Convergence to Equilibrium in Fokker–Planck Equations

Abstract

The present paper is devoted to the investigation of long-time behaviors of global probability solutions of Fokker–Planck equations with rough coefficients. In particular, we prove the convergence of probability solutions under a Lyapunov condition in terms of the Markov semigroup associated to the stationary one. A generalization of earlier results on the existence and uniqueness of global probability solutions is also given.

Appendix A. Stationary Measures and Associated Markov Semigroups

In this section, we recall some results obtained in [12] (also see [6, 7]) concerning Markov semigroups associated to stationary measures.

Let $\mathcal {P}$ be the set of Borel probability measures on $\mathcal {U}$. Let

$$\begin{aligned} \mathcal {M}:=\left\{ \mu \in \mathcal {P}:L\mu =0\,\,\text {in the sense of Definition 1.3}\right\} \end{aligned}$$

be the set of stationary measures of (1.2).

The following results describe the existence and some properties of sub-Markov semigroups associated to a given stationary measure of (1.2).

Proposition A.1

Let $\mu \in \mathcal {M}$.

• (1) If (H) is assumed with $p>n+2$ replaced by $p>n$, then there exists a closed extension $(\overline{\mathcal {L}},\mathcal {D}(\overline{\mathcal {L}}))$ of $(\mathcal {L},C_{0}^{\infty }(\mathcal {U}))$ generating a sub-Markov contractive $C_{0}$-semigroup $(T_{t})_{t\ge 0}$ on $L^{1}(\mathcal {U},\mu )$ such that $\mu$ is sub-invariant for $(T_{t})_{t\ge 0}$, i.e.,

  $$\begin{aligned} \int _{\mathcal {U}}T_{t}\phi d\mu \le \int _{\mathcal {U}}\phi d\mu ,\quad t\ge 0 \end{aligned}$$

  for all $\phi \in L^{\infty }(\mathcal {U},\mu )$ with $\phi \ge 0$.

• (2) If (H) is assumed, then there exist unique sub-probability kernels $K_{t}(\cdot ,dy)$, $t>0$, on $\mathcal {U}$ such that

  $$\begin{aligned} K_{t}(x,dy)=p(t,x,y)dy, \end{aligned}$$

  where p(t, x, y) is locally Hölder continuous and positive on $(0,\infty )\times \mathcal {U}\times \mathcal {U}$, and for each $\phi \in L^{1}(\mathcal {U},\mu )$, the function

  $$\begin{aligned} x\mapsto K_{t}\phi (x):=\int _{\mathcal {U}}\phi (y)p(t,x,y)dy,\quad \mathcal {U}\rightarrow \mathbb {R}\end{aligned}$$

  is a $\mu$-version of $T_{t}\phi$ such that $(t,x)\rightarrow K_{t}\phi (x)$ is continuous on $(0,\infty )\times \mathcal {U}$. In addition, if $\tilde{\mu }\in \mathcal {P}$ is invariant for $(K_{t})_{t\ge 0}$, i.e.,

  $$\begin{aligned} \tilde{\mu }=K^{*}_{t}\tilde{\mu }(dy):=\int _{\mathcal {U}}K_{t}(x,dy)d\nu (x),\quad t\ge 0, \end{aligned}$$

  then $\tilde{\mu }=\mu$.

Proof

See [12, Theorem 2.3] or [26] for (1), and [12, Theorem 4.4] or [5, Theorem 4.1, Corollary 4.3] for (2). $\square$

Here are some remarks, implied by Proposition A.1(2), about the semigroup $(K_{t})_{t\ge 0}$ given in Proposition A.1(2) (see [6, Remark 1.7.6]).

Remark A.1

Assume (H).

• (1) The semigroup $(K_{t})_{t\ge 0}$ is strongly Feller and stochastically continuous.

• (2) The probability measures

  $$\begin{aligned} B\mapsto K_{t}\chi _{B}(x):=\int _{B}p(t,x,y)dy,\quad t>0,\quad x\in \mathcal {U}\end{aligned}$$

  are equivalent. In particular, if $\mu$ is invariant for $(K_{t})_{t\ge 0}$, i.e.,

  $$\begin{aligned} \int _{\mathcal {U}}K_{t}\phi d\mu =\int _{\mathcal {U}}\phi d\mu ,\quad t\ge 0, \end{aligned}$$

  for all $\phi \in L^{1}(\mathcal {U},\mu )$, Doob’s theorem (see e.g. [15, Theorem 4.2.1]) yields

  $$\begin{aligned} \lim _{t\rightarrow \infty }K_{t}\chi _{B}(x)=\mu (B),\quad \forall x\in \mathcal {U}\end{aligned}$$

  for any Borel set $B\subset \mathcal {U}$.

• (3) By the proof of [5, Theorem 4.1], the transition density function p(t, x, y) of $(K_{t})_{t\ge 0}$ is given by

  $$\begin{aligned} p(t,x,y)=p_{t}(x,y)\varrho (y),\quad (t,x,y)\in (0,\infty )\times \mathcal {U}\times \mathcal {U}, \end{aligned}$$

  where $\varrho \in W^{1,p}_{loc}(\mathcal {U})$ is the density of $\mu$, and $(t,x,y)\mapsto p_{t}(x,y)$ is continuous on $(0,\infty )\times \mathcal {U}\times \mathcal {U}$ and satisfies

  $$\begin{aligned} \sup _{x,y\in \mathcal {K}}\sup _{z\in \mathcal {U}}\frac{|p_{t}(x,z)-p_{t}(y,z)|}{|x-y|^{\alpha }}<\infty \end{aligned}$$

  for any $t>0$ and any compact set $\mathcal {K}\subset \mathcal {U}$, where $\alpha >0$ is some constant.

• (4) By [12, Theorem 2.3(iii)], or [6, Theorem 1.5.7(iii)], for any $\phi \in C_{0}^{\infty }(\mathcal {U})$, $T_{t}\phi$ has a continuous modification, which must be $K_{t}\phi$, such that

  $$\begin{aligned} K_{t}\phi (x)\rightarrow \phi (x)\quad \text {as}\quad t\rightarrow 0^{+}\quad \text {locally uniformly in}\quad x\in \mathcal {U}. \end{aligned}$$

In the next result, sufficient conditions for the sub-Markov semigroup $(T_{t})_{t\ge 0}$ in Proposition A.1 being a Markov semigroup are provided.

Proposition A.2

Assume (H) with $p>n+2$ replaced by $p>n$. Let $\mu \in \mathcal {M}$. Then, the following two assertions are equivalent:

• (1) For some (and therefore all) $\lambda >0$, there holds $L^{1}(\mathcal {U},\mu )=\overline{(\mathcal {L}-\lambda I)(C_{0}^{\infty }(\mathcal {U}))}$;

• (2) There exists a unique $C_{0}$-semigroup in $L^{1}(\mathcal {U},\mu )$ whose generator extending $(\mathcal {L},C_{0}^{\infty }(\mathcal {U}))$.

If one of the above two equivalent assertions holds, then the semigroup $(T_{t})_{t\ge 0}$ given in Proposition A.1(1) is a Markov semigroup and $\mu$ is invariant for $(T_{t})_{t\ge 0}$.

Proof

See [12, Proposition 2.6]. $\square$

Set

$$\begin{aligned} \mathcal {M}_{md}=\left\{ \mu \in \mathcal {M}:L^{1}(\mathcal {U},\mu )=\overline{(\mathcal {L}-I)(C_{0}^{\infty }(\mathcal {U}))}\right\} . \end{aligned}$$

(A.1)

The following result holds.

Proposition A.3

Assume (H) with $p>n+2$ replaced by $p>n$. If $\mathcal {M}_{md}\ne \emptyset$, then $\#\mathcal {M}=1$.

Proof

See [12, Theorem 4.1]. $\square$