Stationary measures for stochastic differential equations with degenerate damping

Abstract

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in ${\mathbb {R}}^n$ with a quadratic, conservative nonlinearity B(x, x) and a linear damping term—Ax which is degenerate in the sense that $\textrm{ker} A \ne \emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if A is full rank, but otherwise, energy could potentially accumulate in $\textrm{ker} A$ and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of $\textrm{ker} A$ and many examples where such estimates can be made.

A A basic energy estimate

The following lemma quantifies how $B(x,x) \cdot x = 0$ and the additive nature of the noise imply that the energy level of a trajectory can only change a small amount in a short time.

Lemma A.1

Fix $\epsilon \in (0,1)$. There exist $K_*(\epsilon )\ge 1$, $\tau _*(\epsilon ) \le 1$, and $C > 0$ (which does not depend on $\epsilon$) such that for $0 \le \tau \le \tau _*$ and any $x_0 \in {\mathbb {R}}^n$ with $|x_0| = K \ge K_*$ there holds

$$\begin{aligned} {\textbf{P}}\left( \sup _{0\le t \le \tau } \left| |x_{t}|^2 - K^2\right| \ge \epsilon K^2 \right) \le \frac{C\tau }{\epsilon ^2 K^2}. \end{aligned}$$

(A.1)

Proof

It suffices to show

$$\begin{aligned} {\textbf{P}}\left( \sup _{0 \le t \le \tau } |x_t|^2 \ge (1+\epsilon /2)|x_0|^2 + R^2 \right) \le C \frac{\tau |x_0|^2}{R^4} \end{aligned}$$

(A.2)

and

$$\begin{aligned} {\textbf{P}}\left( \inf _{0 \le t \le \tau }|x_t|^2 \le (1-\epsilon /2)|x_0|^2 - R^2\right) C \frac{\tau |x_0|^2}{R^4} \end{aligned}$$

(A.3)

for some constant C that does not depend on $\epsilon$. Indeed, the desired result follows immediately by taking $R = \sqrt{\epsilon /2}|x_0|$ in (A.2) and (A.3).

We begin with the proof of (A.2). By Itô’s formula and $B(x,x)\cdot x = 0$, we have

$$\begin{aligned} |x_t|^2- |x_0|^2 = 2\int _0^t x_s \cdot \sigma dW_s - 2\int _0^t Ax_s \cdot x_s ds + t \sum _{i,j=1}^n |\sigma _{ij}|^2. \end{aligned}$$

(A.4)

Thus,

$$\begin{aligned} {\textbf{E}}|x_t|^2 \le |x_0|^2 + C_\sigma t, \end{aligned}$$

(A.5)

where we have set $C_\sigma = \sum _{i,j=1}^n |\sigma _{ij}|^2$. Using the martingale inequality followed by Itô isometry and (A.5) in (A.4) gives

$$\begin{aligned}&{\textbf{P}}\left( \sup _{0 \le t \le \tau } |x_t|^2 - |x_0|^2 - C_\sigma t \ge R^2 \right) \le {\textbf{P}}\left( 2\sup _{0 \le t \le \tau }\left| \int _0^t x_s \cdot \sigma dW_s \right| \ge R^2 \right) \nonumber \\&\qquad \le \frac{4}{R^4}{\textbf{E}}\left| \int _0^\tau x_s \cdot \sigma dW_s\right| ^2 \le \frac{C}{R^4} \int _0^\tau {\textbf{E}}|x_s|^2 ds \le C \frac{\tau |x_0|^2}{R^4}, \end{aligned}$$

(A.6)

where in the last inequality we assume that $K_*$ is sufficiently large. The bound (A.2) then follows provided $K_* \ge \sqrt{2C_\sigma /\epsilon }$.

Now we turn to the proof of (A.3). Let ${\tilde{x}}_t = e^{\lambda _A t}x_t$, where $\lambda _A > 0$ is the largest eigenvalue of A. Then,

$$\begin{aligned} d {\tilde{x}}_t = \lambda _A {\tilde{x}}_t dt + e^{\lambda _A t}B(x_t,x_t)dt - A{\tilde{x}}_t dt + e^{\lambda _A t}\sigma dW_t. \end{aligned}$$

(A.7)

Since ${\tilde{x}}_t \cdot e^{\lambda _A t} B(x_t,x_t) = e^{2\lambda _A t} B(x_t,x_t) \cdot x_t = 0$, another application of Itô’s lemma gives

$$\begin{aligned} d|{\tilde{x}}_t|^2 = 2\lambda _A |\tilde{x_t}|^2 dt - 2 A{\tilde{x}}_t \cdot {\tilde{x}}_t dt + 2 e^{\lambda _A t} {\tilde{x}}_t \cdot \sigma dW_t + C_\sigma e^{2\lambda _A t} dt. \end{aligned}$$

(A.8)

Using $\lambda _A |{\tilde{x}}_t|^2 \ge A{\tilde{x}}_t \cdot {\tilde{x}}_t$, (A.8) implies

$$\begin{aligned} |{\tilde{x}}_t|^2 \ge |x_0|^2 - 2 \left| \int _0^t e^{\lambda _A s}{\tilde{x}}_s \cdot \sigma dW_s\right| , \end{aligned}$$

(A.9)

and hence for $0\le t \le \tau$ there holds

$$\begin{aligned} |x_t|^2 \ge e^{-2 \lambda _A \tau }|x_0|^2 - 2\left| \int _0^t e^{2 \lambda _A s}x_s \cdot \sigma dW_s\right| . \end{aligned}$$

(A.10)

Proceeding as in the proof of (A.6) we obtain

$$\begin{aligned} {\textbf{P}}\left( \inf _{0\le t \le \tau } |x_t|^2 \le e^{-2\lambda _A \tau }|x_0|^2 - R^2\right) \le \frac{C}{R^4}\int _0^\tau {\textbf{E}}|x_s|^2 ds \le C \frac{\tau |x_0|^2}{R^4}. \end{aligned}$$

(A.11)

The desired bound follows provided that $\tau _*$ is small enough so that $e^{-2\lambda _A \tau _*} \ge 1-\epsilon /2$. $\square$