Stochastic Processes in Neuroscience

Part II

Wilhelm Stannat

Institut fur MathematikTechnische Universitat Berlin

stannat@math.tu-berlin.de

October 16, 2016

Lecture held in the winter term 2016/17.i

Contents

Chapter 5. Asymptotic Behavior and Stability 15.1. Linearized Stability Analysis 15.2. Lyapunov Stability for Ordinary Differential Equations 35.3. Stability for Stochastic Differential Equations 65.4. Invariant Measures for Stochastic Differential Equations 11

Bibliography 19

v

CHAPTER 5

Asymptotic Behavior and Stability

In the last two chapters we stochastic differential equations and first modelsof oscillators under the influence of noise describing one single neuron. For suchequations it is rarely the case that we can obtain explicit solution formulas, thus anexact (pathwise) description of the dynamics is challenging. However, most of the“interesting” local behavior occurs in the neighborhood of equilibrium pointsand it is of interest how small or large perturbations in the initial condition or dueto noise effect the behavior of solutions. One question might be if solutions staynear equilibrium points or if they even tend to move closer as time elapses. Morallyspeaking, this is the subject of stability theory.

Let us now be more specific and consider for nonlinear b : Rd → Rd, d ≥ 1, thesystem of autonomous ordinary differential equations

(5.1)d

dtXt = b

(Xt

).

A point x∗ ∈ Rd is called equilibrium point of (5.1), if b(x∗) = 0. In particular,Xt = x∗ is a constant solution to (5.1) with initial value X0 = x∗. We define severaltypes of stability in the next section and also introduce the relatively elementarynotion of linear stability the method of linearized stability analysis for equationsof type (5.1). Since the scope of this simple method is rather restricted we also studya nonlinear method due to Lyapunov, which has its counterpart for stochasticdifferential equations. In the last part we generalize these concepts in some senseand study the the law of solutions and their asymptotic behavior, which leads to thenotion of invariant measures. Note that the scope of this chapter is to providea toolbox for applications rather than to introduce a detailed, general theory onthese topics.

5.1. Linearized Stability Analysis

First, we introduce a formal definition of the concept of stability, actually aslightly more general one than we need at this point.

Definition 5.1. An equilibrium point x∗ is called

(1) stable, if for all ε > 0 there exists δ > 0 such that

‖X0 − x∗‖ < δ ⇒ ‖Xt − x∗‖ < ε for all t ≥ 0,

(2) locally asymptotically stable, if it is stable and locally attracting, i. e.there exists δ > 0 such that

‖X0 − x∗‖ < δ ⇒ limt→∞

‖Xt − x∗‖ = 0,

1

2 5. ASYMPTOTIC BEHAVIOR AND STABILITY

(3) globally asymptotically stable, if it is also globally attracting, i. e.

limt→∞

‖Xt − x∗‖ = 0 for all X0 ∈ Rd,

(4) locally/globally exponentially stable, if there exist constants M > 0,κ∗ > 0 such that

‖Xt − x∗‖ ≤Me−κ∗t‖X0 − x∗‖

for all t ≥ 0 and X0 from the δ-ball around x∗ and X0 ∈ Rd in the globalcase, respectively,

(5) unstable, if it is not stable.

Remark 5.2. The stability from i. is also called Lyapunov stability and it fitsfor a more general framework introduced in the next section. By definition of theconcepts above one can see immediately that

loc./glob. exponentially stable ⇒ loc./glob. asymptotically stable ⇒ stable.

For an illustration of the method called linear stability analysis we go back toa system of linear ordinary differential equations given by

(5.2)d

dtXt = AXt ∈ Rd.

In this case it is obvious, that x∗ = 0 is an equilibrium point of (5.2) and we wantto study its stability depending on the properties of the matrix A.

Theorem 5.3. Let x∗ = 0 be the equilibrium point of (5.2).

(1) x∗ = 0 is stable, iff all eigenvalues have a non-positive real part and forall eigenvalues with real part equal to 0 the eigenspace has dimension 1.

(2) x∗ = 0 is unstable, iff there exists an eigenvalue with positive real part orreal part equal to 0 and eigenspace of dimension ≥ 2.

(3) x∗ = 0 is locally asymptotically stable, iff all eigenvalues have negativereal part.

Proof. We only prove the theorem in the special case of A having diagonalform. Then, we can easily compute the solution to (5.2), which is Xt = etAX0, andin terms of the orthonormal basis ek and the eigenvalues λk of A the followingholds

‖Xt − x∗‖ = ‖etAX0‖ ≤(∑

k

|X0 · ek|)

maxk‖etAek‖ =

√d‖X0‖max

keλkt.

If there exist k with Reλk > 0 this obviously tends to infinity as t → ∞. Also inthe case of iii. we have that ‖Xt‖ → 0 as t→∞. If there exists k with Reλk = 0,

then ‖Xt‖ ≤√d‖X0‖ and given ε > 0 the choice of δ = ε/

√d yields stability.

There is an easy consequence concerning the equivalence of the different notionsof stability in Definition 5.1.

Corollary 5.4. The following assertions are equivalent.

(1) All eigenvalues of A have negative real parts.(2) x∗ = 0 is locally asymptotically stable.(3) x∗ = 0 is globally exponentially stable with rate κ∗ given by the maximum

of all of A’s eigenvalues’ real parts.

5.2. LYAPUNOV STABILITY FOR ORDINARY DIFFERENTIAL EQUATIONS 3

Now let us focus again on the nonlinear ordinary differential equation (5.1)where b : Rd → Rd is globally Lipschitz continuous, i. e. there exists L > 0 suchthat ‖b(x)− b(y)‖ ≤ L‖x− y‖ for all x, y ∈ Rd. It is well-known that for everyinitial value x there exists a unique (possibly explosive) solution Xt with X0 = x.Let x∗ be an equilibrium point for this system and suppose in addition that b isalso continuously differentiable in x∗. Then by definition and with b(x∗) = 0

b(x∗ + h) = b(x∗) +Db(x∗)h+R(h) = Db(x∗)h+R(h)

where Db denotes the Jacobian and the error terms R are of higher order, i. e.R(h) ∈ o(‖h‖). In a small neighborhood around x∗ it is reasonable to look at thelinearized equation for Yt = Xt − x∗

(5.3)d

dtYt = JYt, J = Db(x∗).

The main objective is to compare the dynamics of (5.1) with the simpler one (5.3),of course always only in a small neighborhood of the equilibrium point x∗. It isoften convenient to set x∗ = 0 in such results, which can always be achieved by ashift in the phase space. With the help of the stability results for linear systems wecan state the main result below for equilibrium points with an additional property.

Definition 5.5. An equilibrium point x∗ of (5.1) is called hyperbolic if thematrix J has no eigenvalues with zero real part.

Theorem 5.6 (Linearized Stability). Let x∗ = 0 be a hyperbolic equilibriumpoint for (5.1). Then, x∗ is also an equilibrium point for its linearization (5.3) andit is either locally exponentially stable for (5.1), if all eigenvalues of J have negativereal parts, or unstable if at least one eigenvalue has positive real part.

Proof. Exercise.

Remark 5.7. Note that the Theorem 5.6 remains silent on the issue of whathappens if some eigenvalues have zero real parts while all others are negative. Inthese cases, the analysis of the linearization is not enough and nonlinear effects areresponsible for stability or instability of the equilibrium point. In particular forhigh dimensional systems such a scenario does not happen in rare cases.

5.2. Lyapunov Stability for Ordinary Differential Equations

The scope of the linearized stability analysis we just introduced is the behaviorof a system near an equilibrium point when a linear approximation is adequate. Itdoes not however tell us anything when nonlinear effects dominate or about whathappens farther away from equilibrium. For this reason we consider a nonlinearmethod due to Lyapunov which is based on so-called Lyapunov functions that—morally speaking—have x∗ as a global minimum and are monotone decreasing alongtrajectories of Xt.

The following special class of (5.1) is useful to get a good grasp of Lyapunov’sidea. Consider b being the gradient of a potential V ∈ C2(Rd;R), i. e.

(5.4)d

dtXt = −∇V

(Xt

),

where∇V = (∂x1V, . . . , ∂xdV ) denotes the gradient of V . In this case all equilibriumpoints x∗ are zeros of ∇V , in particular they are local extrema. Now suppose that

4 5. ASYMPTOTIC BEHAVIOR AND STABILITY

V has exactly one global minimum x∗ and that ∇V (x) 6= 0 for all x 6= x∗. We cannow look at the time evolution of V (Xt) and it follows that

d

dtV(Xt

)= 〈∇V

(Xt

)〉 ddtXt = −‖∇V

(Xt

)‖2 < 0 if Xt 6= x∗,

where 〈x〉y denotes the usual inner product in Rd. If in addition V is uniformlystrictly convex—since V ∈ C2 this implies the Hessian of V is uniformly positivedefinite—it follows by Taylor’s formula that

〈∇V (x)〉x− x∗ = 〈∇V (x∗) +HV (ξ)(x− x∗)〉x− x∗ ≥ κ∗‖x− x∗‖2

for all x ∈ Rd, where κ∗ > 0, ξ ∈ x∗ + θ(x − x∗) : θ ∈ [0, 1] and HV :=(∂xixjV )ij denotes the Hessian of V . This implies that the Euclidean norm squaredis monotone decreasing in t, even more

d

dt‖Xt − x∗‖2 = 2〈Xt − x∗〉

d

dtXt = −2〈∇V

(Xt

)〉Xt − x∗ ≤ −2κ∗‖Xt − x∗‖2.

With the product rule it follows that

‖Xt − x∗‖2 ≤ e−2κ∗t‖X0 − x∗‖2,

i. e. x∗ is globally exponentially stable with rate κ∗.The idea of Lyapunov was to mimic the roles of V and the Euclidean norm for

general dynamical systems to obtain a quantity, which is monotone decreasing intime. In the following we assume again w. l. o. g. that x∗ = 0.

Definition 5.8. Let U be some open neighborhood of x∗ = 0 and V ∈C1(U, [0,∞)). V is called a Lyapunov function for (5.1), if 〈∇V (x)〉b(x) ≤ 0and V is positive definite, i. e. V (0) = 0 and V (x) > 0 for every non-zero x ∈ U .V is called a strict Lyapunov function, if moreover 〈∇V (x)〉b(x) < 0 for everynon-zero x ∈ U .

Theorem 5.9 (Lyapunov Stability). Let V be a Lyapunov function for (5.1),then the equilibrium point x∗ = 0 is stable.

Proof. Since U is an open neighborhood around 0 we can choose ε > 0 suchthat the closure of the ε-ball Bε(0) around 0 is still contained in U . Set m :=min‖x‖=ε V (x). Since V is continuous and positive definite, m is positive and wecan find δ > 0 such that 0 < max‖x‖<δ V (x) < m. Now let X0 ∈ Bδ(0), thenV (X0) < m and

(5.5)d

dtV(Xt

)= 〈∇V

(Xt

)〉b(Xt

)≤ 0 ⇒ V

(Xt

)≤ V

(X0

)for every t ≥ 0. Thus, V (Xt) < m and ‖Xt‖ 6= ε for all t ≥ 0. Since we startedinside the ε-ball around 0 this shows ‖Xt‖ < ε and therefore x∗ = 0 is stable.

Theorem 5.10 (Asymptotic Stability). Let V be a strict Lyapunov function,then x∗ = 0 is locally asymptotically stable. Moreover, if U = Rd and V is coercive,i. e. lim‖x‖→∞V (x) = +∞, then x∗ = 0 is globally asymptotically stable.

Proof. Consider the first statement, then by Theorem 5.9, x∗ = 0 is stable.W. l. o. g. we can assume V can be extended to the closure U such that all itsproperties still hold. Stability guarantees that there exists δ > 0 such that ifX0 ∈ Bδ(0) we have Xt ∈ U for all times. It remains to prove that Xt → 0 ast→∞.

5.2. LYAPUNOV STABILITY FOR ORDINARY DIFFERENTIAL EQUATIONS 5

Now suppose V (Xt) does not converge to 0 as t → ∞. By (5.5) V (X·) isdecreasing and since V is positive definite and continuous there has to be c > 0such that V (Xt) ≥ c for every t ≥ 0 and also

inft≥0‖Xt‖ ≥ r

for some r > 0. By compactness of U\Br(0) the continuous function 〈∇V (Xt)〉b(Xt)attains its extrema, hence it is bounded away from 0. This implies d

dtV (Xt) ≤ −kfor some k > 0, which contradicts the non-negativity of V .

So far, we have proven via contradiction that V (Xt) → 0 as t → ∞. Givenε > 0 we can define

vε := minx∈U\Bε(0)

V (x) > 0

by compactness, continuity and positive definiteness. Obviously, there exists T > 0such that V (Xt) < vε for all t ≥ T . This implies Xt ∈ Bε(0) since it cannot leaveU due to stability. As ε was arbitrary, asymptotic stability follows.

The second statement remains an exercise.

In order to obtain exponential stability we have to restrict ourselves to a sub-class of Lyapunov functions as defined below.

Definition 5.11. A function V ∈ C1(U, [0,∞)) is called a quadratic Lya-punov function for (5.1), if there exist constants α1, α2, α3 > 0 such that

α1‖x‖2 ≤ V (x) ≤ α2‖x‖2 and 〈∇V (x)〉b(x) ≤ −α3‖x‖2

for all x ∈ U .

Theorem 5.12 (Exponential Stability). The equilibrium point x∗ = 0 is locally

exponentially stable with κ∗ = α3

2α2 and M =√

α2

α1, iff there exists a quadratic

Lyapunov function for (5.1).

Proof. Exercise.

As a concluding remark concerning the two discussed methods one might saythat since it is not always straightforward to obtain a suitable Lyapunov function,the method of linearized stability analysis seems easier to apply. However as afact, whenever linearization works we can always find a V (x) = ‖x‖2, where ‖·‖is some equivalent norm such that 〈x〉Jx ≤ −c‖x‖2 holds for some c > 0 andall x ∈ Rd. On the contrary, there are very simple (d = 1) examples with non-hyperbolic equilibrium points for which the linearization approach is useless.

Example 5.13. Consider d = 2 and b(x, y) = (−x3 + 2y3,−2xy2). The point(0, 0) is an equilibrium point but apparently a non-hyperbolic one, since all eigen-values are 0. Thus, we have to find a suitable Lyapunov function and reason-able (and often useful) guess is the square of the Euclidean norm or more generalV (x, y) := αx2+βxy+y2 with α, β to be determined. We immediately see that thischoice of V is positive definite on U = R2. In our case, we stick with V (x) = x2+y2,hence

〈∇V (x, y)〉b(x, y) = 2x(− x3 + 2y3

)+ 2y

(− 2xy2

)= −2x4 + 4xy3 − 4xy3 = −2x4 ≤ 0.

6 5. ASYMPTOTIC BEHAVIOR AND STABILITY

Theorem 5.9 yields stability for (0, 0). For asymptotic stability or even exponentialstability one might try to look for a different choice of V (see exercises), but an-other method called LaSalle’s Invariance Principle allows to prove asymptoticstability in such cases.

Definition 5.14. (1) A subset S ⊂ Rd is called the ω-limit set of thesolution X to (5.1) with initial value X0, if for every y ∈ S we can finda sequence tn ∞ such that Xtn → y as n → ∞. One also denotesS = ω(X0).

(2) A subset S ⊂ Rd is called a (forward) invariant set, if for all y ∈ Sand X0 = y we have Xt ∈M for all t ≥ 0.

Remark 5.15. In other words, the ω-limit set is the set of all accumulationpoints of a given trajectory with start in X0. Clearly, any ω-limit set is forwardinvariant.

Theorem 5.16 (LaSalle’s Invariance Principle). Suppose there exists a neigh-borhood U of x∗ = 0 and a positive definite function V : U → R with 〈∇V (x)〉b(x) ≤0 for all x ∈ U . Define

s := x ∈ U : 〈∇V (x)〉b(x) = 0.Then, there exists δ > 0 such that for all X0 ∈ Bδ(0) the trajectory Xt converges tothe largest invariant set contained in S as t∞ , in particular ω(X0) is containedin the largest invariant set in S.

Corollary 5.17. If x∗ = 0 is the only invariant set in S, then it is locallyasymptotically stable.

Corollary 5.18. If U = Rd, V is coercive and x∗ = 0 is the only invariantset in S, then it is globally asymptotically stable.

Example 5.19. In Example 5.13 the Lyapunov function V was only positivesemi-definite. Thus, we cannot apply Theorem 5.10 to obtain asymptotic stability.However, the set S of points where 〈∇V (x)〉b(x) = 0 is small, in our case the y-axis,and the vector field b is not parallel to this set. Hence, x∗ = 0 is the only invariantset in S and Theorem 5.16 yields global asymptotic stability.

5.3. Stability for Stochastic Differential Equations

Throughout this section we shall study stability problems for stochastic differ-ential equations of the form

(5.6) dXt = b(Xt

)dt+ σ

(Xt

)dWt, X0 = ξ0,

with b : Rd → Rd, σ : Rd → Rd×n globally Lipschitz, W a n-dimensional Brow-nian motion on (Ω,F ,P) and ξ0 F-measurable. For a fixed, deterministic initialcondition x ∈ Rd we denote by Xx

t the solution to (5.6) at time t ≥ 0 with Xx0 = x.

The question of stability first raises another question, namely if there existsan equilibrium point. One can easily see that in the case of additive noise, i. e.σ = const there is no point x∗ where the right hand side vanishes. Thus, we restrictourselves to multiplicative noise and assume x∗ = 0 and b(0) = 0, σ(0) = 0 suchthat the constant process X0

t = 0 is a solution to (5.6). Recall that the restrictionto 0 is not a loss in generality. There are now several possibilites to define stabilityfor stochastic differential equations.

5.3. STABILITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS 7

Definition 5.20. The equilibrium x∗ = 0 is said to be

(1) stable in probability if for every ε ∈ (0, 1), δ > 0 there exists a r > 0such that

P[supt≥0‖Xx

t ‖ < δ

]≥ 1− ε for all ‖x‖ < r,

(2) asymptotically stable in probability if it is stable in probability andfor every ε ∈ (0, 1) there exists r > 0 such that

P[

limt→∞

‖Xxt ‖ = 0

]≥ 1− ε for all ‖x‖ < r.

Definition 5.21. The equilibrium x∗ = 0 is said to be

(1) p-stable, p > 0 if for every ε > 0 there exists r > 0 such that

supt≥0

E [‖Xxt ‖p] < ε for all ‖x‖ < r,

(2) asymptotically p-stable if it is p-stable and moreover

limt→∞

E [‖Xxt ‖p] = 0,

(3) exponentially p-stable if there are constants M,κ∗ > 0 such that forall t ≥ 0

E [‖Xxt ‖p] ≤M‖x‖pe−κ

∗t.

The special cases p = 1 and p = 2 are most commonly considered and are re-ferred to as stability in the mean and stability in mean square, respectively.In the following, we generalize the Lyaponov function approach to stochastic dif-ferential equations. Apparently, we first have to find a suitable definition of aLyaponov function for (5.6) and Ito’s formula for f ∈ C2(Rd;R) implies the follow-ing.

df(Xt

)=( d∑i=1

bi(Xt

)∂xif

(Xt

)+

1

2

d∑i,j=1

aij(Xt

)∂2xixjf

(Xt

))dt

+

d∑i=1

n∑j=1

∂xif(Xt)σij(Xt)dWjt

= Lf(Xt)dt+ dMft

with

(1) a = σ · σT ,

(2) the infinitesimal generator Lf(x) := 12

∑di,j=1 aij(x)∂2xixjf(x)+

∑di=1 bi(x)∂xif(x)

associated with (5.6), also called Kolmogorov operator and

(3) the local martingale Mft =

∫ t0

∑di,j=1 ∂xif(Xs)σij(Xs)dW

js .

This point of view is strongly related to the martingale problem for stochas-tic differential equations, see [SV79], namely given the infinitesimal generator Lassociated with (5.6), the process

f(Xt

)− f

(X0

)−∫ t

0

Lf(Xs

)ds = Mf

t

is a (local) martingale for every f ∈ C2(Rd).

8 5. ASYMPTOTIC BEHAVIOR AND STABILITY

A reasonable choise for a Lyaponov function V would be a positive definiteV ∈ C2(Rd;R) and LV ≤ 0. However, unlike in the deterministic case, where V issupposed to be sufficiently smooth around x = 0, in the stochastic case there mightnot even exist such functions which are smooth in the origin. For this reason let usdefine the class of functions following [?].

Definition 5.22. Let V : Rd → [0,∞) be twice continuously differentiableeverywhere, except possibly for x = 0 and continuous on all closed sets x : ‖x‖ ≥ε, ε > 0. V is called a Lyaponov function for (5.6) if LV ≤ 0 and V is positivedefinite. Again, V is a strict Lyapunov function if LV < 0 for all x 6= 0.

Lemma 5.23. Let β ∈ R, x 6= 0, then

E[‖Xx

t ‖β]≤ ‖x‖βekt,

where k is a constant depending only on β and the global Lipschitz constant of (5.6).

Proof. let ε > 0, then f(x) = ‖x‖β is twice continuously differentiable onx : ‖x‖ > ε. Define the stopping time

τε := inft ≥ 0 : ‖Xxt ‖ ≤ ε

and Ito’s formula yields

‖Xxt∧τε‖β =‖x‖β + β

∫ t∧τε

0

‖Xxs ‖β−2

[〈b(Xxs

)〉Xx

s +1

2

d∑i=1

aii(Xxs

)]ds

+ β

∫ t∧τε

0

‖Xxs ‖β−2

n∑j=1

〈σ·j(Xxs

)〉Xx

s dWjt

+β

2(β − 2)

∫ t∧τε

0

‖Xxs ‖β−4〈a

(Xsx

)Xxs 〉Xx

s ds.

Now taking expectations and using the global Lipschitz condition together withb(0) = 0 and σ(0) = 0, we obtain

E[‖Xx

t∧τε‖β]≤ ‖x‖β + kE

[∫ t∧τε

0

‖Xxs ‖β ds

]= ‖x‖β + k

∫ t

0

E[‖Xx

s∧τε‖β]ds.

Gronwall’s lemma implies the estimate

(5.7) E[‖Xx

t∧τε‖β]≤ ‖x‖βekt.

Also, it is clear that t ∧ τε < t ⊆ ‖Xxt∧τε‖ ≤ ε and therefore with Markov’s

inequality and (5.7) with β = −1

P [t ∧ τε < t] ≤ P[‖Xx

t∧τε‖−1 ≥ ε−1]≤ ε

‖x‖ekt.

Hence

(5.8) P [t ∧ τε → t as ε→ 0] = 1

and we can combine this with (5.7) to conclude the result.

With the lemma above we can state the main theorem on stability in probabil-ity.

5.3. STABILITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS 9

Theorem 5.24 (Stability). Let V be a Lyapunov function for (5.6). ThenV (Xt) converges P-a. s. as t→∞ and x∗ = 0 is stable in probability. Moreover forany δ > 0

lim‖x‖→0

P[supt≥0‖Xx

t ‖ ≥ δ]

= 0,

i.e., we have some form of local stability of x∗ = 0 with probability 1.

Proof. Let ε > 0 and τε as in Lemma 5.23. Then we can apply Ito’s formulato V (Xx

t∧τε)

V(Xxt∧τε

)= V

(Xxs∧τε

)+

∫ t∧τε

s∧τεLV(Xxr

)dr +MV

t∧τε −MVs∧τε ,

so that by LV ≤ 0 and the optional sampling theorem

E[V(Xxt∧τε

)| Fs∧τε

]≤ V

(Xxs∧τε

),

where Ft is the filtration generated by the Brownian motion W . In particular,V (Xx

t∧τε) is a non-negative supermartingale. Letting ε→ 0 it follows by (5.8) thatP[τ0 <∞

]= 0, thus also V (Xx

t ) is a non-negative supermartingale and the almostsure martingale convergence theorem implies that the limit limt→∞ V (Xx

t ) existsP-a. s.

For the proof of stability, let δ > 0 be given and define the stopping time

τδ := inft ≥ 0 : ‖Xxt ‖ > δ.

Obviously,

P[

sup0≤s≤t

‖Xxs ‖ > δ

]= P [τδ ≤ t]

and furthermore

E[V(Xxt∧τδ

)]≥ E

[1τδ≤tV

(Xxτδ

)]≥ P [τδ ≤ t] inf

‖x‖≥δV (x) =: P [τδ ≤ t]Vδ.

Thus, using Markov’s inequality and the supermartingale property we have shownthat

P[

sup0≤s≤t

‖Xxs ‖ > δ

]≤ V −1δ E

[V(Xxt∧τδ

)]≤ V −1δ V (x)

and since V (0) = 0 and V continuous this implies the result by letting t→∞.

Theorem 5.25 (Asymptotic Stability). Let V be a strict Lyapunov functionfor (5.6). Then, x∗ = 0 is asymptotically stable in probability.

Proof. Let ε, δ > 0 be given, then by Theorem 5.24 we can find r > 0 suchthat

(5.9) P [‖Xxt ‖ < δ] ≥ 1− ε

4

for all ‖x‖ < r. Now choose 0 < α < β < ‖x‖ to be precisely determined later.Again, we need the stopping times from above, namely τα, the first entry time intothe ball of radius α, and τδ, the first exit time from the ball of radius δ. Then, Ito’sformula implies

E[V(Xxt∧τα∧τδ

)]≤ V (x) + E [t ∧ τα ∧ τδ] inf

‖y‖=αLV (y) =: −E [t ∧ τα ∧ τδ]Lα.

By assumption, Lα > 0 is well-defined. Markov’s inequality implies

tP [τα ∧ τδ ≥ t] ≤ E [t ∧ τα ∧ τδ] ≤ L−1α V (x),

10 5. ASYMPTOTIC BEHAVIOR AND STABILITY

hence P [τα ∧ τδ <∞] = 1. But we also have P [τδ <∞] ≤ ε4 by (5.9) and so

P [τα <∞] ≥ 1 − ε4 , in particular we can find θ > 0 large enough such that

P [τα < θ] ≥ 1− ε2 .

The rest of the story of the proof is more or less this one: once the processentered the (possibly very small) ball of radius α it does not leave the ball of radius βwith probability greater than 1−ε. For this purpose define the additional stoppingtimes

σ :=

τα if τα < τδ ∧ θ∞ otherwise,

and

τβ := inft > σ : ‖Xxt ‖ ≥ β.

With the supermartingale property follows E[V (Xx

t∧τβ )]≤ E [V (Xx

t∧σ)] and in

particular we can restrict this to the subset where σ is finite, i. e.

E[1τα<τδ∧θV (Xx

t∧τβ )]≤ E

[1τα<τδ∧θV (Xx

t∧σ)].

We also know that τβ <∞ is a subset of this event and therefore

VβP [τβ <∞] := inf‖y‖=β

V (y)P [τβ <∞] ≤ sup‖y‖=α

V (y) =: V α.

At this point we determine the choice of α, namely small enough such that V −1β V α ≤ε4 and then of course follows P [τβ <∞] ≤ ε

4 . We can put the pieces together asfollows.

P [σ <∞ and τβ =∞] ≥ P [τα < τδ ∧ θ]− P [τβ <∞]

≥ P [τα < θ]− P [τδ <∞]− P [τβ <∞]

≥ 1− ε

2− ε

4− ε

4= 1− ε.

Since β was arbitrary this concludes the proof.

Theorem 5.26 (Exponential p-Stability). The equilibrium x∗ = 0 is exponen-tially p-stable if there exists a Lyapunov function V satisfying

α1‖x‖p ≤ V (x) ≤ α2‖x‖p and LV (x) ≤ −α3‖x‖p

for all x ∈ Rd and constants α1, α2, α3 > 0.

Proof. Exercise.

Example 5.27. As an example we consider a stochastic differential equationwith explicitly known solution since then we can compare the abstract stabilityresults with the properties of the solution formula. Our choice is the geometricBrownian motion in, i. e.

dXt = bXt dt+ σXt dWt

in dimension d = 1. A smooth Lyapunov function as in the previous section fordeterministic systems would be V (x) = x2. Obviously LV (x) = (σ2 + 2b)x2 ≤ 0

iff b < −σ2

2 . Theorem 5.24 yields stability in probability but in comparison to the

5.4. INVARIANT MEASURES FOR STOCHASTIC DIFFERENTIAL EQUATIONS 11

ordinary differential equation with σ = 0 the noise seems to destabilize. However,this is not the case and in comparison with the explicit solution we see why.

Xxt = exp

(bt+ σWt −

σ2

2t)x

= exp((b− σ2

2

)t+ σWt

)x→ 0

P-a. s. even for b < σ2

2 , in particular b > 0 in which the solution of the correspondingordinary differential equation explodes. We can see this stability with a different

choice for the Lyapunov function, namely V (x) = |x|1−2bσ2 =: |x|2α for b < σ2

2 which

is not differentiable in 0 for b > −σ2

2 . With this choice of V follows

LV (x) = α|x|2α(

2b+ σ2(2α− 1))≤ 0

and we obtain even asymptotic stability in probability by Theorem 5.25. Also,Theorem 5.26 yields exponential p-stability for p < 1− 2b

σ2 .

5.4. Invariant Measures for Stochastic Differential Equations

In this section we depart from the pathwise picture of the solution to a stochas-tic differential equation and study how its law evolves as time passes by. A naturalanalogue to an equilibrium point, i. e. pathwise stationarity, is a stationary law.Since the law of a solution to a stochastic differential equation evolves accordingto its transition probabilities, stationarity is described as invariance with respectto the corresponding transition semigroup. We introduce these notions in detailbelow and then study existence of such laws using the Krylov-Bogoliubov theory.

So again we consider throughout the whole section a stochastic differentialequation of the form (5.6), for convenience stated again below,

(5.10) dXt = b(Xt

)dt+ σ

(Xt

)dWt, X0 = ξ0.

We also assume a global Lipschitz condition on b and σ and recall that Xxt denotes

the solution to (5.10) with deterministic initial condition X0 = x ∈ Rd. With thisin mind we can define the transition probabilities

pt(x,A) := P [Xxt ∈ A] , t ≥ 0, A ∈ B(Rd).

Furthermore note that W su := Wu −Ws, u ≥ s, is again a Brownian motion with

start in 0 and its natural filtration is given by

F ts := σ(Wu −Ws : u ∈ [s, t]

), 0 ≤ s ≤ t.

With our assumptions the existence of a unique strong solution Xs,xt , t ≥ s of

(5.11) dXt = b(Xt

)dt+ σ

(Xt

)dW s

t , Xs = x

is guaranteed. Moreover, uniqueness implies the so-called flow-property, i. e. for0 ≤ s ≤ t ≤ u we have that

Xs,xu = X

t,Xs,xtu .

Proposition 5.28 (Markov Property). Let Ft = σ(Ws, s ∈ [0, t]) be thenatural filtration of W . Then the solution of (5.10) satisfies

P [Xt ∈ A | Fs] = P [Xt ∈ A | Xs] = pt−s(Xs, A) P-a. s.

12 5. ASYMPTOTIC BEHAVIOR AND STABILITY

Proof. Clearly we can write (5.10) in its integral form in the following way

Xt = x+

∫ t

0

b(Xu

)du+

∫ t

0

σ(Xu

)dWu

= Xs +

∫ t

s

b(Xu

)du+

∫ t

s

σ(Xu

)dW s

u

where Xu, u ≥ s, is the unique (strong) solution of (5.11) with Xs = Xs, so thatby the flow property

Xu = Xs,Xsu and thus Xt = Xs,Xs

t .

Now, Xs,xt only depends on the Brownian increment W s

u = Wu−Ws, u ≥ s, whichis independent of Fs, so that Xs,x

t is independent of Fs, too. Consequently, for FFs-measurable,

E [F P [Xt ∈ A | Fs]] = E[F E

[1A

(Xt

)| Fs

]]= E

[F E

[1A

(Xs,Xst

)| Fs

]]= E

[F E

[1A

(Xs,Xst

)]]= E [F P [Xt ∈ A | Xs]] .

Furthermore the map (s, x) 7→ Xs,xt is Borel-measurable (proof via Picard-Lindelof

iteration) and thus (s, x) 7→ P [Xs,xt ∈ A], A ∈ B(Rd) is Borel-measurable. It follows

that pt−s(Xs, A) = P [Xt ∈ A | Xs] is σ(Xs)-measurable, hence pt−s(Xs, A) is aversion of P [Xt ∈ A | Xs].

The proof of Proposition 5.28 shows that pt(x, dy) is a transition kernel ofprobability measures (stochastic kernel) from R+ × Rd to B(Rd), i.e.,

(1) A 7→ pt(x,A) is a probability measure on (Rd,B(Rd)) for all (t, x) ∈R+ × Rd and

(2) (t, x) 7→ pt(x,A) is B(R+)⊗ B(Rd)-measurable for all A ∈ B(Rd).Thus, we can look at the following integral operator on the space of bounded Borel-measurable functions given by(

Ptf)(x) :=

∫Rdf(y)pt(x, dy) = E

[f(Xxt

)], x ∈ Rd.

Pt is well-defined for f Borel-measurable and bounded or non-negative. Further-more it satisfies the Chapman-Kolmogorov equation, namely Pt Ps = Pt+sfor all t, s ≥ 0 and P0 = Id. In particular, Pt is a semigroup of linear operators.This can be seen quite easily using the Markov property((

Pt Ps)f)

(x) =(Pt(Psf

))(x) = E

[Psf

(Xxt

)]= E

[E[f(Xxt+s

)| Ft

]]= E

[f(Xxt+s

)]=(Pt+sf

)(x).

Example 5.29. Let us illustrate these objects and properties with a few ex-amples.

(1) Consider Brownian Motion with start in x, i. e. dXt = dWt, X0 = x,then we know that

pt(x,A) = P [x+Wt ∈ A] = N (x, t)(A).

5.4. INVARIANT MEASURES FOR STOCHASTIC DIFFERENTIAL EQUATIONS 13

For the semigroup property observe that Xt+s = x+Wt+s = x+ (Wt+s−Ws) +Ws with two independent increments. Hence(

Pt+sf)(x) = E

[f(x+Wt+s

)]= E

[E[f(x+Wt+s −Ws +Ws

)|Ws

]]= E

[∫Rf(x+ y +Ws

)N (0, t)( dy)

]=

∫R

∫Rf(x+ y + z)N (0, s)( dz)N (0, t)( dy)

=

∫R

(∫Rf(z)N (x+ y, s)( dz)

)N (0, t)( dy)

=(Pt(Psf

))(x) =

((Pt Ps

)f)

(x).

(2) Consider the Ornstein-Uhlenbeck process dXt = −bXt dt+ dWt, X0 = xwith b > 0. The solution is given by the stochastic convolution

Xt = e−btx+

∫ t

0

e−b(t−s) dWs,

where the law of the stochastic integral is given byN (0, qt), qt =∫ t0

e−2bs ds.Thus, the transition kernel is

pt(x,A) = N (e−btx, qt)(A).

One can calculate the semigroup property in a similar fashion as above.((Pt Ps

)f)

(x) = E[Psf

(Xxt

)]=

∫RPsf

(e−btx+ y

)N(0, qt

)( dy)

=

∫R

∫Rf(

e−bs(e−btx+ z

)+ y)N(0, qs

)( dz)N

(0, qt

)( dy)

=

∫R

∫Rf(

e−b(t+s)x+(e−bsz + y

))N(0, qs

)( dz)N

(0, qt

)( dy)

=

∫Rf(e−b(t+s)x+ y

)N(0, e−2bsqt + qs

)( dy) =

(Pt+sf

)(x).

The last line is due to the fact that the convolution of two normal distri-butions is again a normal distribution with variance equal to the sum ofthe variances. Moreover qt+s = e−2bsqt + qs.

In the following we fix a semigroup Pt of stochastic integral operators on Rd.

Definition 5.30. A probability measure µ on B(Rd) is called invariant forPt if ∫

Ptf dµ =

∫f dµ for all f ∈ Bb

(Rd),

i. e. all bounded Borel-measurable functions f . Intuitively this means that if ξ0 ∼ µthe unique (strong) solution Xt of (5.10) has distribution µ for all times and thusthe distribution of Xt is invariant with respect to time.

In the following we study existence of invariant measures for stochastic differ-ential equations of type (5.10) using the Krylov-Bogoliubov theory. One key

14 5. ASYMPTOTIC BEHAVIOR AND STABILITY

ingredient to this theory is Prokhorov’s theorem, which relates the notions of tight-ness and relative weak compactness for probability measures.

Definition 5.31. (1) A family of probability measures Γ ⊆ M1(Rd) iscalled tight if for all ε ∈ (0, 1) there exists a compact Kε ⊆ Rd such thatµ(KCε

)≤ ε for all µ ∈ Γ.

(2) A family of probability measures Γ ⊆M1(Rd) is called relatively weaklycompact if any sequence (µn) in Γ has a weakly convergent subsequence.

Theorem 5.32 (Prokhorov). The following statements are equivalent:

(1) Γ ⊆M(Rd) is weakly relatively compact.(2) Γ ⊆M(Rd) is tight.

The main idea to prove existence of an invariant measure is now to find asequence of measures, which is tight and therefore by Theorem 5.32 has a weaklyconvergent subsequence. The limit of this subsequence is then a candidate forthe invariant measure. The sequence of measures we consider are called meanoccupation times and defined as follows. Let T > 0, x ∈ R, then

µT,x(A) := E

[1

T

∫ T

0

1A

(Xxt

)dt

], A ∈ B

(Rd)

is a probability measure, where the integral yields the total amount of time spentin the set A up to time T , or in other words the occupation time of A.

Remark 5.33.∫f dµT,x = E

[1

T

∫ T

0

f(Xxt

)dt

]=

1

T

∫ T

0

E[f(Xxt

)]=

1

T

∫ T

0

(Ptf

)(x) dt

Theorem 5.34 (Krylov-Bogoliubov). Suppose there exists x ∈ Rd for whichµT,xT≥1 is tight. Then there exists an invariant measure µ for Pt.

Proof. As mentioned above, Prokhorov’s theorem implies the existence of aweakly converging subsequence to a limit measure µ ∈ M1(Rd), i. e. there existsTn ∞, such that limn→∞ µTn,x = µ weakly. This is equivalent to

limn→∞

1

Tn

∫ Tn

0

(Ptf

)(x) dt =

∫f dµ

for all f ∈ Cb(Rd). Let s > 0 be arbitrary. Then Psf ∈ Cb(Rd), hence∫Psf dµ = lim

n→∞

1

Tn

∫ Tn

0

(Pt(Psf

))(x) dt = lim

n→∞

1

Tn

∫ Tn

0

(Pt+sf

)(x) dt

= limn→∞

1

Tn

∫ Tn+s

s

(Ptf

)(x) dt

= limn→∞

[1

Tn

∫ Tn

0

(Ptf

)(x) dt+

1

Tn

(∫ Tn+s

Tn

(Ptf

)(x) dt−

∫ s

0

(Ptf

)(x) dt

)].

The latter term converges to 0 as Tn ∞ and the first one converges to∫f dµ as

noticed above. Thus, µ is invariant for Pt.

5.4. INVARIANT MEASURES FOR STOCHASTIC DIFFERENTIAL EQUATIONS 15

Example 5.35. In this example we illustrate a simple criterion that indeedimplies tightness and therefore existence of an invariant measure. Suppose that forsome p > 0 and x ∈ Rd, the pth moment of ‖Xx

t ‖ is uniformly bounded, i. e.

E [‖Xxt ‖p] ≤ C

for all t ≥ 0. Then the family µT,x is tight and thus there exists an invariantmeasure µ for Pt. This is proven as follows. Let R > 0, then

µT,x(BR

C)=

∫‖x‖>R

dµT,x ≤1

Rp

∫‖x‖p dµT,x

=1

Rp1

T

∫ T

0

E [‖Xxt ‖p] dt ≤

C

Rp

Now for arbitrary ε ∈ (0, 1) we can choose Rε = (C/ε)1p such that µT,xB

CRε

) ≤ ε

for all T ≥ 1. Since the closed ball of radius Rε is compact in Rd, µT,x is tight.As an illustration consider the Ornstein-Uhlenbeck process

dXt = BXt dt+ C dWt

with B,C ∈ Rd×d and W a d-dimensional Brownian motion. By the variation ofconstants method the solution is given by

Xxt = etBx+

∫ t

0

e(t−s)BC dWs,

which is N (etBx,Qt) distributed. Here,

Qt :=

∫ t

0

esBC · CT esBT

ds

By Ito’s formula we obtain

E[‖Xx

t ‖2]

= trace Qt.

The trace is uniformly bounded in t if the operator norm ‖esB‖L(Rd) ≤ e−sκ∗ forsome κ∗ > 0, in particular this is the case if B only has strictly negative eigenvalues.

In the following, we use this criterion to deduce existence of an invariant mea-sure for nonlinear equations (5.10) with a recurrent drift term. This is a commonassumption in models for neurons.

Proposition 5.36. Suppose that the coefficents of (5.10) satisfy

(1) 〈b(x)〉x ≤ α− λ‖x‖2 and(2) trace (σ(x) · σ(x)T ) ≤ 2C

for some α, λ,C > 0. Then for any x ∈ Rd

E[‖Xx

t ‖2]≤ e−2λt‖x‖2 +

α+ C

λ

(1− e−2λt

),

in particular the second moment is uniformly bounded in t. Hence, there exists aninvariant measure for Pt.

Proof. Ito’s formula implies

d‖Xxt ‖2 = 2〈Xx

t 〉dXxt + trace

(σ(Xxt

)· σ(Xxt

)T)dt

≤ 2(α− λ‖Xx

t ‖2)dt+ 2C dt+ 2〈Xx

t 〉σ(Xxt

)dWt.

16 5. ASYMPTOTIC BEHAVIOR AND STABILITY

Therefore, using the product rule

d(e2λt‖Xx

t ‖2)≤ 2(α+ C)e2λtdt+ 2e2λt〈Xx

t 〉σ(Xxt

)dWt,

hence

e2λt‖Xxt ‖2 ≤ ‖x‖2 + 2(α+ C)

∫ t

0

e2λs ds+ 2

∫ t

0

e2λs〈Xxs 〉σ(Xxs

)dWs

Taking expectations on both sides yields the result.

Example 5.37 (FHN-system with Noise). Consider the FitzHugh-Nagumo sys-tem with the usual parameters perturbed by independent Brownian motion in bothvariables.

(5.12) d

[VtWt

]=

[f(Vt)−Wt + I

ε(Vt − γWt

) ] dt+

[σV,V σV,WσW,V σW,W

] (Vt,Wt

)d

[BVtBWt

].

We assume that the trace of the diffusion coefficient is bounded as

trace(σ · σT

)(v, w) =

(σ2V V + σ2

VW + σ2WV + σ2

WW

)(v, w) ≤ 2C

uniformly in v, w. Denote the drift by b(v, w), then

〈b(v, w)〉[vw

]= f(v)v − vw + Iv + εvw − εγw2

≤ −v4

2+

(2 + a2

2+

(ε− 1)2

2εγ

)v2 − εγ

2w2 +

1

2I2 ≤ α− λ(v2 + w2)

for some λ > 0 and α > 0 sufficiently large. To obtain the second line we used theestimates

f(v)v = −v4 + (1 + a)v3 − av2 ≤ −v4

2+

1 + a2

2v2,

(ε− 1)vw ≤ εγ

2w2 +

(ε− 1)2

2εγv2,

Iv ≤ 1

2I2 +

1

2v2.

Proposition 5.36 now implies that

E

[∥∥∥∥[ VtWt

]∥∥∥∥2]≤ e−2λt

∥∥∥∥[v0w0

]∥∥∥∥2 +α+ C

λ

(1− e−2λt

)is uniformly bounded in t. Hence there exists an invariant measure µ for (5.12).

It is difficult to get further information on µ via the Krylov-Bogoliubov theory.However, in special cases the structure of the drift allows to explicitly specify theinvariant measure. The following example shows this for the Andronov-Hopfoscillator.

Example 5.38. Consider the following stochastic differential equation

dZt = b(Zt)dt+ σ

(Zt)dWt,(5.13)

with Zt = (Xt, Yt) ∈ R2, W a two-dimensional Brownian motion and drift given by

b(Xt, Yt

)=

[Xt − Yt −Xt‖Zt‖2Yt +Xt − Yt‖Zt‖2

]

5.4. INVARIANT MEASURES FOR STOCHASTIC DIFFERENTIAL EQUATIONS 17

Note that 〈b(z)〉z = ‖z‖2 − ‖z‖4 ≤ 1 − 12‖z‖

2, hence for bounded trace (σ · σT )there exists an invariant measure. In the particular case σ = σ0Id, σ0 > 0 constant,the (unique) invariant measure is given by

µ( dz) = Z−1 exp( 1

σ20

‖z‖2 − 1

2σ20

‖z‖4)dz

where

Z =

∫R2

exp( 1

σ20

‖z‖2 − 1

2σ20

‖z‖4)dz

is the normalizing constant. It suffices to check that for any f ∈ C20 (R2) we have∫

Lf dµ = 0, a weaker notion often called infinitesimal invariance. Here, L isthe Kolmogorov operator associated with (5.13)(

Lf)(z) =

σ20

2

(∂xxf + ∂yyf

)(z) + 〈b(z)〉∇f(z).

We can verify this infinitesimal invariance with a simple integration by parts. Forthis reason, let us first look at the partial derivatives of the density of µ but becauseit is an exponential we only need the derivate of the exponent. Clearly

∂x

( 1

σ20

‖z‖2 − 1

2σ20

‖z‖4)

=2x

σ20

− 2x

σ20

‖z‖2,(5.14)

∂y

( 1

σ20

‖z‖2 − 1

2σ20

‖z‖4)

=2y

σ20

− 2y

σ20

‖z‖2,(5.15)

So it follows that∫ (Lf)(z)µ( dz) =

σ20

2

∫ (∂xxf + ∂yyf

)(z) + 〈b(z)〉∇f(z)µ( dz)

= −∫〈[x− x‖z‖2y − y‖z‖2

]〉∇f(z) + 〈b(z)〉∇f(z)µ( dz)

=

∫−y∂xf(z) + x∂yf(z)µ( dz)

using integration by parts and (5.14) and (5.15). Another integration by partsyields

=

∫2

σ20

(y(x− x‖z‖2

)− x(y − y‖z‖2

))f(z)µ( dz) = 0.

Thus, µ is indeed invariant for (5.13). Its density depends on σ0 in the followingway.

Z =√

2π32σ0e

1

2σ20 Φ(σ−10

),

where Φ denotes the cumulative density function of the standard normal distri-bution. Clearly, Z → ∞ as σ0 → 0 so that µ concentrates on a set of Lebesguemeasure 0. In fact, it converges to the uniform distribution on the unit circle S1,which is the invariant measure of the deterministic motion. Figure 1 illustrates thisfor different values of σ0 and since µ is rotation invariant a plot of the density aty = 0 suffices.

18 5. ASYMPTOTIC BEHAVIOR AND STABILITY

Figure 1. A plot of the density of µ at y = 0 for three differentvalues of σ0 = 0.1, 0.55, 1 with colors red, green and blue, respec-tively.

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19