A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

Abstract

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation (“the cell problem”), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.

Appendices

A Material from [8]

In this appendix, we collect several useful assumptions and results from [8]. The assumptions listed here are not used in the entirety of this current work. When they are needed for a particular result we explicitly reference them.

Assumption A.1

We assume that the Hamiltonian has the form given in (9) where K and $\psi$ are $C^2$ and K is non-negative. For every $T>0$, we assume the following bounds hold on $[0,T]\times \mathbb {R}^{2n}$:

• 1. There exist $C>0$ and $M>0$ such that

  $$\begin{aligned} \max \{|\partial _t K(t,q,z)|,\Vert \nabla _qK(t,q,z)\Vert \}\le M+CK(t,q,z). \end{aligned}$$

• 2. There exist $c>0$ and $M\ge 0$ such that

  $$\begin{aligned} \Vert \nabla _z K(t,q,z)\Vert ^2+M\ge c K(t,q,z). \end{aligned}$$

• 3. For every $\delta >0$ there exists an $M>0$ such that

  $$\begin{aligned} \max \left\{ \Vert \nabla _z K(t,q,z)\Vert ,\left( \sum _{ij}|\partial _{z_i}\partial _{z_j}K(t,q,z)|^2\right) ^{1/2}\right\} \le M+\delta K(t,q,z). \end{aligned}$$

Assumption A.2

For every $T>0$, we assume that the following hold uniformly on $[0,T]\times \mathbb {R}^n$:

• 1. V is $C^2$ and $\nabla _q V$ is bounded

• 2. $\gamma$ is symmetric with eigenvalues bounded below by some $\lambda >0$.

• 3. $\gamma$, F, $\partial _t\psi$, and $\sigma$ are bounded.

• 4. There exists $C>0$ such that the (random) initial conditions satisfy

  $$K^\epsilon (0,x^\epsilon _0)\le C$$

  for all $\epsilon >0$ and all $\omega \in \mathrm {\Omega }$.

Assumption A.3

We assume that for every $T>0$ there exists $c>0$, $\eta >0$ such that

$$\begin{aligned} K(t,q,z)\ge c\Vert z\Vert ^{2\eta } \end{aligned}$$

on $[0,T]\times \mathbb {R}^{2n}$.

Assumption A.4

We assume that $\gamma$ is $C^1$ and is independent of p.

Assumption A.5

We assume that K has the form

$$\begin{aligned} K(t,q,z)=\tilde{K}(t,q,A^{ij}(t,q)z_iz_j) \end{aligned}$$

where $\tilde{K}(t,q,\zeta )$ is $C^2$ and non-negative on $[0,\infty )\times \mathbb {R}^n\times [0,\infty )$ and A(t, q) is a $C^2$ function whose values are symmetric $n \times n$-matrices. We also assume that for every $T>0$, the eigenvalues of A are bounded above and below by some constants $C>0$ and $c>0$ respectively, uniformly on $[0,T]\times \mathbb {R}^n$.

We will write $\tilde{K}^\prime$ for $\partial _\zeta \tilde{K}$ and will use the abbreviation $\Vert z\Vert _A^2$ for $A^{ij}(t,q)z_iz_j$ when the implied values of t and q are apparent from the context.

Assumption A.7

We assume that, for every $T>0$, $\nabla _q V$, F, and $\sigma$ are Lipschitz in x uniformly in $t\in [0,T]$. We also assume that A and $\gamma$ are $C^2$, $\psi$ is $C^3$, and $\partial _t\psi$, $\partial _{q^i}\psi$, $\partial _{q^i}\partial _{q_j}\psi$, $\partial _t\partial _{q^i}\psi$, $\partial _t\partial _{q^j}\partial _{q^i}\psi$, $\partial _{q^l}\partial _{q^j}\partial _{q^i}\psi$, $\partial _t\gamma$, $\partial _{q^i} \gamma$, $\partial _t\partial _{q^j}\gamma$, $\partial _{q^i}\partial _{q^j}\gamma$, $\partial _t A$, $\partial _{q^i} A$, $\partial _t \partial _{q^i}A$, and $\partial _{q^i}\partial _{q^j} A$ are bounded on $[0,T]\times \mathbb {R}^{2n}$ for every $T>0$.

Note that, combined with Assumptions A.1–A.4, this implies $\tilde{\gamma }$, $\tilde{\gamma }^{-1}$, $\partial _t\tilde{\gamma }^{-1}$, $\partial _{q^i}\tilde{\gamma }^{-1}$, $\partial _t\partial _{q^j}\tilde{\gamma }^{-1}$, and $\partial _{q^i}\partial _{q^j}\tilde{\gamma }^{-1}$ are bounded on compact t-intervals.

Lemma A.1

Under Assumptions A.1 and A.2, for any $T>0$, $p>0$ we have

$$\begin{aligned} E\left[ \sup _{t\in [0,T]}\Vert q_t^\epsilon \Vert ^p\right] <\infty . \end{aligned}$$

Lemma A.2

Under Assumptions A.1–A.3, for any $T>0$, $p>0$ we have

$$\begin{aligned} \sup _{t\in [0,T]}E[\Vert p_t^\epsilon -\psi (t,q_t^\epsilon )\Vert ^{p}]=O(\epsilon ^{p/2}) \text { as }\epsilon \rightarrow 0^+ \end{aligned}$$

and for any $p>0$, $T>0$, $0<\beta <p/2$ we have

$$\begin{aligned} E\left[ \sup _{t\in [0,T]}\Vert p_t^\epsilon -\psi (t,q_t^\epsilon )\Vert ^{p}\right] =O(\epsilon ^\beta ) \text { as }\epsilon \rightarrow 0^+. \end{aligned}$$

The following is a slight variant of the result from [8], but the proof is identical.

Lemma A.3

Let $T>0$ and suppose we have continuous functions $\tilde{F}(t,x):[0,\infty )\times \mathbb {R}^{n+m}\rightarrow \mathbb {R}^n$, $\tilde{\sigma }(t,x):[0,\infty )\times \mathbb {R}^{n+m}\rightarrow \mathbb {R}^{n\times k}$, and $\psi :[0,\infty )\times \mathbb {R}^n\rightarrow \mathbb {R}^m$ that are Lipschitz in x, uniformly in $t\in [0,T]$.

Let $W_t$ be a k-dimensional Wiener process, $p\ge 2$ and $\beta >0$ and suppose that we have continuous semimartingales $q_t$ and, for each $0<\epsilon \le \epsilon _0$, $\tilde{R}_t^\epsilon$, $x_t^\epsilon =(q_t^\epsilon ,p_t^\epsilon )$ that satisfy the following properties:

• 1. $q_t^\epsilon =q_0^\epsilon +\int _0^t\tilde{F}(s,x_s^\epsilon )ds+\int _0^t\tilde{\sigma }(s,x_s^\epsilon )dW_s+\tilde{R}^\epsilon _t$.

• 2. $q_t=q_0+\int _0^t\tilde{F}(s,q_s,\psi (s,q_s))ds+\int _0^t\tilde{\sigma }(s,q_s^\epsilon ,\psi (s,q_s))dW_s$.

• 3. $E[\Vert q_0^\epsilon -q_0\Vert ^p]=O(\epsilon ^\beta )\text { as }\epsilon \rightarrow 0^+$.

• 4. $E\left[ \sup _{t\in [0,T]}\Vert \tilde{R}_t^\epsilon \Vert ^p\right] =O(\epsilon ^\beta )\text { as }\epsilon \rightarrow 0^+$.

• 5. $\sup _{t\in [0,T]}E[\Vert p_t^\epsilon -\psi (t,q_t^\epsilon )\Vert ^p]=O(\epsilon ^\beta )\text { as }\epsilon \rightarrow 0^+$.

• 6. $E\left[ \sup _{t\in [0,T]}\Vert q_t^\epsilon \Vert ^p\right] <\infty$ for all $\epsilon >0$ sufficiently small.

• 7. $E\left[ \sup _{t\in [0,T]}\Vert q_t\Vert ^p\right] <\infty$.

Then

$$\begin{aligned} E\left[ \sup _{t\in [0,T]}\Vert q_t^\epsilon -q_t\Vert ^p\right] =O(\epsilon ^\beta )\text { as }\epsilon \rightarrow 0^+. \end{aligned}$$

B Polynomial Boundedness of $\tilde{\chi }$

Changing variables, $\tilde{\chi }$ can be expressed as

$$\begin{aligned} \tilde{\chi }(t,q,\zeta )&=\frac{1}{2b_1(t,q)}\zeta \int _0^1\exp [\beta (t,q) \tilde{K}(t,q,s\zeta )]\\&\times \int _0^{1} r^{(m-2)/2}\exp [-\beta (t,q) \tilde{K}(t,q,rs\zeta )]\left( G(t,q,rs\zeta )-\tilde{G}(t,q)\right) dr ds. \end{aligned}$$

Applying the DCT to this expression several times, one can prove that $\tilde{\chi }$ is $C^{1,2}$. Using the fact that $\tilde{K}$ and $\partial _{q^i}\tilde{K}$ are polynomially bounded in $\zeta$, uniformly in $(t,q)\in [0,T]\times \mathbb {R}^n$, one can see that $\tilde{\chi }(t,q,\zeta )$ is bounded on $[0,T]\times \mathbb {R}^n\times [0,\zeta _0]$ for any $\zeta _0>0$. From Assumption 4.2, there exists $\zeta _0$ and $C>0$ such that $\tilde{K}^\prime (t,q,\zeta )\ge C$ for all $(t,q,\zeta )\in [0,T]\times \mathbb {R}^n\times [\zeta _0,\infty )$.

By combining (42) with (40), one finds that for $\zeta \ge \zeta _0$, $\tilde{\chi }$ can alternatively be written as

$$\begin{aligned} \tilde{\chi }(t,q,\zeta )=&\tilde{\chi }(t,q,\zeta _0)+\frac{1}{2b_1(t,q)}\int _{\zeta _0}^\zeta \zeta _1^{-m/2}\exp [\beta (t,q) \tilde{K}(t,q,\zeta _1)]\\&\times \int _{\zeta _1}^\infty \exp [-\beta (t,q)\tilde{K}(t,q,\zeta _2)]\zeta _2^{(m-2)/2}(\tilde{G}(t,q)-G(t,q,\zeta _2))d\zeta _2 d\zeta _1. \end{aligned}$$

Therefore, if we can show that the second term has the polynomial boundedness property then so does $\tilde{\chi }$, and hence $\chi$.

Letting $\tilde{C}$ denote a constant that potentially changes in each line and choosing $\zeta _0$ as in Assumption 4.2, we have

for some $q>0$. To obtain the first inequality, we use polynomial boundedness of $\partial _{q^i}\tilde{K}$. For the second, we used Assumption (A.3) together with the fact that $\tilde{K}^\prime \ge C>0$ on $[0,T]\times \mathbb {R}^n\times [\zeta _0,\infty )$.

Therefore we obtain

$$\begin{aligned} \Vert \tilde{\chi }(t,q,\zeta )\Vert \le \tilde{C}\left( 1+\int _{\zeta _0}^\zeta P(\tilde{K}(t,q,\zeta _1))d\zeta _1\right) \end{aligned}$$

for some polynomial P(x) with positive coefficients that are independant of t and q. Polynomial boundedness of $\tilde{K}$ then implies

$$\begin{aligned} \Vert \tilde{\chi }(t,q,\zeta )\Vert \le \tilde{C}\left( 1+ \int _{\zeta _0}^\zeta Q(\zeta _1)d\zeta _1\right) \end{aligned}$$

for some polynomial $Q(\zeta )$. This proves the desired polynomial boundedness property for $\tilde{\chi }$.