A Varadhan Estimate for Big Order Differential Generators ^†

Abstract

We give a logarithmic estimate of an elliptic semi-group generated by a big-order generator by using the Malliavin Calculus of Bismut type and large deviation estimates.

1. Introduction and Main Results

Let us consider a compact Riemannian manifold M of dimension d endowed with its normalized Riemannian measure $dx$$(x\in M)$. Associated with it, we consider the Laplace–Beltrami operator $\Delta$ and the heat semi-group associated with it, $P_t$

$$\partial /(\partial t)P_{t}f=-\Delta /2P_{t}f$$

(1)

if f is a smooth function on M.

The heat semi-group is represented by a heat-kernel if $t>0$

$$P_{t}f\left(x\right)={\int }_M{p}_t(x,y)f\left(y\right)dy$$

(2)

where $(x,y)\to p_t(x,y)$ is smooth positive.

Associated with the Riemannian structure, we consider the Riemannian distance $(x,y)\to d_R(x,y)$ which is continuous positive. Varadhan’s type estimate states that

$$li{m}_{t\to 0}2tlog{p}_t(x,y)=-d_R^2(x,y)$$

(3)

For a subelliptic operator, we can consider the associated semi-group. By Hoermander’s theorem, there is still an associated heat-kernel. There is the generalization in this case of the Riemannian distance called the Sub-Riemannian distance $d_{s.R}(x,y)$ which is still continuous and positive finite. Under some technical conditions, using a mixture of the Malliavin calculus and large deviation estimates, we have shown in [1] that

$${\overline{lim}}_{t\to 0}2tlog{p}_t(x,y)\le -d_{s.R}^2(x,y)$$

(4)

Our goal is to repeat the strategy of [1] for a non-Markovian semi-group. We consider some vector fields $X_i$ smooth without divergence on the manifold and we consider the operator

$$L={(-1)}^k\sum _{i=1}^m{X}_i^{2k}$$

(5)

for some strictly positive integer k. We suppose that at each point x of M, the vector fields span the tangent space $T_x\left(M\right)$. In such a case, the operator is elliptic positive symmetric. According to abstract theory [2], it permits a self-adjoint extension and is essentially self-adjoint.

We consider the heat semi-group associated with it:

$$\partial /(\partial t)P_t^{L}f=-L{P}_t^{L}f$$

(6)

if f is a smooth function on M and $P_0^{L}f=f$. The main difference with the case of the Laplacian is that the semi-group does not preserve the positivity. Classically in analysis, the semi-group $P_t^L$ has a heat-kernel $p_t^L(x,y)$ which changes sign [2]. We have shown this result using the tools of Malliavin calculus for non-Markovian semi-groups (see [3] for a review).

Associated with L is a Hamiltonian H. It is an application on $T^{*}\left(M\right)$, the cotangent bundle of M given by the following if $\xi \in T_x^{*}\left(m\right)$:

$$(x,\xi )\to \sum _{i=1}^m<\xi ,X_i\left(x\right){>}^{2k}$$

(7)

Due to the hypothesis of ellipticity, we have

$$\left|H(x,\xi )\right|\ge {C\left|\xi \right|}^{2k}$$

(8)

According to the theory of large deviation, we introduce the associated Lagrangian. It is a function from $T\left(M\right)$ the tangent bundle of M into $\mathbb{R}$

$$L(x,p)=\underset{\xi }{sup}(<\xi ,p>-H(x,\xi ))$$

(9)

where $p\in T_x\left(M\right)$. It $t\to \gamma \left(t\right)$ is a finite energy curve on M. We define its action as

$$S\left(\gamma \right)={\int }_0^{1}L(\gamma \left(t\right),d/dt\gamma \left(t\right))dt$$

(10)

and we put

$$l(x,y)=\underset{\gamma \left(0\right)=x;\gamma \left(1\right)=y}{inf}S\left(\gamma \right)$$

(11)

By standard methods, due to the Estimate (8), $(x,y)\to l(x,y)$ is continuous.

The goal of this note is to show a Varadhan-type estimate for $p_t^L(c,y)$:

Theorem When $t\to 0$, we have uniformly

$$\overline{lim}{t}^{1/(2k-1)}log\left|p_t^L(x,y)\right|\le -l(x,y)$$

(12)

This estimate has to be compared with the standard estimates of harmonic analysis (see, for instance, [4]). We adapt the method of [1] in this non-Markovian context. Let us remark that we have already obtained similar estimates in [3,5,6] for right-invariant elliptic operators on compact Lie groups by mixing tools of the Malliavin calculus for non-Markovian semi-groups and Wentzel–Freidlin estimates for non-Markovian semi-groups.

With respect to [4], the variational problem associated with the semi-group appears. The asymptotic is global, unlike the standard asymptotics of semi-classical analysis, which are unstable. The main novelty with respect of the traditional results of stochastic analysis is the new exponent in the asymptotic, which is due to the fact that we consider a big-order generator.

In the next part, we prove the theorem by using rough estimates of the heat-kernel which are obtained using Malliavin calculus and large deviation estimates on the semi-group.

2. Proof of the Theorem

Since L is symmetric,

$${\int }_{M}f\left(x\right)P_t^{L}g\left(x\right)dx={\int }_{M}g\left(x\right)P_t^{L}f\left(x\right)dx$$

(13)

such that $p_t^L(x,y)=p_t^L(y,x)$.

Let us recall some results of [3]. In part 4 of [3], we showed using the intrinsic Malliavin calculus on the semi-group generated by L that

$$|p_t^l(x,y)|\le C{t}^{-l}$$

(14)

for $t\le 1$ uniformly in $(x,y)$.

Moreover, in part 5 of [3], we showed that if O is an open ball uniformly in x, that

$$\overline{lim}{t}^{1/(2k-1)}log|P_t^l|\left(1_O\right]\left(x\right)\le -\underset{y\in O}{inf}l(x,y))$$

(15)

This means, in other words, that if $\eta$ is small for small t, then

$${\int }_O\left|p_t^L(x,y)\right|dy\le exp\left[(-\underset{y\in O}{inf}l(x,y)+\eta )t^{-(1/(2k-1\left)\right)}\right]$$

(16)

We have shown in [3] Lemma 9 the following result. For all $\delta$ and all C, there exists $s_{\delta }$ such that if $s\le s_{\delta }$, then

$$|P_{st}^L|\left[1_{B{(x,\delta )}^c}\right]\left(x\right)\le exp[-C{t}^{-1/(2k-1)}]$$

(17)

This means that

$${\int }_{B{(x,\delta )}^c}\left|p_{st}^L\right|(x,y)dy\le exp[-C{t}^{-1/(2k-1)}]$$

(18)

The two previous estimates are uniform. Equations (14), (15), and (18) will allow us to conclude. By the the semi-group property

$$p_t^L(x,y)={\int }_M{p}_{(1-s)t}^L(x,z)p_{st}^L(z,y)dz,$$

(19)

we deduce that

$$|p_t^L(x,y)|\le A+B,$$

(20)

where

$$A={\int }_{B(y,\delta )}|p_{(1-s)t}^L\left|\right|p_{\left(st\right)}^L(z,y)|dz$$

(21)

and

$$B={\int }_{B{(y,\delta )}^c}|p_{(1-s)t}^L\left|\right|p_{\left(st\right)}^L(z,y)|dz$$

(22)

If s is small enough,

$$B\le C{\left((1-s)t\right)}^{-l}exp[-C{t}^{-1/(2k-1)}]$$

(23)

If t is small enough,

$$A\le C{\left(st\right)}^{-l}exp[-(\underset{z\in B(y,\delta )}{inf}l(x,z)+\eta )t^{-(1/(2k-1\left)\right)}]$$

(24)

The conclusion holds if we choose a very small $\delta$ such that ${inf}_{z\in B(y,\delta )}l(x,z)$ is close to $l(x,y)$ because $(x,z)\to l(x,z)$ is continuous.