On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

Marco Carfagnini; Maria Gordina

REVIEW 3 major objections 4 minor 19 references

On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Brownian motion on the Heisenberg group has an Onsager-Machlup functional equal to minus half the squared horizontal energy, so small-tube probabilities around horizontal curves are governed by the same energy law as in Euclidean space.

desk verdict First genuine attempt at an Onsager-Machlup functional for a hypoelliptic diffusion, with a plausible formula, but the proof relies on unsupported small-ball comparisons across equivalent metrics and matched radii — deserves a serious referee, not desk rejection. read the letter →

arxiv 1908.09182 v4 pith:KUBKI2J7 submitted 2019-08-24 math.PR

Marco Carfagnini , Maria Gordina This is my paper · ORCID

classification math.PR MSC 58J6560J6035R0353C17

keywords Onsager-MachlupfunctionalHeisenberggrouphypoellipticBrownianmotionCarnot-CarathéodorydistanceCameron-Martin-Girsanovtheoremsub-Riemanniangeometrytubeprobabilityasymptoticsstochasticexponential

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that Brownian motion on the Heisenberg group has an Onsager-Machlup functional of the same quadratic form as Euclidean Brownian motion. For any two horizontal finite-energy curves $\varphi$ and $\psi$, the ratio of the probabilities that the hypoelliptic Brownian path stays inside an $\varepsilon$-tube around $\varphi$ versus around $\psi$ converges, as $\varepsilon$ tends to zero, to $\exp(-\frac{1}{2}\|\varphi\|^2_{\mathcal{H}(H)} + \frac{1}{2}\|\psi\|^2_{\mathcal{H}(H)})$. This is the first Onsager-Machlup result for a hypoelliptic diffusion, and it matters because the Heisenberg group is the simplest nontrivial sub-Riemannian space, where the Riemannian comparison and curvature techniques used in the classical results are unavailable. The proof replaces those tools with the group's left-invariance and a Cameron-Martin-Girsanov shift of the two-dimensional Brownian motion that drives the hypoelliptic process.

What carries the argument

The argument runs on three pieces working together. First, because the Carnot-Carathéodory distance is left-invariant and equivalent to the homogeneous distance $\rho$, the tube probability around a curve $\varphi$ is rewritten as the probability that the left-translated process $u^\varphi_t=\varphi(t)^{-1}g_t$ stays within $\varepsilon$ of the identity. Second, for horizontal $\varphi$ the left Maurer-Cartan form of $u^\varphi$ is explicit: its horizontal part is the shifted Brownian motion $W_t-\varphi(t)$, and its vertical part is a stochastic-area integral plus a drift built from $\varphi'$. Third, the proof applies the Cameron-Martin-Girsanov theorem to the driving two-dimensional Brownian motion, implemented by the stochastic exponential $E^\gamma_1$ for a horizontal curve $\gamma$ whose Maurer-Cartan form is $c_\gamma=c_\varphi-c_\psi$, so that the law of $u^\varphi$ under a tilted measure matches the law of an auxiliary process $z$ that stays close to $u^\psi$. A conditioning lemma (Lemma 4.8) turns conditional exponential moments into the exponential of inner products, and the energy difference $-\frac{1}{2}\|\varphi\|^2+\frac{1}{2}\|\psi\|^2$ emerges.

What would settle it

Compute the small-ball probabilities $P(\sup_{t\in[0,1]} \rho(u^\psi_t,e)<\varepsilon)$ and $P(\sup_{t\in[0,1]} d_{cc}(g_t,\psi(t))<\varepsilon)$ to the same order in $\varepsilon$ for one explicit horizontal path $\psi$, such as a line segment. If their ratio does not tend to 1 as $\varepsilon\to 0$, or if the limit changes when $\varepsilon$ is replaced by the mismatched radius appearing in the inclusion argument, then the asymptotic interchange behind Lemma 4.7 fails and the claimed ratio in Proposition 4.9 is not established.

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Extended reading notes

Core claim

The central claim is Theorem 3.13: if $H$ is the Heisenberg group with Carnot-Carathéodory distance $d_{cc}$, $g_t$ is the hypoelliptic Brownian motion, and $\varphi,\psi$ belong to the Cameron-Martin space $\mathcal{H}(H)$ of horizontal finite-energy curves, then the small-tube probability ratio satisfies $$\lim_{\varepsilon\to 0} \frac{P\left(\sup_{t\in[0,1]} d_{cc}(g_t,\varphi(t))<\varepsilon\right)}{P\left(\sup_{t\in[0,1]} d_{cc}(g_t,\psi(t))<\varepsilon\right)}=\exp\left(-\frac{1}{2}\|\varphi\|^2_{\mathcal{H}(H)}+\frac{1}{2}\|\psi\|^2_{\mathcal{H}(H)}\right).$$ In particular, the Onsager-Machlup functional is $L(p,v)=-\frac{1}{2}\|v\|^2_{H_p}$ for horizontal tangent vectors $v\in H_p$. In words, among horizontal paths, the most probable path in a small tube is the one with the smallest squared horizontal energy, exactly as in the flat Euclidean case.

Load-bearing premise

The proof leans on treating the Carnot-Carathéodory tube probability and the homogeneous-distance tube probability as asymptotically identical at small radius, although the distance equivalence only gives inclusions with mismatched radii; if those asymptotic probabilities differ, the theorem's ratio is not established.

Editorial extensions

If this is right

For any two horizontal finite-energy paths, the small-tube probability ratio has a finite, nonzero limit that depends only on the difference of their Cameron-Martin energies, so the Heisenberg Brownian motion has genuine Onsager-Machlup asymptotics in the sup-norm and Carnot-Carathéodory sense.
The most probable horizontal path in the small-tube limit minimizes the squared horizontal norm $\int_0^1 |c_\varphi(s)|^2\,ds$; the vertical coordinate enters only through the horizontality condition, not through an additional potential or curvature term.
The class of processes with a known Onsager-Machlup functional is extended from elliptic diffusions on Riemannian manifolds to a hypoelliptic diffusion on a sub-Riemannian manifold, with the Heisenberg group as the first example.
The Cameron-Martin space $\mathcal{H}(H)$ plays the same role as in the Euclidean case: exactly the horizontal finite-energy curves are the paths for which the tube probability has a nontrivial limiting ratio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same left-translation plus Girsanov argument goes through on any Carnot group, the Onsager-Machlup functional would again be minus half the squared horizontal norm, making the Euclidean formula a universal sub-Riemannian law.
The proof routes around curvature by using the group product as a substitute for parallel transport; one testable consequence is that the same construction should work for hypoelliptic diffusions on any nilpotent Lie group with a comparable left-invariant distance, regardless of curvature data.
A numerical check of the radius-matching step would be to simulate the ratio of the Carnot-Carathéodory and homogeneous tube probabilities for decreasing $\varepsilon$; any systematic drift away from 1 would locate the failure in the unproved small-ball regular variation rather than in the Girsanov computation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear —,

First thing to know: this is the first genuine attempt at an Onsager-Machlup functional for a hypoelliptic diffusion, and the formula it claims — L(p,v) = −½‖v‖² for horizontal v, with no curvature correction — is exactly what the elliptic analogy suggests for the Heisenberg group. The formula may well be true. The machinery, however, does not prove it in this version.

The novel part is real: the authors avoid Riemannian geometry entirely, using left-invariance and the explicit group structure to reduce the tube problem to two-dimensional Brownian motion plus a conditional exponential moment. The definition of the Cameron-Martin space of finite-energy horizontal curves is clean, and the Maurer-Cartan computations in Section 4.1 are correct and useful. If the paper is revised, these pieces will survive.

The soft spot is where the proof changes distance and then changes process. Equation (3.15) asserts that the Carnot-Carathéodory tube and the rho-tube have the same limit. From the equivalence c·rho ≤ d_cc ≤ C·rho you only get inclusions at radii ε/c and Cε. For that to yield equality of limits you need the small-ball probability to be regular in ε — i.e., P(aε)/P(ε) → 1 — and for a hypoelliptic diffusion with decay like exp(−K/ε²) that is false. The same defect appears in Lemma 4.7: the inclusions land at radius sqrt(3ε² + εCγ) ~ sqrt(ε), not ε, so the claimed equality of limits in (4.15) and (4.16) is not a consequence of the lemma. The conditional-expectation estimate in Proposition 4.9 inherits the problem. This is a load-bearing gap, not a cosmetic one.

I don't see circularity or any attempt to fit the answer; the paper is honest and the computations are straightforward. The result is plausible and the topic is important. The right response is not to throw the paper out but to send it to a referee who knows small-ball estimates for hypoelliptic processes, with the clear expectation that the tube-comparison estimates need real work.

Referee Report

3 major / 4 minor

Summary. The paper claims to determine the Onsager-Machlup functional for the hypoelliptic Brownian motion on the Heisenberg group. For horizontal finite-energy curves φ,ψ starting at the identity, the main result (Theorem 3.13) states that the ratio of Carnot-Carathéodory tube probabilities converges to exp(-1/2||φ||^2_{H(H)} + 1/2||ψ||^2_{H(H)}). The proof proceeds in two stages: first, by left-invariance and the assumed asymptotic equivalence of the Carnot-Carathéodory and homogeneous distances (Eq. (3.15)), the problem is reduced to tube probabilities for the processes u^φ_t = φ(t)^{-1}g_t around the identity. Second, a Girsanov transformation and an auxiliary process z_t are used to compare u^φ and u^ψ, leading via Lemma 4.8 to the claimed exponential ratio. The paper also introduces the notion of finite-energy horizontal curves and the Cameron-Martin space H(H).

Significance. If the result were established, it would be a notable first example of an Onsager-Machlup functional in a hypoelliptic setting, with the natural candidate L(p,v)=-1/2||v||^2_{H_p}. The use of the group structure and Girsanov transformation is a sensible strategy, and the proposed functional is consistent with the Euclidean and Riemannian cases. However, the proof as written has decisive gaps at exactly the points where small-ball asymptotics need to be controlled: the equivalence of tube probabilities under different distances and the comparison of the auxiliary processes u^ψ and z are not justified by the estimates provided. Because these gaps affect the central claim, the manuscript is not ready for publication in its present form.

major comments (3)

[Section 3.2, Eq. (3.15)] The equality of limits of tube probabilities under d_cc and ρ does not follow from the bi-Lipschitz equivalence (3.12). Equivalent metrics only give inclusions of the form {d_cc<ε}⊆{ρ<c^{-1}ε} and {ρ<ε}⊆{d_cc<Cε}, which determine the small-ball limit only up to a scale factor in the radius. The equality in (3.15) requires a further regularity property of the small-ball probability under radius scaling, and no such property is stated or proved. Since (3.15) is used in (4.3) and (4.8) as the first reduction step, this gap is load-bearing for Theorem 3.13.
[Lemma 4.7] The proof of Lemma 4.7 establishes only one-sided inclusions at mismatched radii: if ρ(u^ψ,e)<ε, then ρ(z,e)<r(ε), and conversely, where r(ε)=sqrt(3ε^2+εC_γ). Since r(ε)/ε→∞ as ε→0, these inclusions do not imply that the limits (4.15) and (4.16) exist together or are equal. For small-ball probabilities of the expected polynomial order P(ε)≍ε^a, the ratio P(r(ε))/P(ε)≍(r(ε)/ε)^a diverges, so the asserted equality would generally fail. The lemma needs a genuine small-ball asymptotic comparison under radius scaling, which is not provided; the current argument cannot support the substitution of P(ρ(z,e)<ε) for P(ρ(u^ψ,e)<ε) in Proposition 4.9.
[Proposition 4.9] The step after the identity for ∫⟨cγ,dW⟩ is not justified. The proof claims that the conditional exponential moments of ∫⟨cγ(s),dW(s)⟩ given the tube A_ε = {sup ρ(u^φ,e)<ε} converge to exp(c⟨γ,φ⟩), which is equivalent to asserting that the stochastic integral ∫⟨cγ(s),dW_s-φ'(s)ds⟩ is asymptotically negligible on A_ε. No estimate for this term on the tube is given; this is a non-trivial tube estimate of the same nature as Lemma 4.7, and it is exactly the kind of small-ball control the paper lacks. Consequently, the application of Lemma 4.8 is not supported, and Proposition 4.9, which is the core of Theorem 3.13, is not established.

minor comments (4)

[Eq. (4.11)] The third coordinate of γ is defined with an integral from 0 to 1, which appears to be a typo for an integral from 0 to t. As written, γ has zero third component derivative and is not a horizontal curve in general; please correct the formula, since the proof of Proposition 4.9 relies on γ∈H(H).
[Lemma 4.7, proof] In the final line of the proof, 'sup_{t∈[0,1]} ρ(u^ψ_t(t),e)' should read 'sup_{t∈[0,1]} ρ(u^ψ_t,e)'.
[References] The citation for [7] contains 'XXX' in place of the arXiv identifier or page range and should be completed.
[Proposition 4.9, display] In the chain of equalities, the stochastic integral in the second line is written with subscript u^φ_t while the integration variable is s; this should be u^φ_s for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Onsager-Machlup functional is derived from the definition via Girsanov, with self-citations that are not load-bearing and only unproven analytic estimates that are correctness concerns, not circular reasoning.

full rationale

The paper's central claim, Theorem 3.13, is derived by reducing the tube probability for the hypoelliptic Brownian motion to a Girsanov-transformed process and then evaluating the conditional expectation of an exponential martingale. The final functional L(p,v) = -1/2 ||v||^2_Hp follows by exact algebraic identities (1/2 ||gamma||^2 - <gamma, phi> = -1/2 ||phi||^2 + 1/2 ||psi||^2) rather than by fitting or by assuming the conclusion. The cited results by the authors ([1], [2], and [5]) are not load-bearing: Lemmas 3.2 and 3.3 are proved in the text, and [1]-[2] are only mentioned as potentially useful for future work. The proof does contain analytical gaps: Eq. (3.15) asserts equality of limits under the equivalent distances d_cc and rho without proving the required regular variation of small-ball probabilities, and Lemma 4.7 only establishes inclusions at mismatched radii (epsilon versus sqrt(3epsilon^2 + epsilon C_gamma)), so the claimed equality of the limits in (4.15) and (4.16) is not rigorously demonstrated. These are unproved estimates or missing regularity arguments, not circularity: no equation is defined in terms of the target result, no fitted parameter is renamed as a prediction, and no load-bearing conclusion is imported from a self-citation. The derivation is self-contained in structure and would be correct if the missing tube estimates were supplied.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim does not fit data and has no free parameters or invented entities. It relies on standard theorems and on two unproven small-ball asymptotic assumptions that are load-bearing: the interchangeability of equivalent distances in (3.15) and the equivalence claimed in Lemma 4.7.

assumptions (4)

domain assumption The small-ball tube probabilities under the Carnot-Carathéodory and homogeneous distance rho have the same asymptotic ratio (Eq 3.15).
Invoked in (3.15) to replace d_cc with rho in the main ratio. Requires regular variation of the small-ball probability under scaling, not proven and generally false for exponential rates.
ad hoc to paper Lemma 4.7: the tube probabilities for u^psi and z_t are asymptotically equivalent.
The lemma as stated is vacuous (both limits are 0) and its proof only shows inclusions at mismatched radii sqrt(3epsilon^2+epsilon C_gamma) vs epsilon. It is the load-bearing bridge in Proposition 4.9.
ad hoc to paper In Proposition 4.9, the stochastic integral of the horizontal part of the Maurer-Cartan form is small on the tube event.
Asserted without a detailed proof; conditioning on the tube does not obviously make the Ito integral small because the quadratic variation of the driving Brownian motion does not vanish.
standard math The Cameron-Martin-Girsanov theorem applies with gamma in H(H) and the stochastic exponential E^gamma is a true martingale.
Used in Section 4.2 and Appendix A; standard for finite energy gamma.

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Cite this review

Pith. "Pith review of On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group." pith.science (2026). https://pith.science/paper/KUBKI2J7

@misc{pith2026190809182,
  author       = {Pith},
  title        = {Pith review of: On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KUBKI2J7}},
  note         = {Machine review of arXiv:1908.09182}
}

read the original abstract

Onsager-Machlup functionals are used to describe the dynamics of a continuous stochastic process. For a stochastic process taking values in a Riemannian manifold, they have been studied extensively. We describe the Onsager-Machlup functional with respect to the sup norm for a hypoelliptic Brownian motion on a Heisenberg group. Unlike in the Riemannian case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not easily available in the sub-Riemannian setting. In addition, we study fine properties of trajectories of the hypoelliptic Brownian motion, including a new notion of horizontal continuous curves.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

5, 1362 – 1422

Fabrice Baudoin, Qi Feng, and Maria Gordina, Integration by parts and quasi- invariance for the horizontal wiener measure on foliated co mpact manifolds , Journal of Functional Analysis 277 (2019), no. 5, 1362 – 1422

work page 2019
[2]

Fabrice Baudoin, Maria Gordina, and Tai Melcher, Quasi-invariance for heat kernel measures on sub-Riemannian inﬁnite-dimensional Heisenbe rg groups , Trans. Amer. Math. Soc. 365 (2013), no. 8, 4313–4350. MR 3055697

work page 2013
[3]

R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under trans- lations, Ann. of Math. (2) 45 (1944), 386–396. MR MR0010346 (6,5f)

work page 1944
[4]

Mireille Capitaine, On the Onsager-Machlup functional for elliptic diﬀusion pro cesses, S´ eminaire de Probabilit´ es, XXXIV, Lecture Notes in Math. , vol. 1729, Springer, Berlin, 2000, pp. 313–328. MR 1768071

work page 2000
[5]

Driver and Maria Gordina, Heat kernel analysis on inﬁnite-dimensional Heisenberg groups, J

Bruce K. Driver and Maria Gordina, Heat kernel analysis on inﬁnite-dimensional Heisenberg groups, J. Funct. Anal. 255 (2008), no. 9, 2395–2461. MR MR2473262

work page 2008
[6]

Takahiko Fujita and Shinichi Kotani, The Onsager-Machlup function for diﬀusion processes, J. Math. Kyoto Univ. 22 (1982/83), no. 1, 115–130. MR 648558

work page 1982
[7]

Hara and Y

K. Hara and Y. Takahashi, Stochastic analysis in a tubular neighborhood or Onsager- Machlup functions revisited , arXiv e-prints (2016), XXX

work page 2016
[8]

Keisuke Hara and Yoichiro Takahashi, Lagrangian for pinned diﬀusion process , Itˆ o’s stochastic calculus and probability theory, Springer, Tok yo, 1996, pp. 117–128. MR 1439520

work page 1996

Show all 19 references

[9]

119 (1967), 147–171

Lars H¨ ormander, Hypoelliptic second order diﬀerential equations , Acta Math. 119 (1967), 147–171. MR 0222474 (36 #5526)

1967
[10]

24, North-Holland Publishing Co., Amsterdam, 1989

Nobuyuki Ikeda and Shinzo Watanabe, Stochastic diﬀerential equations and diﬀusion processes, second ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam, 1989. MR MR1011252 (90m:60069)

1989
[11]

Shreve, Brownian motion and stochastic calculus , second ed., Graduate Texts in Mathematics, vol

Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus , second ed., Graduate Texts in Mathematics, vol. 113, Spring er-Verlag, New York,
[12]

Terry Lyons and Ofer Zeitouni, Conditional exponential moments for iterated Wiener integrals, Ann. Probab. 27 (1999), no. 4, 1738–1749. MR 1742886

1999
[13]

Machlup and L

S. Machlup and L. Onsager, Fluctuations and irreversible process. II. Systems with kinetic energy, Physical Rev. (2) 91 (1953), 1512–1515. MR 0057766

1953
[14]

(2) 91 (1953), 1505–

, Fluctuations and irreversible processes , Physical Rev. (2) 91 (1953), 1505–

1953
[15]

Paolo Pigato, Tube estimates for diﬀusion processes under a weak H¨ ormande r condi- tion, Ann. Inst. Henri Poincar´ e Probab. Stat.54 (2018), no. 1, 299–342. MR 3765891

2018
[16]

L. A. Shepp and O. Zeitouni, Exponential estimates for convex norms and some applications, Barcelona Seminar on Stochastic Analysis (St. Feliu de Gu ´ ıxols, 1991), Progr. Probab., vol. 32, Birkh¨ auser, Basel, 1993, pp. 203– 215. MR 1265050

1991
[17]

Takahashi and S

Y. Takahashi and S. Watanabe, The probability functionals (Onsager-Machlup func- tions) of diﬀusion processes , Stochastic integrals (Proc. Sympos., Univ. Durham, 24 CARF AGNINI AND GORDINA Durham, 1980), Lecture Notes in Math., vol. 851, Springer, B erlin, 1981, pp. 433–

1980
[18]

Ofer Zeitouni, On the Onsager-Machlup functional of diﬀusion processes aro und non- C 2-curves, Ann. Probab. 17 (1989), no. 3, 1037–1054. MR 1009443 Department of Mathematics, University of Connecticut, Sto rrs, CT 06269, U.S.A. E-mail address : marco.carfagnini@uconn.edu Depa...

1989
[1991]

MR 1121940 (92h:60127)

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

5, 1362 – 1422

Fabrice Baudoin, Qi Feng, and Maria Gordina, Integration by parts and quasi- invariance for the horizontal wiener measure on foliated co mpact manifolds , Journal of Functional Analysis 277 (2019), no. 5, 1362 – 1422

work page 2019

[2] [2]

Fabrice Baudoin, Maria Gordina, and Tai Melcher, Quasi-invariance for heat kernel measures on sub-Riemannian inﬁnite-dimensional Heisenbe rg groups , Trans. Amer. Math. Soc. 365 (2013), no. 8, 4313–4350. MR 3055697

work page 2013

[3] [3]

R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under trans- lations, Ann. of Math. (2) 45 (1944), 386–396. MR MR0010346 (6,5f)

work page 1944

[4] [4]

Mireille Capitaine, On the Onsager-Machlup functional for elliptic diﬀusion pro cesses, S´ eminaire de Probabilit´ es, XXXIV, Lecture Notes in Math. , vol. 1729, Springer, Berlin, 2000, pp. 313–328. MR 1768071

work page 2000

[5] [5]

Driver and Maria Gordina, Heat kernel analysis on inﬁnite-dimensional Heisenberg groups, J

Bruce K. Driver and Maria Gordina, Heat kernel analysis on inﬁnite-dimensional Heisenberg groups, J. Funct. Anal. 255 (2008), no. 9, 2395–2461. MR MR2473262

work page 2008

[6] [6]

Takahiko Fujita and Shinichi Kotani, The Onsager-Machlup function for diﬀusion processes, J. Math. Kyoto Univ. 22 (1982/83), no. 1, 115–130. MR 648558

work page 1982

[7] [7]

Hara and Y

K. Hara and Y. Takahashi, Stochastic analysis in a tubular neighborhood or Onsager- Machlup functions revisited , arXiv e-prints (2016), XXX

work page 2016

[8] [8]

Keisuke Hara and Yoichiro Takahashi, Lagrangian for pinned diﬀusion process , Itˆ o’s stochastic calculus and probability theory, Springer, Tok yo, 1996, pp. 117–128. MR 1439520

work page 1996