Skip to main content

Self-repelling diffusions on a Riemannian manifold

Abstract

Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space \(M\times \mathbb {R}^{n}\); which is obtained via a natural change of variable from a self-repelling diffusion taking the form

$$\begin{aligned} dX_{t}= \sigma dB_{t}(X_t) -\int _{0}^{t}\nabla V_{X_s}(X_{t})dsdt,\qquad X_{0}=x \end{aligned}$$

where \(\{B_t\}\) is a Brownian vector field on M, \(\sigma >0\) and \(V_x(y) = V(x,y)\) is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability \(\mu \) given as the product of the normalized Riemannian measure on M and a Gaussian measure on \(\mathbb {R}^{n}\). We then prove an exponential decay to this invariant probability in \(L^{2}(\mu )\) and in total variation.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. This is a classical result and can easily be verified as follows. Formula (52) shows that the set \(C^{\infty }_b({\mathbb R}^n)\) of bounded \(C^{\infty }\) functions with bounded derivatives is stable under \((P_t^{OU})\); hence a Core by Proposition 6. Furthermore for each \(f \in C^{\infty }_b({\mathbb R}^n)\) it is easy to construct a sequence \(f_n \in C^{\infty }_c({\mathbb R}^n)\) such that \(f_n \rightarrow f\) and \(L_{OU} f_n \rightarrow L_{OU} f\) in \(L^2(e^{-\Phi }).\)

References

  1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, F., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logaritheoremique, Panorama et synthèse, no. 10. SMF (2000)

  2. Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 9(36), 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  3. Azéma, J., Duflo, M., Revuz, D.: Mesure Invariante des processus de markov récurrent, Convergence of Probability Measures. Séminaire de probabilités (Strasbourg) 3, 24–33 (1969)

    MATH  Google Scholar 

  4. Benaïm, M., Ledoux, M., Raimond, O.: Self-interacting diffusions. Probab. Theory Related Fields 122, 1–41 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benaïm, M., Raimond, O.: Self-interacting diffusions II: convergence in law. Ann. Inst. H. Poincaré 6, 1043–1055 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benaïm, M., Raimond, O.: Self-interacting diffusions III: symmetric interactions. Ann. Probab. 33(5), 1716–1759 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cranston, M., Le Jan, Y.: Self-attracting diffusions: two case studies. Math. Ann. 303, 87–93 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cranston, M., Mountford, T.: The strong law of large number for a Brownian polymer. Ann. Probab. 24(3), 1300–1323 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Springer, New York (2004)

    Book  MATH  Google Scholar 

  10. Da Prato, G., Röckner, M.: Cores for generators of some Markov semigroups. In: Proceedings of the Centennial Conference ”Alexandra Muller” Mathematical Seminar, vol. 87, Iasi, p. 97 (2011)

  11. Dolbeault, J., Klar, A., Mouhot, C., Schmeiser, C.: Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Appl. Math. Res. Express 2, 165–175 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercitivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc. (2015). ISSN 0002-9947

  13. Durrett, R.T., Rogers, L.C.G.: Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92(3), 337–349 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New yORK (1986)

    Book  MATH  Google Scholar 

  15. Grothaus, M., Klar, A., Maringer, J., Stilgenbauer, P.: Geometry, mixing properties and hypocoercitivity of a degenerate diffusion arising in technical textile industry (2012, arXiv preprint)

  16. Grothaus, M., Stilgenbauer, P.: Hypocoercitivity for Kolmogorov backward evolution equations and applications. J. Funct. Anal. 267, 3515–3556 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Herrmann, S., Roynette, B.: Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325(1), 81–96 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Herrmann, S., Scheutzow, M.: Rate of convergence of some self-attracting diffusions. Stoch. Process. Appl. 111(1), 41–55 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, Berlin (1976)

  20. Hsu, Elton P.: Stochastic Analysis on Manifold. American Mathematical Society, Providence (2002)

    Book  Google Scholar 

  21. Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30(3), 235–254 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997)

    MATH  Google Scholar 

  23. Kleptsyn, V., Kurtzmann, A.: Ergodicity of self-attracting motions. Electron. J. Probab. 17(50), 1–37 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Kurtzmann, A.: The ODE method for some self-interacting diffusions on \( \mathbb{R}^{d}\). Ann. Inst. Henri Poincaré Probab. Stat. 46(3), 618–643 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mountford, T., Tarrès, P.: An asymptotic result for Brownian polymer. Ann. Inst. Henri Poincaré Probab. Stat. 44(3), 29–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pemantle, R.: A survey of random processes with reinforcement. Probab. Survey 4, 1–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Phillips, R.S.: The adjoint semi-group. Pac. J. Math. 5(2), 269–283 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  28. Portegies, J.W.: Embeddings of riemannian manifolds with heat kernels and eigenfunctions. Commun. Pure Appl. Math. 69(3), 478–518 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Raimond, O.: Self attracting diffusions: case of the constant interaction. Probab. Theory Related Fields 107(2), 177–196 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Raimond, O.: Self-interacting diffusions: a simulated annealing version. Probab. Theory Related Fields 144, 247–279 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, North-Holland Mathematical Library, Amsterdam (1999)

    Book  MATH  Google Scholar 

  32. Tarrès, P., Tóth, B., Valkó, B.: Diffusivity bounds for 1D Brownian polymers. Ann. Probab. 40(2), 695–713 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950) (2009)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carl-Erik Gauthier.

Additional information

We acknowledge financial support from the Swiss National Science Foundation Grant 200020-149871/1. We thank B. Colbois and H. Donnelly for useful discussions on eigenfunctions of the Laplace operator and P. Monmarché for useful discussion about hypocoercivity.

Appendix: A deterministic study

Appendix: A deterministic study

In this Appendix, we study on \(\mathbb {S}^{1}\times \mathbb {R}^{2}\) the ODE

$$\begin{aligned} \left\{ \begin{array}{l l l} \dot{X}_{t}&{}=&{} (\sin (X_{t})U_{t}-\cos (X_{t})V_{t})\\ \dot{U}_{t}&{}=&{} \cos (X_{t}) \\ \dot{V}_{t}&{}=&{} \sin (X_{t}) \end{array} \right. \end{aligned}$$
(74)

in order to prove Theorem 3. Since the vectorial field F defined by

$$\begin{aligned} F(X,U,V)=\begin{pmatrix} (\sin (X)U-\cos (X)V)\\ \cos (X)\\ \sin (X) \end{pmatrix} \end{aligned}$$
(75)

is smooth and sub-linear,it induces a smooth flow \(\psi :\mathbb {R}\times (\mathbb {S}^{1}\times \mathbb {R}^{2})\rightarrow \mathbb {S}^{1}\times \mathbb {R}^{2}\). A first and important observation is

Proposition 7

If the initial condition for the ODE (74) is

$$\begin{aligned} (X_{0},U_{0},V_{0})=(X_{0},\cos (X_{0}),\sin (X_{0})), \end{aligned}$$

then

$$\begin{aligned} \psi _{t}(X_{0},U_{0},V_{0})=(X_{0},\cos (X_{0})(t+1),\sin (X_{0})(t+1)) \, \forall t\in \mathbb {R}. \end{aligned}$$

In particular, the line

$$\begin{aligned} \{(X,Y,Z)\in \mathbb {S}^{1}\times \mathbb {R}^{2}\, : \, X=X_{0},\,\exists t\in \mathbb {R}\quad \text {such that } (Y,Z)=(\cos (X_{0})t,\sin (X_{0})t)\} \end{aligned}$$

is invariant under \(\psi \).

Proof

By the hypothesis, we have \(\dot{X}(0)=0\). Hence \(X(t)=X_{0}\) for all \(t\in \mathbb {R}\). Therefore, \(U(t)=\cos (X_{0})(t+1)\) and \(V(t)=\sin (X_{0})(t+1)\)

An immediate consequence is

Corollary 3

If \(\dot{X}(0)>0\) (respectively \(\dot{X}(0)<0\)), then \(\dot{X}(t)>0\) (respectively \(\dot{X}(t)<0\)) for all t.

Proof

We proceed by contradiction. Hence, by continuity of \(\dot{X}\), there exists \(t_{0}\) such that \(\dot{X}(t_{0})=0\). Then the two last Propositions imply that \(\dot{X}(t)=0\) for all t. In particular \(\dot{X}(0)=0\), which is a contradiction.

Let

$$\begin{aligned} \begin{pmatrix} x\\ u\\ v \end{pmatrix}= \Xi \left( \begin{pmatrix} X\\ U\\ V \end{pmatrix}\right) = \begin{pmatrix} X\\ \cos (X)U+\sin (X)V\\ -\sin (X)U+\cos (X)V \end{pmatrix}. \end{aligned}$$
(76)

Note that (uv) is obtained from (UV) by a rotation of angle \(-X\). Then, in the new variable, the ODE (74) becomes the ODE

$$\begin{aligned} \dot{x}(t)=- v(t) \end{aligned}$$
(77)
$$\begin{aligned} \left\{ \begin{array}{l l l} \dot{u}(t)&{}=&{}1- v(t)^{2} \\ \dot{v}(t)&{}=&{} u(t)v(t) \end{array} \right. \end{aligned}$$
(78)

Let

$$\begin{aligned} H(u,v) = \left\{ \begin{array}{l} \frac{1}{2}(u^2 + v^2 - \log (v^2)),\quad \text{ if } v \ne 0,\\ \infty , \quad \text{ if } v = 0. \end{array} \right. \end{aligned}$$
(79)

Proposition 8

The function H is a first integral for the ODE (78).

Proof

Let \(v_{0}\ne 0\). Deriving H with respect to t and applying the chain rule, we obtain

$$\begin{aligned} \frac{d}{dt} H(u,v)= & {} (u\dot{u}+v\dot{v})-\frac{\dot{v}}{v}\\= & {} (u-uv^{2}-vuv)-u\\= & {} 0 \end{aligned}$$

\(\square \)

Note that H is convex, reaches its global minimum in \((0,\pm 1)\) and takes the value 1 / 2 at these points.

For \(c\in [1/2,\infty [\), let

$$\begin{aligned} H_{c}^{+}=H^{-1}(c)\cap \{ v>0\} ,\quad H_{c}^{-}=H^{-1}(c)\cap \{v<0\} \end{aligned}$$

and set \(H_{\infty }=\{v=0\}\). Then, we define \(\mathbb {T}_{c}^{\alpha }=\mathbb {S}^{1}\times H_{c}^{\alpha }\) for \(\alpha \in \{+,-\}\) and \(T_{\infty }=\mathbb {S}^{1}\times H_{\infty }\).

Since the function H is strictly convex on \(\{v>0\} \) and \(\{ v<0\}\), we observe that \(T_{1/2}^{\alpha }\) is a closed curve, \(T_{c}^{\alpha }\) a torus and \(T_{\infty }\) a cylinder.

A first result is

Proposition 9

Let (x(t), u(t), v(t)) be a solution of the ODE defined by (77) and (78).

  1. (i)

    \(\mathbb {T}_{1/2}^{\alpha }\) is a periodic orbit with period \(2\pi \), \(\alpha \in \{+,-\}\)

  2. (ii)

    On \(T_{\infty }\), the dynamic takes the form \((x(t),u(t),v(t))=(x(0),u(0)+t,0).\)

For \(c>1/2\), let \(T_{c}\) be the period of (78) on \(H_{c}^{\alpha }\)

  1. (iii)

    If \(\frac{x(T_{c})}{2\pi }\in \mathbb {Q}\), then every trajectory on \(T_{c}^{\alpha }\) is periodic with period \(qT_{c}\) if the irreducible fraction of \(\frac{x(T_{c})}{2\pi }\) writes \(\frac{p}{q}\).

  2. (iv)

    If \(\frac{x(T_{c})}{2\pi }\notin \mathbb {Q}\), then every trajectory on \(\mathbb {S}^{1}\times H^{-1}(c)\) is dense either on \(T_{c}^{+}\) or \(T_{c}^{-}\).

Proof

Points (i) and (ii) follow immediately from (77), (78) and the function H.

Without loss of generality, we assume that \(x(0)=0\). Let \(c>1/2\). Because for \(m\in \mathbb {N}^{*}\), we have

$$\begin{aligned} x(mT_{c})= & {} \int _{0}^{mT_{c}}\dot{x}(t)dt=-\int _{0}^{mT_{c}}v(t)dt\nonumber \\= & {} -m\int _{0}^{T_{c}}v(t)dt\nonumber \\= & {} m\int _{0}^{T_{c}}\dot{x}(t)dt,\nonumber \\= & {} m x(T_{c}) \end{aligned}$$
(80)

we obtain that when (u(t), v(t)) is back to its initial condition, then x(t) does a rotation of angle \(x(T_{c})\). Hence if \(\frac{x(T_{c})}{2\pi }=\frac{p}{q}\), with \(q\in \mathbb {N}^{*}\), \(p\in \mathbb {Z}\) and such that the fraction is irreducible, then

$$\begin{aligned} 2p\pi= & {} qx(T_{c})\\= & {} x(qT_{c}). \end{aligned}$$

This proves (iii).

If \(\frac{x(T_{c})}{2\pi }\notin \mathbb {Q}\), then \((x(qT_{c}))_{q\in \mathbb {N}}\) is dense on \(\mathbb {S}^{1}\). Now, assume without lost of generality that \(v(0)<0\) and let T be the first time such that \(x(T)=2\pi \). We claim that \((u(nT),v(nT))_{n\in \mathbb {N}}\) is dense on \( H_{c}^{-}\). Indeed, if it is not the case, then it is periodic since \( H_{c}^{-}\) is a closed simple curve. This implies that (x(t), u(t), v(t)) is periodic with period \(n_{0}T\). Thus, there exists \(q\in \mathbb {N}\) such that \(n_{0}T=qT_{c}\). Therefore, by (80), we have \(2n_{0}\pi =x(qT_{c})=qx(T_{c})\); so that \(\frac{x(T_{c})}{2\pi }=\frac{n_{0}}{q}\). This is a contradiction.

The density of \((x(qT_{c}))_{q\in \mathbb {N}}\) on \(\mathbb {S}^{1}\) and the one of \((u(nT),v(nT))_{n\in \mathbb {N}}\) on \( H_{c}^{-}\) implies the density of \(((x(t),u(t),v(t)))_{t\geqslant 0}\) on \(T_{c}^{-}\). This proves (iv). \(\square \)

From now, we assume without lost of generality that \(v(0)<0\) (the case \(v(0)>0\) being symmetric). In order to derive properties of \(c\mapsto T_{c}\) (see Proposition (9)), we change the time scale by use of \(t\mapsto x(t)\). This is possible because it is strictly increasing. We denote by y the inverse function of x. Since we have assumed that \(x(0)=0\), it follows that \(y(0)=0\).

Set \(u_{2}(t)=u(y(t))\) and \(v_{2}(t)=v(y(t))\). Therefore \((u_{2},v_{2})\) is solution to the ODE

$$\begin{aligned} \left\{ \begin{array}{l l l} \dot{u}_{2}(t)&{}=&{}\left( v_{2}(t)-\frac{1}{v_{2}(t)}\right) \\ \dot{v}_{2}(t)&{}=&{}- u_{2}(t) \end{array} \right. \end{aligned}$$
(81)

with initial condition (u(0), v(0)). Observe that H is still a first integral for this system.

Proposition 10

Let (x(t), u(t), v(t)) be a solution to the ODE defined by equation (77) with initial condition \((0,u_{0},v_{0})\) and let \((t,u_{2}(t),v_{2}(t))\) where \((u_{2}(t),v_{2}(t))\) is the solution to the ODE defined by Eq. (81) with initial condition \((u_{0},v_{0})\).

Then (x(t), u(t), v(t)) is periodic in \(\mathbb {S}^{1}\times \mathbb {R}^{2}\) iff \((t,u_{2}(t),v_{2}(t))\) is periodic in \(\mathbb {S}^{1}\times \mathbb {R}^{2}\).

Further, if T is the period of (x(t), u(t), v(t)), then x(T) is the period of \((t,u_{2}(t),v_{2}(t))\).

Proof

Straightforward.\(\square \)

Denote by \(T_{c,2}\) the period of \((u_{2}(t),v_{2}(t))\), where \(c=H(u_{2}(0),v_{2}(0))>1/2\). Then

$$\begin{aligned} T_{c,2}= x(T_{c}). \end{aligned}$$
(82)

An immediate consequence of Propositions 9 and 10 is that \((t,u_{2}(t),v_{2}(t)) \) is periodic if and only if

$$\begin{aligned} \frac{T_{c,2}}{2\pi }\in \mathbb {Q}. \end{aligned}$$
(83)

In the rest of this Appendix, we study the “period-function”

$$\begin{aligned} f:(1/2, +\infty )\rightarrow \mathbb {R}_{+}:c\mapsto T_{c,2}. \end{aligned}$$
(84)

First notice that (0, 1) and \((0,-1)\) are stationary points for the ODE (81).

Let \((u_{0},v_{0})\in \mathbb {R}\times (0,\infty )\). By symmetry of H along the line \(v_{2} = 0\), what follow remains true for \(v_{0}<0\).

Set \(c=H(u_{0},v_{0})\). Since H is a first integral, then \(H(u_{2}(t),v_{2}(t))=c\) for all t.

Fig. 5
figure 5

Graph of the function \(\phi \)

Using the fact that \(\dot{v}_{2}=-u_{2}\), we have that

$$\begin{aligned} \frac{1}{2}\dot{v}_{2}^{2}+\left( \frac{v_{2}^{2}}{2}-\log (v_{2})\right) =c. \end{aligned}$$
(85)

Set \(\phi (v)=(\frac{v^{2}}{2}-\log (v)).\) Below (Fig. 5) is given its graph.

Since the curve \(H^{-1}(c)\) is symmetric along the line \(u_{2} = 0\), we have that

$$\begin{aligned} \frac{T_{c,2}}{2}=\int _{c_{1}}^{c_{2}}\frac{dv}{\sqrt{2(c-\phi (v))}}, \end{aligned}$$
(86)

i.e.

$$\begin{aligned} T_{c,2}=\sqrt{2}\int _{c_{1}}^{c_{2}}\frac{dv}{\sqrt{(c-\phi (v))}}, \end{aligned}$$
(87)

where \(0<c_{1}<1<c_{2}<\infty \) are the roots of the function \(v\mapsto \phi (v)-c\).

Denote by h the inverse function of \(\phi \) restricted to \([1,\infty )\) and by g the inverse function of \(\phi \) restricted to (0, 1). By a change of variable, we then obtain

$$\begin{aligned} \int _{1}^{c_{2}}\frac{dv}{\sqrt{(c-\phi (v))}}=\int _{\frac{1}{2}}^{c}\frac{h'(v)dv}{\sqrt{(c-v)}} \end{aligned}$$
(88)

and

$$\begin{aligned} \int _{c_{1}}^{1}\frac{dv}{\sqrt{(c-\phi (v))}}=-\int _{\frac{1}{2}}^{c}\frac{g'(v)dv}{\sqrt{(c-v)}}. \end{aligned}$$
(89)

Therefore

$$\begin{aligned} f(c)=T_{c,2}=\sqrt{2}\int _{\frac{1}{2}}^{c}\frac{(h'-g')(v)}{\sqrt{(c-v)}}dv=\int _{\mathbb {R}}\Lambda (v)A(c-v)dv=(\Lambda *A)(c), \end{aligned}$$
(90)

where \(*\) stands for the convolution product, \(\Lambda (v)=\sqrt{2}(h'-g')(v)\mathbf {1}_{v>1/2}\) and \(A(v)=\frac{1}{\sqrt{v}}\mathbf {1}_{v>0}\).

Hence

$$\begin{aligned} f'(c)=(\Lambda *A')(c). \end{aligned}$$
(91)

Since \(g(v)\in (0,1)\) and \(h(v)>1 \) for \(v\in (1/2,c)\), then \(g'(v)=\frac{1}{\phi '(g(v))}<0\) and \(h'(v)=\frac{1}{\phi '(h(v))}>0\). Using the fact that \(A'(v)=-\frac{1}{2}\mathbf {1}_{v>0}\frac{1}{\sqrt{v^{3}}}\), we have

$$\begin{aligned} f'(c)<0 \quad \text {for all}\quad 1/2<c<\infty . \end{aligned}$$
(92)

Our next goal is now to study the limiting behaviour \(c\rightarrow 1/2\) and \(c\rightarrow \infty \)

Lemma 15

Let \(c>1/2\) and let \(c_{1}\) and \(c_{2}\) the two roots of the function \(v\mapsto \phi (v)-c\). Then

$$\begin{aligned} T_{c,2}\geqslant 2\sqrt{2}[\sqrt{\frac{c_{1}}{1+c_{1}}}+\sqrt{\frac{c_{2}}{1+c_{2}}}]. \end{aligned}$$

Proof

By convexity of \(\phi \), we have \(\frac{\phi (v)-\phi (c_{1})}{v-c_{1}}\geqslant \phi '(c_{1})\). Hence

$$\begin{aligned} \sqrt{c-\phi (v)}\leqslant \sqrt{-\phi '(c_{1})}\sqrt{v-c_{1}} . \end{aligned}$$

Therefore

$$\begin{aligned} \int _{c_{1}}^{1}\frac{dv}{\sqrt{c-\phi (v)}}\geqslant \frac{1}{\sqrt{-\phi '(c_{1})}}\int _{c_{1}}^{1}\frac{dv}{\sqrt{v-c_{1}}}=2\frac{\sqrt{1-c_{1}}}{\sqrt{-\phi '(c_{1})}}. \end{aligned}$$

Since \(-\phi '(v)=\frac{1}{v}- v\), \(-\phi '(c_{1})=(1- c_{1}^{2})/c_{1}\) and thus

$$\begin{aligned} \int _{c_{1}}^{1}\frac{dv}{\sqrt{c-\phi (v)}}\geqslant 2\sqrt{\frac{c_{1}}{(1+c_{1})}}. \end{aligned}$$

Once again convexity of \(\phi \) implies \(\frac{\phi (c_{2})-\phi (v)}{c_{2}-v}\leqslant \phi '(c_{2})\), so that \(c-\phi (v)\leqslant \phi '(c_{2})(c_{2}-v). \) By proceeding as above, we obtain

$$\begin{aligned} \int _{1}^{c_{2}}\frac{dv}{\sqrt{c-\phi (v)}}\geqslant 2\sqrt{\frac{c_{2}}{(1+c_{2})}} . \end{aligned}$$

Hence

$$\begin{aligned} f(c)=T_{c,2}=\sqrt{2}[\int _{c_{1}}^{1}\frac{dv}{\sqrt{c-\phi (v)}}+\int _{1}^{c_{2}}\frac{dv}{\sqrt{c-\phi (v)}}]\geqslant 2\sqrt{2}[\sqrt{\frac{c_{1}}{1+c_{1}}}+\sqrt{\frac{c_{2}}{1+c_{2}}}]. \end{aligned}$$

\(\square \)

Lemma 16

\(\lim _{c\rightarrow 1/2} f(c)=\sqrt{2}\pi .\)

Proof

We have \(c_{1},c_{2}\rightarrow 1\) as \(c\rightarrow 1/2\). Thus, it implies that \(\log (v)\approx (v-1)-\frac{1}{2}(v-1)^{2}\) for \(v\in (c_{1},c_{2})\) and therefore

$$\begin{aligned} \phi (v)=\frac{1}{2}(v-1+1)^{2}-\log (v)\approx \frac{1}{2}+(v-1)^{2}. \end{aligned}$$

But

$$\begin{aligned} \int _{c_{1}}^{c_{2}}\frac{dv}{\sqrt{c-\frac{1}{2}-(v-1)^{2}}}= & {} \frac{1}{\sqrt{c-\frac{1}{2}}}\int _{c_{1}-1}^{c_{2}-1}\frac{dv}{\sqrt{1-(v/\sqrt{c-\frac{1}{2}})^{2}}}\\= & {} \int _{\frac{c_{1}-1}{\sqrt{c-1/2}}}^{\frac{c_{2}-1}{\sqrt{c-1/2}}}\frac{du}{\sqrt{1-u^{2}}}\\= & {} \arcsin \left( \frac{c_{2}-1}{\sqrt{c-1/2}}\right) +\arcsin \left( \frac{1-c_{2}}{\sqrt{c-1/2}}\right) \end{aligned}$$

Since for c sufficiently close to 1 / 2, \(c=\phi (1+c_{j}-1)\approx \frac{1}{2}+ (c_{j}-1)^{2}\), then \(\lim _{c\rightarrow 1/2}\frac{\vert c_{j}-1\vert }{\sqrt{c-\frac{1}{2}}}= 1, j=1,2\).

Thus, \(\lim _{c\rightarrow 1/2}\int _{c_{1}}^{c_{2}}\frac{dv}{\sqrt{c-(v-1)^{2}}}= \pi \) and therefore

$$\begin{aligned} \lim _{c\rightarrow 1/2}f(c)=\lim _{c\rightarrow 1/2}T_{c,2}=\lim _{c\rightarrow 1/2} \sqrt{2}\int _{c_{1}}^{c_{2}}\frac{dv}{\sqrt{c-\frac{1}{2}-(v-1)^{2}}}= \sqrt{2}\pi . \end{aligned}$$
(93)

\(\square \)

Remark 11

One can prove that \(\sqrt{2}\pi \) is the period of the orbits from the linear ODE

$$\begin{aligned} \left\{ \begin{array}{l l l} \dot{u}(t)&{}=&{} 2v(t)\\ \dot{v}(t)&{}=&{}-u(t). \end{array} \right. \end{aligned}$$
(94)

But this is nothing else than the linearized system at (0, 1) from the ODE (81).

Summarizing all these information concerning \(T_{c,2}\), we obtain

Proposition 11

The “period-function” \(f:(1/2,\infty )\rightarrow \mathbb {R}_{+}:c\mapsto T_{c,2}\) is continuous, decreasing, bounded from below by \(2\sqrt{2}\) and converge to \(\sqrt{2}\pi \) when c tends to 1 / 2.

Proof

The decreasing property comes from (92) whereas the continuity follows from (90). While \(c_{1}\) converges to 0 and \(\frac{c_{2}}{1+c_{2}}\) converges to 1 when c tends to \(\infty \), then Lemma 15 combined with the decreasing property implies that \(f(c)\geqslant 2\sqrt{2}\) for all \(c>1/2\).

Since f is decreasing, then \(\sup _{c>1/2}f(c)=\lim _{c\rightarrow 1/2}f(c)=\sqrt{2}\pi \). Below (Fig. 6) is the graph of the period-function.\(\square \)

Fig. 6
figure 6

Graph of the function \(c\mapsto T_{c,2}\)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benaïm, M., Gauthier, CE. Self-repelling diffusions on a Riemannian manifold. Probab. Theory Relat. Fields 169, 63–104 (2017). https://doi.org/10.1007/s00440-016-0717-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-016-0717-1

Keywords

  • Self-interacting diffusions
  • Strong Feller property
  • Degenerate diffusions
  • Hypocoercivity
  • Invariant probability measure

Mathematics Subject Classification

  • Primary 58J65
  • 60K35
  • 60H10
  • 60J60
  • Secondary 37A25
  • 37A30