}+V)g =0\) with a final nonnegative boundary condition and a Markov generator \( {\mathsf {L}}:= \partial _t + {\mathsf {b}}\!\cdot \!\nabla + \Delta _{ {\mathsf {a}}}/2\). It is well-known that when the drift \( {\mathsf {b}}\), the diffusion matrix \( {\mathsf {a}}\) and the scalar potential V are regular enough and not growing too fast, the classical solution g of this PDE, is represented by the Feynman-Kac formula \( g_t(x)=E_R[\exp \left( \int _{[t,T]} V(s,X_s)\,ds\right) g(X_T)\mid X_t=x] \) where R is the Markov measure with generator \( {\mathsf {L}}\). We do not assume that g, \( {\mathsf {b}}\) and V are regular, and only require that their growth is controlled by a finite entropy condition. These hypotheses are less restrictive than the standard assumptions of the theory of viscosity solutions, and allow for instance V to belong to some Kato class. We prove that g defined by the Feynman-Kac formula belongs to the domain of the extended generator \( {\mathcal {L}}\) of the Markov measure R and satisfies the trajectorial identity: \( [({\mathcal {L}} +V)g] (t,X_t)=0,\ dtdP\text {-}{a.e.}\) where the path measure P is defined by \( P:= f(X_0)\exp \left( \int _{[0,T]}V(t,X_t)\,dt\right) g(X_T)\ R, \) with \( f:{\mathbb {R}}^n\rightarrow [0, \infty )\) another nonnegative function. We also show that the forward drift \( {\mathsf {b}}^P\) of P satisfies \( {\mathsf {b}}^P(t,X_t)=[ {\mathsf {b}}+ {\mathsf {a}}{\widetilde{\nabla }}\log g](t,X_t),\) \(dtdP\text {-}{a.e.},\) where \({\widetilde{\nabla }}\) is some extension of the standard derivative. Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov’s theorem and the Hamilton-Jacobi-Bellman equation satisfied by \(\log g\).

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Notes

  1. Beware, with our notation the role of the wave function \(\Psi \) is played by g, not \(\psi ,\) see (1.17) below.

  2. This is a “local” definition in the sense that this notion probably appears somewhere else with another name.

References

  1. Albeverio, S.: Theory of Dirichlet forms and applications, in Ecole d’Eté de Probabilités de Saint-Flour XXX-2000. Lecture Notes in Mathematics, vol. 1816. Springer, Berlin (2003)

  2. Bernton, E., Ghosal, P., Nutz, M.: Entropic optimal transport: Geometry and large deviations. Preprint arXiv:2102.04397

  3. Bohm, D.: A suggested interpretation of the quantum theory in tems of “hidden’’ variables I. II. Physical Review 85(166–179), 180–193 (1952)

    MathSciNet  Article  Google Scholar 

  4. Cattiaux, P., Conforti, G., Gentil, I., Léonard, C.: Time reversal of diffusion processes under a finite entropy condition. Preprint arXiv:2104.07708

  5. Cattiaux, P., Léonard, C.: Minimization of the Kullback information of diffusion processes. Ann. Inst. H. Poincaré. Probab. Statist. 30, 83–132 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Cattiaux, P., Léonard, C.: Minimization of the Kullback information for some Markov processes. In Seminar on Probability, tome 30 (Univ. Strasbourg, Strasbourg, 1996), volume 1626 of Lecture Notes in Math., pages 288–311. Springer, Berlin, (1996)

  7. Chung, K.L., Zambrini, J.C.: Introduction to random time and quantum randomness. World Scientific Publishing Co., Inc. (2003)

  8. Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation, volume 312 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, (1995)

  9. Conforti, G.: A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost. Probability Theory and Related Fields 174(1), 1–47 (2019)

    MathSciNet  Article  Google Scholar 

  10. Cont, R., Fournié, D.-A.: Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41(1), 109–133 (2013)

    MathSciNet  Article  Google Scholar 

  11. Cruzeiro, A.B., Wu, L., Zambrini, J.-C.: Bernstein processes associated with a Markov process. In: Rebolledo, R. (ed.) Stochastic analysis and mathematical physics, ANESTOC’98. Proceedings of the Third International Workshop, Trends in Mathematics, pp. 41–71. Birkhäuser, Boston (2000)

    Chapter  Google Scholar 

  12. Dellacherie, C., Meyer, P.-A.: Probabilités et Potentiel. Ch. XII à XVI. Théorie du potentiel associée à une résolvante, théorie des processus de Markov. Hermann. Paris, (1987)

  13. Doob, J.L.: Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957)

    MathSciNet  Article  Google Scholar 

  14. Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Classics in Mathematics. Springer, 2nd edition, (2000). (reprint of the 1984 first edition)

  15. Dürr, D., Teufel, S.: Bohmian mechanics. The physics and mathematics of quantum theory. Springer-Verlag, Berlin (2009)

  16. Evans, L.C.: Partial Differential Equations, volume 19 of Graduate Series in Mathematics. American Mathematical Society, (1998)

  17. Fang, S.: Inégalité du type de Poincaré sur l’espace des chemins riemanniens. C. R. Acad. Sci. Paris 318, 257–260 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Feynman, R.: Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  Google Scholar 

  19. Feynman, R., Hibbs, A.: Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics. McGraw-Hill, (1965)

  20. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, volume 25 of Applications of Mathematics. Springer, second edition, (2006)

  21. Föllmer, H.: An entropy approach to the time reversal of diffusion processes. In Stochastic Differential Systems - Filtering and Control, volume 69 of Lecture Notes in Control and Information Sciences, pages 156–163. Springer, (1985)

  22. Föllmer, H.: Time reversal on Wiener space. In Stochastic Processes - Mathematics and Physics, volume 1158 of Lecture Notes in Math., pages 119–129. Springer, Berlin, (1986)

  23. Föllmer, H.: Random fields and diffusion processes, in École d’été de Probabilités de Saint-Flour XV-XVII-1985-87. Lecture Notes in Mathematics, vol. 1362. Springer, Berlin (1988)

  24. Hsu, E.: Analysis on path and loop spaces. volume 5 of IAS/Park City Math. Series. Amer. Math. Soc., (1997)

  25. Jacod, J.: Calcul stochastique et problèmes de martingales, volume 714 of Lecture Notes in Mathematics. Springer, (1979)

  26. Kac, M.: On the distribution of certain Wiener functionals. Trans. Amer. Math. Soc. 65, 1–13 (1949)

    MathSciNet  Article  Google Scholar 

  27. Kac, M.: On some connections between probability theory and differential and integral equations. In Neyman, J. (Ed.), Proc. Second Berkeley Symp. Math. Stat. Prob., pages 189–215. Univ. of California Press, (1951)

  28. Khas’minskii, R.: On positive solutions of the equation \({A}u+{V}u=0\). Theory Probab. Appl. 4, 309–318 (1959)

    Article  Google Scholar 

  29. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel, Dordrecht (1987)

    Book  Google Scholar 

  30. Kunita, H.: Absolute continuity of Markov processes and generators. Nagoya Mathematical Journal 36, 1–26 (1969)

    MathSciNet  Article  Google Scholar 

  31. Kunita, H.: Stochastic flows and stochastic differential equations, volume 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, (1997)

  32. Léonard, C.: Girsanov theory under a finite entropy condition. In Séminaire de probabilités, vol. 44., pages 429–465. Lecture Notes in Mathematics 2046. Springer, (2012)

  33. Léonard, C.: Some properties of path measures. In Séminaire de probabilités, vol. 46., 207–230. Lecture Notes in Mathematics 2123. Springer, (2014)

  34. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4), 1533–1574 (2014)

    MathSciNet  Article  Google Scholar 

  35. Lowther, G.: Cadlag modifications. Almost Sure - A random mathematical blog. https://almostsuremath.com/2009/12/18/cadlag-modifications/

  36. Ma, Z.-M., Röckner, M.: Introduction to the theory of (non-symmetric) Dirichlet forms. Universitext, Springer (1992)

    Book  Google Scholar 

  37. Meyer, P.-A., Zheng, W.A.: Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré. Probab. Statist. 20(4), 353–372 (1984)

    MathSciNet  MATH  Google Scholar 

  38. Meyer, P.-A., Zheng, W.A.: Construction de processus de Nelson réversibles. In Séminaire de probabilités. Tome 19, volume 1123 of Lecture Notes in Mathematics, 12–26. Springer, (1985)

  39. Nelson, E.: Dynamical theories of Brownian motion. Princeton University Press (1967)

  40. Oshima, Y.: Semi-Dirichlet forms and Markov processes, volume 48 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., (2013)

  41. Revuz, D., Yor, M.: Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer, 3rd edition, (1999)

  42. Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144, 144–153 (1931)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Stannat, W.: The theory of generalized Dirichlet forms and its applications in analysis and stochastics, volume 678 of Mem. Amer. Math. Soc. American Mathematical Society, (1999)

  45. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with continuous coefficients, I. Communications on Pure and Applied Mathematics 22(3), 345–400 (1969)

    MathSciNet  Article  Google Scholar 

  46. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with continuous coefficients, II. Communications on Pure and Applied Mathematics 22(4), 479–530 (1969)

    MathSciNet  Article  Google Scholar 

  47. Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Number 233 in Grundlehren der mathematischen Wissenschaften. Springer Verlag, (1979)

  48. von Renesse, M.: An optimal transport view on Schrödinger’s equation. Canad. Math. Bull. 55(4), 858–869 (2011)

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  50. Zambrini, J.-C.: The research program of stochastic deformation (with a view toward geometric mechanics). A series of lectures. Springer, In Stochastic Analysis (2015). arXiv:1212.4186

  51. Zhang, X.: Clark-Ocone formula and variational representation for Poisson functionals. Ann. Probab. 37(2), 506–529 (2009)

    MathSciNet  Article  Google Scholar 

  52. Zheng, W.A.: Tightness results for laws of diffusion processes. Application to stochastic mechanics. Ann. Inst. H. Poincaré. Probab. Statist. 21(2), 103–124 (1985)

    MathSciNet  MATH  Google Scholar 

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Appendices

Appendix A. Carré du champ

Lemma A.2 below is a simplified version of [4, Lemma 3.9], which was used during the proof of Lemma 4.30. For the confort of the reader, we give its detailed proof which is slightly simpler, but essentially the same as [4]’s one.

Let \(Q\in \mathrm {M}(\Omega )\) be a conditionable path measure. Its forward carré du champ is defined by

$$\begin{aligned} \Gamma ^Q(u,v):= {\mathcal {L}}^Q(uv)-u {\mathcal {L}}^Q v-v {\mathcal {L}}^Q u, \quad \ 0\le t\le T, \end{aligned}$$

for any functions uv in \({{\,\mathrm{dom}\,}} {\mathcal {L}}^Q\) such that their product uv also belongs to \({{\,\mathrm{dom}\,}} {\mathcal {L}}^Q\).

The quadratic covariation [u(X), v(X)] is a Q-semimartingale. We denote by \(\langle u(X),v(X)\rangle ^Q\) its bounded variation part, i.e.

$$\begin{aligned} d[u(X),v(X)]_t=d\langle u(X),v(X)\rangle ^Q_t+ dM_t ^{Q, [u,v]}, \qquad {\overline{Q}}\text {-}{a.e.}\end{aligned}$$

where, here and below, or stands for any local Q-martingale. As next lemma indicates, we are interested in situations where the bounded variation process \(\langle u(X),v(X)\rangle ^Q\) is predictable (as a continuous process). Therefore, in the whole article \(\langle u(X),v(X)\rangle ^Q\) is the usual sharp bracket (sometimes called conditional quadratic variation) of stochastic process theory.

Lemma A.1

For any \(u,v\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q\) such that \(uv\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q,\) the process \( \langle u(X),v(X)\rangle ^Q \) is absolutely continuous \(Q\text {-}{a.e.}\) and

$$\begin{aligned} d\langle u({\overline{X}}),v({\overline{X}})\rangle ^Q_t=\Gamma ^Q(u,v)(t, X _{ [0,t]})\,dt, \qquad Q\text {-}{a.e.}\end{aligned}$$

Proof

As a definition of the forward generator \( d(uv)({\overline{X}}_t)= {\mathcal {L}}^Q_t(uv)(t, X _{ [0,t]})\,dt+dM ^{ uv}_t. \) Comparing this expression with (4.23), the Doob-Meyer decomposition theorem gives the announced result. \(\square \)

We say that a process Y can be localized as a bounded (resp. integrable) process if there exists a sequence of stopping times \(( \sigma _k)\) tending almost surely to infinity and such that for each k, the stopped process \(Y^{ \sigma _k}\) is bounded almost surely (resp. integrable).

Lemma A.2

For any conditionable path measure \(Q\in \mathrm {M}(\Omega )\), almost every \(t\in [0,T],\) and any locally bounded functions \(u,v\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q\) such that \(uv\in {{\,\mathrm{dom}\,}} {\mathcal {L}}^Q\), and \(M ^{ Q,[u,v]}\) as defined at Lemma A.1 can be localized as an integrable Q-martingale, there exist an increasing sequence \((\tau _k)\) of Q-integration times of u and v,  and a sequence \((h_n)\) of positive numbers such that \(\lim _{k\rightarrow \infty } \tau _k=\infty ,\) \(Q\text {-}{a.e.}\), \(\lim _{n\rightarrow \infty } h_n=0\) and

$$\begin{aligned} \begin{aligned} \Gamma ^Q (u,v)&(t, X _{ [0,t]})\\&= \lim _{k\rightarrow \infty } \lim _{n\rightarrow \infty } \frac{1}{h_n}E_Q \left. \Big [\{u( {\overline{X}}^{ \tau _k}_{t+h_n})-u( {\overline{X}}^{ \tau _k}_t)\}\{v( {\overline{X}}^{ \tau _k}_{t+h_n})-v( {\overline{X}}^{ \tau _k}_t)\}\,\right| \, X_{[0,t]} \Big ],\quad Q\text {-}{a.e.}\end{aligned} \end{aligned}$$

Proof

Since u and v are assumed to be locally bounded, \(u({\overline{X}})\) and \(v({\overline{X}})\) can be localized as bounded processes. Furthermore, the processes and can also be localized as bounded processes. It follows that the local Q-martingales \(M^u, M^v\), (where \(M^u_t:=u({\overline{X}}_t)-\int _0^t {\mathcal {L}}^Qu(s,X _{ [0,s]})\,ds\)) can also be localized as bounded processes. Localizing as in the proof of Proposition 3.14, it is enough to show that

$$\begin{aligned} \begin{aligned}&\lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |E_Q \big [h ^{ -1}\{u( X_{t+h})-u( X_t)\}\{v( X_{t+h})-v( X_t)\}\,\big |\, X _{[0,t]}\big ]\\&- \Gamma ^Q(u,v)(t,X _{ [0,t]})\,\Big |\,dt =0, \end{aligned} \end{aligned}$$
(A.3)

and we can assume that all the above mentioned processes are bounded.

For each \(0\le t\le T-h\) with \(0<h\le T,\)

$$\begin{aligned}{}[u( X_{ t+h})-&u( X_t)][v( X _{ t+h})-v( X_t)]\\&=\Big [ \int _t ^{ t+h} dM^u_s + \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\Big ] \Big [ \int _t ^{ t+h} dM^v_s + \int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds\Big ]\\&= A_t^h+B_t^h+C_t^h+D_t^h, \qquad Q\text {-}{a.e.}, \end{aligned}$$

where

$$\begin{aligned} A_t^h&= \int _t ^{ t+h} dM^u_s\ \int _t ^{ t+h} dM^v_s,B_t^h= \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\ \int _t ^{ t+h} dM^v_s,\\ C_t^h&= \int _t ^{ t+h} {\mathcal {L}}^Q v({\overline{X}}_s)\,ds\ \int _t ^{ t+h} dM^u_s,D_t^h= \int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\ \int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds. \end{aligned}$$

Let us control \(A^h_t.\) Denoting \(U_{t,s}:=M^u_s-M^u_t\) and \(V_{t,s}:=M^v_s-M^v_t,\)

$$\begin{aligned} A_t^h&= \int _t ^{ t+h} d(U_{t,s}V_{t,s})\\&= \int _t ^{ t+h} U_{t,s} dM^v_s + \int _t ^{ t+h} V_{t,s} dM^u_s + \int _t ^{ t+h} dM ^{Q, [u,v]}_s + \int _t ^{ t+h} d \langle M^u,M^v\rangle ^Q _s, \end{aligned}$$

and with Lemma A.1

$$\begin{aligned} h ^{ -1}E_Q\left. [A^h_t\,\right| \, X _{ [0,t]}]= h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\mid X _{ [0,t]}]\,ds. \end{aligned}$$
(A.4)

Remark that the boundedness properties obtained above by localization, together with the extra assumption that \(M ^{Q, [u,v]}\) is integrable, justify the cancelation of the expectations of the martingale terms.

Let us control \(B^h\):

$$\begin{aligned} h ^{ -1}E_Q \int _0 ^{ T-h}|B_t^h|\,dt&\le E_Q \int _0 ^{ T-h} h ^{ -1}\Big |\int _t ^{ t+h} {\mathcal {L}}^Qu({\overline{X}}_s)\,ds\Big |\ |M^v_{ t+h}-M^v_t|\, dt \\&= E_Q \int _0 ^{ T-h} \Big | k^h*( {\mathcal {L}}^Qu)({\overline{X}}_t)\Big |\ |M^v_{ t+h}-M^v_t|\, dt\\&= o _{ h\rightarrow 0^+}(1) , \end{aligned}$$

where we took \(k^h:= h ^{ -1} {\mathbf {1}}_{ [-h,0]}\) as our convolution kernel. The last identity is a consequence of Lemma 2.11 under the assumption \( {\mathcal {L}}^Qu({\overline{X}})\in L^1({\overline{Q}})\) (because is bounded), the uniform boundedness and right-continuity of \(M^v\) and the dominated convergence theorem.

Similarly, \(\lim _{h\rightarrow 0^+}h ^{ -1}E_Q \int _0 ^{ T-h}|C_t^h|\,dt=0.\)

The control of \(D^h\) is analogous:

$$\begin{aligned} h ^{ -1}E_Q \int _0 ^{ T-h}|D_t^h|\,dt&\le E_Q \int _0 ^{ T-h} \Big | k^h*( {\mathcal {L}}^Qu)({\overline{X}}_t)\Big |\ \Big |\int _t ^{ t+h} {\mathcal {L}}^Qv({\overline{X}}_s)\,ds\Big |\, dt\\&= o _{ h\rightarrow 0^+}(1) , \end{aligned}$$

thanks to the uniform boundedness of \(\int _{[0,T]}| {\mathcal {L}}^Qv({\overline{X}}_s)|\,ds.\)

Putting everything together, we obtain

$$\begin{aligned}&\lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |E_Q \big [h ^{ -1}\{u( X_{t+h})-u( X_t)\}\{v( X_{t+h})-v( X_t)\}\,\Big |\, X _{ [0,t]}\big ]\\&\qquad -h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\,\big |\, X _{ [0,t]}]\,ds\Big |\,dt =0. \end{aligned}$$

On the other hand, with Corollary 2.16 we obtain

$$\begin{aligned} \lim _{h\rightarrow 0^+}E_Q\int _0 ^{ T-h} \Big |h ^{ -1}\int _t ^{ t+h} E_Q[\Gamma ^Q(u,v)({\overline{X}}_s)\mid X _{ [0,t]}]\,ds - \Gamma ^Q(u,v)(t,X _{ [0,t]})\Big |\,dt =0. \end{aligned}$$

The limit (A.3) follows from these last two limits. \(\square \)

Appendix B. About Nelson velocities

This section refers to the diffusion measure Q of Sect. 4. Its content is not used directly in this article. We propose it to the reader to stress the importance for our purpose of considering the relative momentum field \( \beta ^{ Q|R}\) rather than the absolute velocity \( {\mathsf {v}}^Q.\)

Denoting by \(\mathrm {Id}\) the identity mapping on \({\mathbb {R}}^n,\) we see that the vector field \( {\mathsf {v}}^Q\) appearing in the martingale problems satisfy \( {\mathsf {v}}^Q= {\mathcal {L}}^Q[\mathrm {Id}]\). Because of the identification \( {\mathcal {L}}^Q= L^Q\) which was obtained at Sect. 2, one suspects that \( {\mathsf {v}}^Q\) should satisfy

$$\begin{aligned} {\mathsf {v}}^Q_t= L^Q_t[\mathrm {Id}]= \lim _{h\rightarrow 0^+}E_Q\left. \Big ( \frac{X _{ t+h}-X_t}{h}\,\right| \, X _{ [0,t]}\Big ), \end{aligned}$$

whenever this expression is meaningful. The r.h.s. of this identity is the forward Nelson velocity of Q. But in general it is not well defined, due to a possible lack of integrability. In order to give sense to limits of this type in a general setting, one must introduce integration times and work as in Proposition 3.14. Next result presents a situation where integration times can be avoided.

Proposition B.1

Under the hypothesis (4.9), suppose that \( {\mathsf {a}}\) is bounded from above. Then, the limit

$$\begin{aligned} {\mathsf {v}}^{ Q|R}(t, \omega )= \lim _{h\rightarrow 0^+}E_Q\left. \Big ( \frac{X _{ t+h}-X_t}{h} - \frac{1}{h} \int _{ [t,t+h]} {\mathsf {b}}({\overline{X}}_s)\,ds\,\right| \, X _{ [0,t]}=\omega _{ [0,t]}\Big ), \quad (t, \omega )\in {\overline{\Omega }}, \end{aligned}$$

takes place in \(L^2({\overline{Q}}).\)

Proof

Under (4.9), we know that \(E _{ {\overline{Q}}}|\beta ^{ Q|R}|^2_ {\mathsf {a}}< \infty .\) Because of the assumed upper boundedness of \( {\mathsf {a}}\), this implies that \(E _{ {\overline{Q}}}| {\mathsf {v}}^{ Q|R}|^2< \infty ,\) where \( {\mathsf {v}}^{ Q|R}:= {\mathsf {a}}\beta ^{ Q|R}.\) Rewrite the assertion: \(Q\in {{\,\mathrm{MP}\,}}( {\mathsf {a}}, {\mathsf {b}}+ {\mathsf {v}}^{ Q|R})\) as:

$$\begin{aligned} X _{ t+h}-X_t-\int _{ [t,t+h]} {\mathsf {b}}({\overline{X}}_s)\,ds =\int _{ [t,t+h]} {\mathsf {v}}^{ Q|R}({\overline{X}}_s)\,ds +M^Q _{ t+h}-M^Q_t, \end{aligned}$$

where \(M^Q\) is a local Q-martingale. The assumption \(E _{ {\overline{Q}}}| {\mathsf {v}}^{ Q|R}|^2< \infty ,\) expressed with the Euclidean norm rather than the Riemannian norm permits us to apply the convolution Lemma 2.11 to \(v=a \beta \) componentwise with \(p=2\). The critical step where this is used is Jensen’s inequality right below (2.12). With this at hand, proceeding as in the proof of Proposition 2.18 leads us to the announced result. \(\square \)

Remarks B.2

 

  1. (a)

    In the setting of this proposition, if the Nelson velocity \( L^R[\mathrm {Id}]\) is ill defined because \(E_R\int _{[0,T]}| {\mathsf {b}}_t|\,dt= \infty \), it might happen that \( L^Q[\mathrm {Id}]\) is also ill defined. Nevertheless, we have: \(\int _{[0,T]}| {\mathsf {b}}_t|\,dt<\infty , R\text {-}{a.e.},\) and \(Q\in {{\,\mathrm{MP}\,}}( {\mathsf {a}}, {\mathsf {v}}^Q)\) where \( {\mathsf {v}}^Q= {\mathsf {b}}+ {\mathsf {v}}^{ Q|R}\) satisfies \(\int _{[0,T]}| {\mathsf {v}}^Q_t|\,dt< \infty , Q\text {-}{a.e.}\)

  2. (b)

    Requiring that the diffusion matrix field \( {\mathsf {a}}\) is upper bounded is not a strong restriction for the applications, because in general temperature is upper bounded.

  3. (c)

    If \( {\mathsf {a}}\) is only locally bounded, then there exists a sequence \((h_n)\) of positive numbers such that \(\lim _{n\rightarrow \infty } h_n=0\) and the limit

    $$\begin{aligned}&{\mathsf {v}}^{ Q|R}(t, \omega ) = \lim _{k\rightarrow \infty } \lim _{n\rightarrow \infty } E_Q\\&\left. \Big ( \frac{X ^{ \tau _k}_{ t+h_n}-X^{\tau _k} _t}{h_n} -\frac{1}{h_n} \int _{ [t,t+h_n]}{\mathbf {1}}_{ \left\{ s\le \tau _k\right\} } {\mathsf {b}}({\overline{X}}_s)\,ds\,\right| \, X _{ [0,t]}=\omega _{ [0,t]}\Big ) \end{aligned}$$

    holds \({\overline{Q}}\text {-}{a.e.},\) where for each integer \(k\ge 1,\) \( \tau _k:= \inf \{t\in [0,T]: |X_t|\ge k\}.\) The proof of this statement is similar to Proposition 3.14’s proof.

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Léonard, C. Feynman-Kac formula under a finite entropy condition. Probab. Theory Relat. Fields (2022). https://doi.org/10.1007/s00440-022-01155-8

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Keywords

  • Diffusion processes
  • Feynman-Kac formula
  • Hamilton-Jacobi-Bellman equation
  • Relative entropy
  • Extended generator
  • Stochastic derivative
  • Entropic optimal transport
  • Kato class

Mathematics Subject Classification

  • 35K20
  • 60H30
  • 60J60