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
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.
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Notes
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 \(L^2(e^{-\Phi }).\)
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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
in order to prove Theorem 3. Since the vectorial field F defined by
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
then
In particular, the line
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
Note that (u, v) is obtained from (U, V) by a rotation of angle \(-X\). Then, in the new variable, the ODE (74) becomes the ODE
Let
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
\(\square \)
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
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).
-
(i)
\(\mathbb {T}_{1/2}^{\alpha }\) is a periodic orbit with period \(2\pi \), \(\alpha \in \{+,-\}\)
-
(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 \(H_{c}^{\alpha }\)
-
(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 \(\frac{p}{q}\).
-
(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
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
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
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
An immediate consequence of Propositions 9 and 10 is that \((t,u_{2}(t),v_{2}(t)) \) is periodic if and only if
In the rest of this Appendix, we study the “period-function”
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.
Using the fact that \(\dot{v}_{2}=-u_{2}\), we have that
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
i.e.
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
and
Therefore
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
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
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
Proof
By convexity of \(\phi \), we have \(\frac{\phi (v)-\phi (c_{1})}{v-c_{1}}\geqslant \phi '(c_{1})\). Hence
Therefore
Since \(-\phi '(v)=\frac{1}{v}- v\), \(-\phi '(c_{1})=(1- c_{1}^{2})/c_{1}\) and thus
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
Hence
\(\square \)
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
But
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
\(\square \)
Remark 11
One can prove that \(\sqrt{2}\pi \) is the period of the orbits from the linear ODE
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.\(\square \)
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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
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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

