NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

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1 NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs when an unstabe deterministic system of ordinary differentia equations is stabiized by the addition of romness into the system. Noise-induced stabiization is quite an intriguing surprising phenomenon as one s first intuition is often that noise wi ony serve to further destabiize the system. In this paper, we investigate under what conditions one-dimensiona, autonomous stochastic differentia equations are stabe, where we take the notion of stabiity to be that of goba stochastic boundedness. Specificay, we find the minimum amount of noise necessary for noiseinduced stabiization to occur when the drift noise coefficients are power, onentia, or ogarithmic functions.. Introduction This work is motivated by the intriguing phenomenon of noise-induced stabiization. Noise-induced stabiization occurs when the addition of romness to an unstabe deterministic system of ordinary differentia equations (ODEs) resuts in a stabe system of stochastic differentia equations (SDEs). We are particuary motivated to find the minimum amount of noise required for the phenomenon of noise-induced stabiization to occur for one-dimensiona, autonomous SDEs of the form dx(t) = b(x(t))dt + σ(x(t))db(t). Here b(x) is the drift coefficient, which pushes the soution deterministicay in some direction, σ(x) is the noise coefficient, which contros the strength of the noise, B(t) is Brownian motion, a stochastic process where for each fixed t, B(t) is a normay distributed rom variabe with mean 0 variance t. We assume that b(x) σ(x) are West Virginia University, Morgantown, WV 6506 Mercyhurst University, Erie, PA Bethany Lutheran Coege, Mankato, MN Department of Mathematics Statistics, Vaparaiso University, Vaparaiso, IN 46383

2 ALLEN, GEBHARDT, KLUBALL, KOLBA continuous functions that there exists 0 0 such that σ(x) 0 for a x 0. Our sense of stabiity comes from that of goba stochastic boundedness, which is defined more formay beow 3. Definition. X(t) is stabe if for a initia conditions a ɛ > 0, there exists some bound R such that for a t 0. P ( X(t) R) > ɛ In the deterministic setting where X(t) is a soution to an ODE, the definition of stabiity reduces to that of boundedness. Hence, the unstabe deterministic systems that we consider either bow up in finite time or wer off to infinity. We say that noise-induced stabiization occurs when the addition of noise to an unstabe ODE resuts in a stabe SDE. Work by Scheutzow 5 has shown sufficient conditions for the occurrence of noise-induced stabiization. In this paper, we find necessary sufficient conditions for noise-induced stabiization to occur when the drift noise coefficients are restricted to certain forms. Section discusses usefu background information on the techniques that we use to prove noise-induced stabiization. Sections 3, 4, 5, 6 present prove our resuts concerning noise-induced stabiization for when the drift noise coefficients are genera power functions, poynomias, onentia functions, ogarithmic functions, respectivey. Section 7 discusses the probem of the minimum amount of noise required to stabiize an unstabe system when the noise coefficient is not restricted to a particuar form.. Background In this section, we discuss preiminary information on the methods used to find our resuts. In particuar, our work uses a we-known resut from to determine the stabiity of SDEs, specificay whether noise-induced stabiization occurs. For ease, we wi refer to this resut as the Stochastic Stabiity Theorem. Stochastic Stabiity Theorem. Consider the genera form for autonomous first-order SDEs: dx(t) = b(x(t))dt + σ(x(t))db(t) where b(x) σ(x) are continuous functions there exists 0 0 such that σ(x) 0 for a x 0.

3 NOISE-INDUCED STABILIZATION 3 Define the foowing quantities: x b(z) s(x) = σ (z) dz when x > x b(z) σ (z) dz when x < x s(y)dy for x > S(x) = x s(y)dy for x < m(x) = s(x)σ (x) x m(y)dy M(x) = x m(y)dy for x > for x < The SDE is stabe if any ony if there exists 0 such that S( ) =, S( ) =, M( ) <, M( ) <. Note that since the ower imit of integration does not affect the convergence or divergence of these integras, can be any rea number greater than or equa to 0. Due to the terms defined in the Stochastic Stabiity Theorem, there are severa reoccurring, non-trivia integras in our proofs. Hence, we prove beow the convergence or divergence of these types of integras to use as a reference. Lemma. Consider the integra where n is any rea number. e cxn dx () If c 0, then the integra is infinite. () If c < 0 n 0, then the integra is infinite. (3) If c < 0 n > 0, then the integra is finite Proof. () If c 0, then cx n 0 for a x, which impies e cxn for a x. Thus, e cxn dx dx =.

4 4 ALLEN, GEBHARDT, KLUBALL, KOLBA () If c < 0 n 0, then x n for a x, which impies cx n c for a x, which in turn impies that e cxn e c for a x. Thus e cxn dx e c dx =. (3) By the Tayor series ansion for e y, e y yk for any y > 0 k! any nonnegative integer k. Hence, k! for any y > 0 e y y k any nonnegative integer k. If c < 0 n > 0, then e cxn = e c xn where c x n > 0 for a x > 0. Hence, e cxn k! = k! for ( c x n ) k c k x nk any nonnegative integer k for a x > 0. Since it hods for any nonnegative integer k n > 0, we can choose k 0 such that nk 0 >. Thus, k e cxn 0! dx c k 0 x nk 0 dx k 0! = c k 0 (nk0 )x nk 0 Consider the integra = k 0! c k 0 (nk0 ) <. e cxn dx. This integra is we-defined ony when n is an integer, so we wi now restrict to this case. Coroary. Consider the integra e cxn dx, where n is an even integer. () If c 0, then the integra is infinite for any n. () If c < 0 n 0, then the integra is infinite. (3) If c < 0 n > 0, then the integra is finite Proof. If n is an even integer, then e cxn = e c( x)n e cxn dx = e cxn dx.

5 NOISE-INDUCED STABILIZATION 5 Thus, we get the same cassification of the integra being finite versus infinite as in the previous emma. Coroary. Consider the integra e cxn dx, where n is an even integer. () If c 0, then the integra is infinite for any n. () If c > 0 n 0, then the integra is infinite. (3) If c > 0 n > 0, then the integra is finite Proof. If n is an odd integer, then e cxn = e c( x)n e cxn dx = e cxn dx. Thus, we get amost the same cassification of the integra being finite versus infinite as in the previous emma, with just the sign of c reversed. 3. Power Function Stabiization In this section, we consider ODE s where the drift coefficient is a genera power function i.e. dx(t) = r X(t) q dt. These ODE s are unstabe for any r 0. Theorem. Consider the SDE dx(t) = b(x(t))dt + σ(x(t))b(t) where b(x(t)) = a X(t) q if q 0 σ(x(t)) = r X(t) p if p 0. If q < 0 then the drift coefficient takes the foowing form: { r X(t) q for X(t) b(x(t)) = r for X(t) <. Likewise, if p < 0, then the noise coefficient takes the foowing form: { a X(t) p for X(t) σ(x(t)) = a for X(t) <. Aso, r, a, q, p are any rea numbers. Noise-induced stabiization occurs if ony if r 0 one of the foowing sets of conditions is met: { p > max, q + } for any a 0 or

6 6 ALLEN, GEBHARDT, KLUBALL, KOLBA Figure. This image depicts an unstabe power ODE that is stabiized with the sufficient amount of noise. p = q + { a q > r for 0 < q a r for q >. Figure shows three separate graphs depicting the phenomenon of noise-induced stabiization. The graph on the far eft shows an ODE that diverges off to infinity for a positive initia vaues; therefore, the ODE is unstabe. The midde graph depicts the noise vaue of X(t)dB(t) added to the previousy unstabe ODE. This specific amount of noise added does not stabiize the system as a < r. The fina image depicts 3X(t)dB(t) added to the unstabe ODE. It can be seen that the SDE converges. This occurs since a > r. Proof. The Stochastic Stabiity Theorem wi be used to show under what conditions noise-induced stabiization occurs. With = evauation of the s(x) term gives x r z q dz when x > a s(x) = z p x r z q dz when x < a z p { r x a = zq p dz when x > x ( z)q p dz when x < r a where the integration depends on the vaue of q p. The non-critica q+ case where p wi be considered first.

7 NOISE-INDUCED STABILIZATION 7 Simpification of Proof: Y (t) = X(t) must have the same exact stabiity as X(t) since they have the same magnitude. dy (t) = dx(t) = r X(t) q dt a X(t) p db(t) = r Y (t) q dt a Y (t) p db(t) = r Y (t) q dt a Y (t) p db(t) Hence, the stabiity of X(t) with r, q, a, p must be equivaent to the stabiity with r, q, p, a. Thus, when proving Theorem 3, it suffices to just prove the case with r > 0. Non-Critica Case: Assume q p et c = n = q p +. Then { c(x n ) when x > s(x) = c(( x) n ) when x < { c cx n when x > = c c( x) n when x < r a (q p+) therefore S(x) = { c x cyn dy when x > c x c( y)n dy when x <. Evauation of the m(x) term gives then m(x) = M(x) = { { Letting x = gives c x p cx n a when x > c ( x) p c( x) n a when x < c x a y p cy n dy when x > c x a ( y) p c( y) n dy when x < S( ) = c cy n dy () M( ) = a c y p cy n dy ()

8 8 ALLEN, GEBHARDT, KLUBALL, KOLBA etting x = gives S( ) = c = c = c c( y) n dy c( y) n dy cy n dy (3) M( ) = a c ( y) p c( y) n dy = a c ( y) p c( y) n dy = a c y p cy n dy (4) Non-Critica Case Assume r > 0, p > q+, p > which means r n = q p + < 0 c = > 0. It foows that c a (q p+) cyn > 0 therefore c cy n > on the interva from to. The ower bound of cy n aows for the foowing comparison of (): S( ) > c dy =. Therefore, S( ) =. The upper bound of cy n aows for the foowing comparison of (4): M( ) a c y p cdy = y p dy a > After canceation of the onentia terms, the integra takes the form y p dy. This integra converges if p < which is the case since it was assumed that p >. Therefore, M( ) <. The constants n c are both negative so 0 > cy n c on the interva. Then > cy n c. The ower bound of

9 cy n aows for the comparison S( ) c NOISE-INDUCED STABILIZATION 9 cdy = dy = of () the concusion that S( ) =. Therefore, S( ) =. The upper bound of cy n enabes the comparison M( ) < a c dy < yp of (3). Once again, the integra y p dy converges since p >. Therefore, M( ) < 0. Through appication of the Stochastic Stabiity Theorem, it has been shown that noise-induced stabiization occurs when r > 0, p > q+, p >. Recaing the proof simpification, it can aso be concuded that noise-induced stabiization occurs when r > 0, p > q+, p >. Non-Critica Case : Assume r > 0 p < q+. Then n = q p+ > r 0, c = < 0, c > 0. Therefore, the integra contained a (q p+) in () is of the form cy n dy where c is negative n is positive. Lemma.3.4 indicates that integras of this form con- verge. Therefore, S( ) <. Appication of the Stochastic Stabiity Theorem shows that noise-induced stabiization does not occur when r > 0 p < q+. The proof simpification indicates that noiseinduced stabiization aso does not occur when r > 0 p < q+ Non-Critica Case 3: Assume r > 0 > p > q+. Then n is a negative constant c is a positive constant. Therefore, cy n > on the interva from to. This aows for the foowing comparison of (4): M( ) < a c y p dy. Since p <, the integra y p dy must diverge. Therefore, M( ) = so noise-induced stabiization does not occur when r > 0 > p > q+. The simpification of the proof indicates that stabiization aso does not occur when r > 0 > p > q+ Critica Case: Assume q p =. Then the s(x) term takes the..

10 0 ALLEN, GEBHARDT, KLUBALL, KOLBA form { r x a s(x) = z dz when x > r x a ( z) dz when x < { r n(x) when x > = a r n( x) when x < a {x r a for x > = ( x) r a for x < { x s(y)dy for x > S(x) = x s(y)dy for x < { x = y r a dy for x > r ( y) a dy for x <. x Evauation of the m(x) term gives { x r m(x) = a a p for x > ( x) r a a p for x < M(x) = { a x y r a x a p dy for x > r ( y) a p dy for x <. Evauation of the terms at x = x = gives S( ) = S( ) = = = M( ) = a y r a dy, (5) ( y) r a dy ( y) r a dy y r a dy, (6) y r a p dy, (7)

11 NOISE-INDUCED STABILIZATION M( ) = ( y) r a = ( y) r a = a a p dy a p dy y r a p dy. (8) Critica Case : Assume r > 0, p = q+ >, a r a q > r. It foows from the assumptions that 0 < r for q >. Therefore, a r < r which means the integras in both (5) a a (6) S( ) = S( ) = y r a dy y r a dy must diverge. Likewise, if q, it must be the case that q < r < r < q so that the integras must sti diverge. Therefore, a a S( ) = S( ) =. Note, that in the case where r is negative, r > r. This is why the magnitude of r must be considered in a a the stabiization conditions. The assumption that r < a q p = q+ is equivaent to the assumption that r p < for q. Since r is positive it can further a be impied that r p <. The fact that both of these terms are a stricty ess than negative means the integras in both (7) (8) M( ) = a M( ) = a y r a p dy y r a p dy must converge. For q >, it must be the case that q > r which a impies r p < r p < so the integras once again converge. a a Therefore, M( ) M( ) are both finite. Through appication of the Stochastic Stabiity Theorem, it has been

12 ALLEN, GEBHARDT, KLUBALL, KOLBA shown that noise-induced stabiization occurs when r > 0, p = q+ >, a r a q > r. Recoection of the proof simpification motivates the concusion that stabiization aso occurs when r > 0, p = q+ >, a r a q > r. Critica Case : Assume r > 0, p = q+, a < r. Then it must be the case that r < which impies the integra contained a in (5) must converge. Therefore, S( ) < so noise-induced stabiization does not occur. Likewise, if r < 0, p = q+ a < r, then r > r < so the integra contained in (6) a a must converge. Therefore, S( ) > noise-induced stabiization does not occur when r > 0, p = q+, a < r. The proof simpification aows for the additiona concusion that stabiization does not occur when r > 0, p = q+, a < r. Critica Case 3: Assume r > 0, p = q+ a q r. Then it must be the case that r p which impies the integra contained in (7) a diverges. Therefore, M( ) = noise-induced stabiization does not occur. Now assume r < 0, p = q+ a q r. Then it foows that r p which impies the integra contained in (8) diverges. a Therefore, M( ) = noise-induced stabiization does not occur when r > 0, p = q+ a q r. The proof simpification indicates that stabiization aso does not occur when r > 0, p = q+ a q r. Critica Case 4: Assume p = q+ assume a q > r. These assumptions impy r < q 0. However, this is not possibe. Therefore, a there are no cases where both p = q+ a q > r. 4. Poynomia Stabiization In this section, we investigate stabiizing dx(t) = b(x(t))dt, where b(x) is any poynomia of degree q. If r is the coefficient of the highest degree term, this ODE is unstabe when q is even for any r 0 when q is odd for any r > 0. The emma beow, as we as the foowing coroaries, describe genera resuts about poynomias which wi be used in the proof of noiseinduced stabiization for when the drift noise coefficients are poynomias.

13 NOISE-INDUCED STABILIZATION 3 Lemma. Let f(x) be any poynomia of degree q, where the coefficient of the highest degree term is r > 0. Then, there exists k 0 such that for a x k. r(x k) q f(x) r(x + k) q Proof. If q = 0, then f(x) = r the caim is true for k = 0. Hence, assume q. Since f(x) is a poynomia of degree q, it has the foowing form: f(x) = rx q + r x q + r x q r q x + r q ( = r x q + r r xq + r r xq r q r x + r ) q r where r, r,..., r q, r q are any rea constants. Let r max = max{ r r, r r,..., r q r, r q r } k = max{q r max, }. Then for x k, factoring gives Hence for x k, rx q kx q f(x) rx q + kx q rx q (x k) f(x) rx q (x + k). r(x k) q f(x) r(x + k) q. Coroary 3. Let f(x) be any poynomia of degree q, where the coefficient of the highest degree term is r < 0. Then, there exists k 0 such that r(x + k) q f(x) r(x k) q for a x k. Coroary 4. Let f(x) be any poynomia of degree q, where the coefficient of the highest degree term is r > 0. Then, there exists k 0 such that if q is even, if q is odd, for a x k. r(x + k) q f(x) r(x k) q r(x k) q f(x) r(x + k) q

14 4 ALLEN, GEBHARDT, KLUBALL, KOLBA Figure. This image depicts an unstabe poynomia ODE that is stabiized with the sufficient amount of noise. Coroary 5. Let f(x) be any poynomia of degree q, where the coefficient of the highest degree term is r < 0. Then, there exists k 0 such that if q is even, if q is odd, for a x k. r(x k) q f(x) r(x + k) q r(x + k) q f(x) r(x k) q Theorem. Let b(x) be any poynomia of degree q, where the coefficient of the highest degree term is r, et σ(x) be any poynomia of degree p, where the coefficient of the highest degree term is a 0. Then noise-induced stabiization of the SDE dx(t) = b(x(t))dt + σ(x(t))db(t) occurs if ony if the corresponding ODE is unstabe one of the foowing conditions is met: p > q+ or p = q+ { a > r for q = a r for q 3. Figure shows three separate graphs depicting the phenomenon of noise-induced stabiization where the drift noise coefficients are poynomias. The graph on the far eft shows an ODE that diverges off to infinity for a positive initia vaues; therefore, the ODE is unstabe. The midde graph depicts the noise vaue of (X(t) + )db(t) added to the previousy unstabe ODE. This specific amount of noise added does not stabiize the system as p < q+. The fina image depicts

15 NOISE-INDUCED STABILIZATION 5 (X(t) + ) db(t) added to the unstabe ODE. It can be seen that the SDE converges. This occurs since p > q+. Proof. Suppose r > 0. By Lemma Coroary 4, there exist some k b k σ such that x s(x) x x s(x) x r(z+k b ) q a (z k σ) p dz r(z k b ) q a (z+k σ) p dz for even q, x s(x) x x s(x) x r(z k b ) q a (z+k σ) dz p r(z+k b ) q a (z k σ) p dz r(z+k b ) q a (z k σ) dz p r(z+k b ) q a (z k σ) p dz r(x k b ) q a (x+k σ) dz p r(z k b ) q a (z+k σ) p dz when x > max (k b, k σ ) when x < max (k b, k σ ) when x > max (k b, k σ ) when x < max (k b, k σ ) when x > max (k b, k σ ) when x < max (k b, k σ ) when x > max (k b, k σ ) when x < max (k b, k σ ) (9) (0) () () for odd q. These bounds aow for simper anaysis of conditions set by the Stochastic Stabiity Theorem. First, consider the case where q is even. Suppose p > q+. By (9), x S( ) r(z + k b ) q dz dx. a (z k σ ) p Let y = z k σ. Then, by Lemma, there exists a k > 0 such that x r(y q + S( ) k y q ) dy dx. a y p By dividing through integrating, it can be found that r x q p+ r k x q p S( ) c c dx a q p + a q p r where c = q p+ r k c a q p+ = q p. Since q p <, a q p r k therefore > 0, the smaest r k x q p can be is. a (q p) a q p

16 6 ALLEN, GEBHARDT, KLUBALL, KOLBA Therefore, S( ) c c r x q p+ dx. a q p + Since q p+ < 0, this integra diverges by Lemma. By comparison, S( ) =. With these bounds on s(x), S( ) x r(z + k b ) q dz dx. a (z k σ ) p Let y = z k σ. Then, by Coroary 4, there exists a k > 0 such that x r(y q + S( ) k y q ) dy dx. a y p By methods simiar to those used previousy, it can be found that r x q p+ S( ) c dx a q p + where c = c = r ( ) q p+ a q p+ r k a ( ) q p q p, this integra diverges to. Thus, S( ) =. Consider M( ). By Lemma, x M( ) a (x k σ ) p. By Coroary r(z q + k z q ) dz dx. a z p There argest the onentia term can be is, therefore where c = M( ) c c a (x k σ ) r q p+ c a q p+ = r k a p dx q p q p. This integra converges for p > q+, which is aways true since p > q 0. Thus, M( ) <. Proof that M( ) < foows simiary. By Coroary 4, x r(z q + M( ) k z q ) dz dx a (x + k σ ) p a z p. The argest the onentia term can be is c where c = Thus, M( ) c dx a (x + k σ ) p, r a ( ) q p+ q p+.

17 where c = NOISE-INDUCED STABILIZATION 7 r k ( ) q p. This integra converges for p >, a q p which again is aways true. Therefore M( ) <, noiseinduced stabiization does occur when p > q+, r > 0, q is even. Now, assume q is odd. The proofs that S( ) = M( ) < are identica to the case where q is even. Now consider S( ). By (), S( ) x r(z k b ) q dz dx. a (z + k σ ) p Let y = z + k σ. Then, by Coroary 4, there exists a k > 0 such that x r(y q S( ) k y q ) dy dx. a y p By dividing through integrating, it can be found that r x q p+ r k x q p S( ) c c dx a q p + a q p r ( ) where c = q p+ r k c a q p+ = ( ) q p a q p is odd,. Therefore, r k a x q p q p S( ) c c r x q p+ dx. a q p +. Since q p By Coroary, this integra diverges to negative infinity, therefore S( ) =. Now consider M( ). By Coroary 4, x r(z q M( ) k z q ) dz dx a (x + k σ ) p a z p. The argest the onentia term can be is. Therefore, M( ) c c dx a (x + k σ ) p, r ( ) where c = q p+ r k c a q p+ = ( ) q p. Again, this a q p integra converges therefore M( ) <. Now, it is sufficient to say that noise-induced stabiization occurs when p > q+, r > 0, q is odd.

18 8 ALLEN, GEBHARDT, KLUBALL, KOLBA Suppose p = q. By (0), S( ) can be bounded such that x r(z k b ) q S( ) dz dx. a (z + k σ ) p Let y = z + k σ,, by Lemma, there exists a k > 0 such that x r(y q S( ) k y q ) dy dx. a y p By dividing through integrating, it can be found that x r x S( ) a dz r k a z dz dx rx = c c x r k a dx, a where c = r a c = r k a. This integra converges, therefore S( ) <. Suppose p < q. By the same bounds on s(x), S( ) x r(z k b ) q dz a (z + k σ ) p dx. Let y = z + k σ. By Lemma, there exists a k > 0 such that x r(y q S( ) k y q ) dy dx a y p r x q p+ r k x q p = c c dx, a q p + a q p r where c = q p+ r k c a q p+ = q p. For sufficienty a q p arge x, r x q p+ r x q p+ S( ) c c dx a q p + a q p + r x q p+ = c c dx. a q p + By Lemma this integra converges, thus S( ) <. Therefore noise-induced stabiization does not occur when p < q+ r > 0.

19 NOISE-INDUCED STABILIZATION 9 Suppose p = q+, suppose a > r when q = a r when q 3. By (9), S( ) x r(z + k b ) q dz dx. a (z k σ ) p Let y = z k σ. By Lemma, there exists a k > 0 such that x r(y q + S( ) k z q ) dy dx a y p = c c x r r k a dx, a x where c = r a c = r k. This integra diverges for a a r, therefore S( ) =. Now, consider S( ). Note that since p = q+, since p is an integer, q must be odd. By (), S( ) x r(z k b ) q dz dx. a (z + k σ ) p Let y = z + k σ. By Coroary 4, there exists a k > 0 such that x r(z q S( ) k z q ) dz dx a z p c c x r r k a dx, a x where c = r a c = r k. This integra diverges to a negative infinity for a r, therefore S( ) =. Consider M( ). By Lemma, M( ) x a (x k σ ) p r(z q + k z q ) dz dx. a z p The onentia term can be partiay bounded such that, M( ) c r x a a (x k σ ) p dx,

20 0 ALLEN, GEBHARDT, KLUBALL, KOLBA where c = r a. For sufficienty arge x, a (x k σ) p p a x p. Therefore, M( ) c = c p r x a a xp dx p a x r a p dx. This integra converges for a q > r, therefore M( ) <. Consider M( ). By Coroary 4, x r(z q M( ) k z q ) dz dx a (x + k σ ) p a z p = c r r k c x a a (x + k σ ) p dx a x, where c = r a r k c = r k. The argest can a a x be is. Aso, for sufficienty arge x, M( ) c p r c x a a xp dx = c p r c x a a p dx This integra converges, therefore M( ) <. Now, suppose a r when q = a < r when q 3. Consider S( ). By the bounds on s(x), x r(z k b ) q S( ) dz dx. a (z + k σ ) p Let y = z + k σ. By Lemma, there exists a k > 0 such that x r(y q S( ) k z q ) dy dx a y p = c c x r r k a dx, a x where c = r a c = r k. Since r k, a a x S( ) c c x r a dx. This integra converges when a r, therefore S( ) <. Thus, when p = q+, r > 0, noise-induced stabiization occurs if ony

21 NOISE-INDUCED STABILIZATION if a > r when q = a r when q 3. Let Y (t) = X(t), suppose q is even. Since Y (t) is bounded if ony if X(t) is bounded, Y (t) must have the exact same stabiity as X(t). It foows that dy (t) = dx(t) = b(x(t)dt + σ(x(t))db(t) = b( Y (t)dt + σ( Y (t))db(t). Since the highest degree term of the drift coefficient determines the stabiity, since q is even, the stabiity of X(t) with eading coefficient r is equivaent to its stabiity with eading coefficient r. Since the noise coefficient, σ(x), is aways squared in the terms defined by the Stochastic Stabiity Theorem, the sign of the coefficient pays no roe in the stabiity of the SDE. Note that this ony hods for even q since the origina ODE is stabe for r < 0 q odd. 5. Exponentia Function Stabiization This section considers ODE s where the drift coefficient is an onentia function, i.e. dx(t) = r(x(t)) q dt. where r q are any rea number. The ODE is unstabe for any r Genera Exponentia Stabiization. Theorem 3. Consider the SDE dx(t) = r(x(t)) q dt + a(x(t)) p db(t) where r, a, q, p are rea numbers. Noise-induced stabiization occurs if ony if r 0, a 0 one of the foowing sets of conditions is met: r > 0 p > max{0, q } or r < 0 p > max{0, q }. Figure 3 shows three separate graphs depicting the phenomenon of noise-induced stabiization. The graph on the far eft shows an ODE that diverges off to infinity for a positive initia vaues; therefore, the ODE is unstabe. The midde graph depicts the noise vaue of 0 X(t) db(t) added to the previousy unstabe ODE. This specific amount of noise added does not stabiize the system as p <. The 4

22 ALLEN, GEBHARDT, KLUBALL, KOLBA Figure 3. This image depicts an unstabe onentia ODE that is stabiized with the sufficient amount of noise. fina image depicts X(t) db(t) added to the unstabe ODE. It can be seen that the SDE converges. This occurs since p >. Proof. The Stochastic Stabiity Theorem wi be used to show under what conditions noise-induced stabiization occurs. Simpification of Proof: Y (t) = X(t) must have the same exact stabiity as X(t) since they have the same magnitude. dy (t) = dx(t) = r(x(t)) q dt a(x(t)) p db(t) = r( Y (t)) q dt a( Y (t)) p db(t) = r(y (t)) q dt a(y (t)) p db(t) Hence, the stabiity of X(t) with r must be equivaent to the stabiity with r, but with q, p, a substituted for q, p, a. Thus, when proving Theorem 3, it suffices to just prove the case with r > 0.

23 NOISE-INDUCED STABILIZATION 3 With =, the s(x) term takes the form x r qz a s(x) = dz for x > pz x r qz dz for x < a pz { r x a = (q p)zdz for x > r x (q p)zdz for x < a { r x a = kzdz for x > r x kzdz for x < a where k = q p. The integration depends on the vaue of k. The non-critica case where k 0 wi first be considered. Non-Critica Case: Assume k 0. { r s(x) = (kx k) for x > a k r(kx k) for x < a k r k r a = kx for x > k a k r k r kx for x < a k a k S(x) = = where c = m(x) term gives m(x) = M(x) = = r k a k r k a k { x c r x c r r k a k { x r ky dy for x > a k x r ky dy for x < a k ky dy for x > a k ky dy for x < a k c = r k a k. Moving on to the a c r kx px for x > a k a c r kx px for x < a k { x a c { x a c r ky pydy for x > a k x r ky a c pydy for x < a k r ky py dy for x > a k x r ky py a c dy for x <. a k

24 4 ALLEN, GEBHARDT, KLUBALL, KOLBA Setting x equa to positive infinity gives r S( ) = c a k ky dy (3) M( ) = a c r ky py dy. (4) a k Setting x equa to negative infinity gives r S( ) = c a k ky dy r = c a k ky dy r = c a k ky dy (5) M( ) = a c = a c = a c = a c r a k ky py dy r ky py dy a k r ky + py dy a k ( ) r ky + a kp a k r y dy. (6) Non-Critica Case : Assume r > 0 p > max{0, q }. Since p > max{0, q }, it must be the case that k = q p < 0 p > 0. Since k is negative r is positive, it foows that the constant r a k is positive therefore the onentia terms r ky a k r ky must be greater than on the interva from to. a k This aows for the comparison S( ) c dy = S( ) c dy = the concusion that (3) (5) diverge. Therefore S( ) = S( ) =. The onentia decay term ky bounds

25 NOISE-INDUCED STABILIZATION 5 the onentia term r ky between r a k a k (ower bound) (upper bound). This aows for the foowing comparison of (4) to a smaer integra: M( ) a c < py dy The smaer integra has the form pydy with where the constant p is negative. Then Lemma indicates that the integra converges M( ) <. Since k is positive, the Tayor series ansion indicates ky > ky on the interva from to. Now, assuming k + a pk > 0, the r foowing comparison of (6) to a smaer magnitude integra can be made: M( ) a c = a c >. r a k r a k ( ky + a pk r ( k + a pk r ) y ) y Since it was assumed that k + a pk > 0, the smaer magnitude integra takes the form r cy n dy where c is negative n =. Then Lemma indicates that integras of this form converge so M( ) < if k + a pk > 0. r Now, assume k + a pk 0 which is equivaent to the assumption that r k a pk. This inequaity aows for the comparison ky r a pk the foowing comparison of (6) to a smaer magnitude r integra: M( ) a c ( r a pk a k r y ) + a pk r y. Recaing that a pk > 0 considering the Tayor Series Expansion, the comparison a pk r ( ) ( ) y > a pk y + a pk r r r y can

26 6 ALLEN, GEBHARDT, KLUBALL, KOLBA be made. This aows for the foowing comparison of the smaer magnitude integra: M( ) a c ( r a pk a k r y + = ( r a pk a c a k r >. ( a pk r ) y ) ) y + a pk r y The fina integra takes the form cy n dy where c = r a k ( a pk r is a negative constant n = is a positive constant. Then Lemma indicates that integras of this form converge so then M( ) <. Appication of the Stochastic Stabiity Theorem indicates that noiseinduced stabiization occurs when r > 0 p > max{0, q }. The simpification of the proof indicates that noise-induced stabiization aso occurs when r > 0 p > max{0, q }. Non-Critica Case : Assume r > 0 q > p. Since q > p, it must be the case that k = q p > 0 r < 0. The Tayor series a k ansion indicates that ky > ky on the interva which aows for the foowing comparison of (3) to a smaer magnitude integra: r S( ) c a k ky dy < The smaer magnitude integra takes the form cy n dy where c = r is a negative constant n =. Then Lemma indicates a that integras of this form converge so S( ) <. Appication of the Stochastic Stabiity Theorem indicates that noiseinduced stabiization does not occur when r > 0 p < q. The simpification of proofs indicates that noise-induced stabiization aso does not occur when r > 0 p < q. Non-Critica Case 3: Assume r > 0 q < p 0. It foows from the assumptions that k = q p < 0 which impies r < 0. The onentia decay term ky bounds r ky between a k r a k a k )

27 NOISE-INDUCED STABILIZATION 7 which aows for the foowing comparison of (4) to a smaer magnitude integra: M( ) a c r pydy. a k Since p is a non-negative constant, it must be the case that py on the interva which aows for the second comparison M( ) r dy a c a k where (4) is being compared to an even smaer magnitude integra which diverges. Therefore, M( ) =. Appication of the Stochastic Stabiity Theorem indicates that noiseinduced stabiization does not occur when r > 0 q < p 0. The simpification of proof indicates that noise-induced stabiization does not occur when r > 0 q < p 0 which can aso be stated as when r < 0 q > p 0. Critica Case: Assume q = p so k = 0 consider when x >. Then the s(x) term takes the form r s(x) = a r = x 0dz (x ) a r rx = a so then the S(x) term takes the form r x ry S(x) = dy a a ( ) r a rx r = a r a a = a r rx r + a a a r a

28 8 ALLEN, GEBHARDT, KLUBALL, KOLBA Considering when r < 0 setting x = gives S( ) = a r = a r <. r a r a + a r The Stochastic Stabiity Theorem indicates that noise-induced stabiization does not occur when r < 0 p = q. The simpification of proof indicates that noise-induced stabiization aso does not occur when r < 0 p = q. The genera onentia stabiization theorem shows that sufficienty arge onentia growth can stabiize any onentia drift coefficient. Recognizing the difference between the drift coefficient eriencing onentia growth as opposed to onentia decay, the foowing sections consider them separatey. 5.. Exponentia Growth Stabiization. In this section, stabiization of onentia growth in the drift coefficient is considered which means r q have the same sign. Theorem 4. Consider the SDE dx(t) = r qx(t) dt+σ(x(t))db(t) where a X(t) m px(t) for sgn(r)x σ(x(t)) = a px(t) for 0 sgn(r)x < a nx(t) for sgn(r)x < 0 r, a, q, p, m, n R. More specificay, consider when r, q n have the same sign p = q. Noise-induced stabiization occurs if ony if r 0, a 0, one of the foowing sets of conditions is met: m >, n > q, m =, n > q, a r, m >, n = q a n < r or m =, n = q <, a r.

29 NOISE-INDUCED STABILIZATION 9 Simpification of Proof: Y (t) = X(t) must have the same exact stabiity as X(t) since they have the same magnitude. dy (t) = dx(t) r(x(t)) q dt a X(t) m (X(t)) p db(t) for sgn(r)x(t) > r(x(t)) q dt a(x(t)) p db(t) = for 0 sgn(r)x(t) < r(x(t)) q dt a(x(t)) n db(t) for sgn(r)x(t) < r( Y (t)) q dt a Y (t) m ( Y (t)) p db(t) for sgn(r) Y (t) > r( Y (t)) q dt a( Y (t)) p db(t) = for 0 sgn(r) Y (t) < r( Y (t)) q dt a( Y (t)) n db(t) for sgn(r) Y (t) < r(y (t)) q dt a Y (t) m (Y (t)) p db(t) for sgn( r)y (t) > r(y (t)) q dt a(y (t)) p db(t) = for 0 sgn( r)y (t) < r(y (t)) q dt a(y (t)) n db(t) for sgn( r)y (t) < Hence, the stabiity of X(t) with r q, m, a, p, n must be equivaent to the stabiity with r, q, a, m, p, n. Thus, when proving Theorem 3, it suffices to just prove the case with r > 0. Proof. The Stochastic Stabiity Theorem wi be used to show the conditions for noise-induced stabiization. With = r > 0, the s(x) term takes the form { r x a s(x) = z m dz for x > r x (q n)z dz for x < a where the integration depends on the vaue of m.

30 30 ALLEN, GEBHARDT, KLUBALL, KOLBA Non-Critica Case: Assuming m n q gives r a s(x) = ( m) (x m ) for x > r ((q n)x n q for x < a (q n) r r a = ( m) a ( m) x k for x > r n q r (q n)x for x < a (q n) a (q n) r c a = ( m) x m for x > r c (q n)x for x < a (q n) x c S(x) = r y m dy for x > a( m) x c r (q n)y dy for x < a (q n) where c = the m(x) term gives r a ( m) c = c m(x) = x m r a a ( m) x m px r c (q n)x nx a a (q n) r n q. Evauation of a (q n for x > for x < x c M(x) = a y m r a ( m) y m py dy for x > x r (q n)y ny c dy for x <. a a (n q) Letting x = gives r S( ) = c a ( m) y m dy (7) M( ) = c a y m r a ( m) y m py dy. (8)

31 NOISE-INDUCED STABILIZATION 3 Likewise, etting x = gives r S( ) = c a (q n) r = c a (n q) = c M( ) = c a = c a = c a (q n)y dy (q n)y r (n q)y a (n q) r a (n q) r a (n q) dy dy (9) (q n)y ny dy (q n)y ny r (n q)y + ny a (n q) dy dy. (0) Non-Critica Case : Assume r > 0, q > 0, n > 0, m > n > q. The integra in (7) takes the form cy n dy where c = r is a positive constant so the integra must diverge by Lemma a ( m). Therefore, S( ) =. Since m >, then m < 0 so the term y m r bounds a ( m) y m r between a ( m) y m. This aows for the foowing comparison of (8) to the smaer magnitude integra: M( ) c a r y m py. a ( m) The onentia decay term py aows for the further comparison of (8) to an even smaer magnitude integra: M( ) c a r y m dy. a ( m) Once again, since m >, it must be the case that this fina integra converges. Therefore, M( ) <. The conditions for x < are the same as for Case of the genera onentia theorem. Therefore, S( ) = M( ) <. Appication of the Stochastic Stabiity Theorem indicates that noise-induced stabiization occurs when r > 0, q > 0, m > n > q. The proof simpification indicates that stabiization aso occurs when r > 0, q > 0, m >, n > q.

32 3 ALLEN, GEBHARDT, KLUBALL, KOLBA Non-Critica Case : Assume r > 0, q > 0, m <. Then the integra in (7) takes the form cy n r dy where c = is a ( m) a negative constant n = m is a positive constant. Then Lemma indicates that integras of this form converge so S( ) < noise-induced stabiization does not occur when r > 0, q > 0, m <. Then the proof simpification aows for the concusion that noise-induced stabiization aso does not occur when r > 0, q > 0, m <. Non-Critica Case 3: Assume r > 0 0 < n < q. It foows from the assumption that n q is a negative constant. Therefore, the onentia decay term (n q)y bounds the onentia term r (n q)y a (n q) between (upper bound) ower bound on the interva r a (n q) from to. This fact aows for the foowing comparison of (0) M( ) c a ny dy The integra (0) is compared to has the form cy n dy where c = n n =. Then Lemma indicates that integras of this form diverge so M( ) = noise-induced stabiization does not occur when r > 0 0 < n < q. Additionay, the proof simpification indicates that noise-induced stabiization aso does not occur when r > 0 0 < n < q. Critica Case: Now, considering when m = x, the s(x) term takes the form r x s(x) = z dz a r = n(x) then = x r a S(x) = a x y r a dy.

33 Evauation of m(x) gives Letting x = gives NOISE-INDUCED STABILIZATION 33 m(x) = a x r a px M(x) = a x S( ) = M( ) = a y r a pydy. y r a dy () y r a pydy. () Critica Case : Assume r > 0, m =, n > q > 0, a r. Once again, the conditions for x < are the same as for Case of the previous theorem. Therefore, S( ) = M( ) <. The assumption that a r is equivaent to the assumption that r. Then the integra in () diverges S( ) =. The a assumptions aso impy that r 0. Then y r a a on the interva which aows for the foowing comparison of () to a arger magnitude integra. M( ) pydy a The smaer magnitude integra has the form cy n dy where c = p is a negative constant n =. Then Lemma indicates that integras of this form converge so M( ) <. Appication of the Stochastic Stabiity Theorem indicates that noise-induced stabiization occurs when r > 0, m =, n > q > 0, a r. The proof simpification further concudes that stabiization aso occurs r > 0, m = q, n > > 0, a r. Critica Case : Assume r > 0, m =, a < r. It foows from the assumptions that r <. Therefore, the integra contained in a () converges. So S( ) = noise-induced stabiization does not occur when r > 0, m =, a < r. The proof simpification indicates that noise-induced stabiization aso does not occur when r > 0, m =, a < r.

34 34 ALLEN, GEBHARDT, KLUBALL, KOLBA Semi-Critica Case: Consider when r > 0 n = q. Then for x <, the s(x) term takes the form r x s(x) = dz a a r = r r x r S(x) = a a Evauation of the m(x) term gives m(x) = a r a a x, y dy. ( ) r a n x M(x) = r x ( ) r a a a n y dy. Setting x = gives r S( ) = a r a y dy. (3) M( ) = r ( a n r ) y dy. (4) a a Semi-Critica Case : Assume r > 0, m >, n = q > 0 a n < r. It was proven in Non-Critica Case that S( ) = M( ) < under these conditions. The integra in (3) has the form cy n dy where c = r is positive n =. Then Lemma indicates that integras of this form diverge. Therefore, S( ) =. The assumption a that a n < r is equivaent to the assumption that n r < 0. Therefore, the integra contained in (4) takes the form a cy n dy where c = n r is a negative constant n =. Then Lemma indicates a that integras of this form converge so M( ) <. Appication of the Stochastic Stabiity Theorem indicates that noise-induced stabiization occurs when r > 0, m >, n = q > 0 a n < r. The proof simpification aows for the concusion that stabiization aso occurs when r > 0, m > q, n > > 0, a n < r. Semi-Critica Case : Assume r > 0, m =, 0 < n = q <, a r. It was proven in Critica Case that S( ) =

35 NOISE-INDUCED STABILIZATION 35 M( ) < under these conditions. It was proven in Semi-Critica Case that S( ) = under these conditions. Since a r, it must be the case that r. Then the assumption that n < impies a n r < 0. The integra contained in (4) has the form cy n dy a where c = n r is negative n. Then Lemma indicates that a integras of this form converge. Therefore M( ) < the Stochastic Stabiity Theorem indicates that noise-induced stabiization occurs when r > 0, m =, 0 < n = q <, a r. The proof simpification aows for the concusion that stabiization aso occurs when r > 0, m = q, 0 < n = <, a r Semi-Critica Case 3: Assume r > 0, n = q > 0 a n r. Then n r 0. The integra contained in (4) takes the form a cy n dy where c = n r 0. Then Lemma indicates that a integras of this form diverge so M( ) = noise-induced stabiization does not occur when r > 0, n = q > 0 a n r. Then the proof simpification indicates that stabiization aso does not occur when r > 0, n = q > 0, a n r. Semi-Critica Case 4: Assume r > 0, m =, a r, n = q r. It foows that since n, it must be the case a that n r 0. Since (4) takes the form cy n dy where a c = n r 0, the integra must diverge by Lemma. Therefore, a noise-induced stabiization must not occur when r > 0, m =, a r, n = q. Then stabiization aso does not occur when r > 0, m =, a r, q n =. The onentia growth theorem s on the genera onentia theorem by defining the noise coefficient piece-wise therefore aowing for the critica case where p = q Exponentia Decay Stabiization. In this section, stabiization of onentia decay in the drift coefficient is considered which means r q have the opposite sign. Theorem 5. Consider the SDE dx(t) = r qx(t) dt+a X(t) p db(t) where r q have opposite signs. Noise-induced stabiization occurs if ony if r 0, a 0, p >. Proof. The Stochastic Stabiity Theorem wi be used to show under what conditions noise-induced stabiization occurs.

36 36 ALLEN, GEBHARDT, KLUBALL, KOLBA Simpification of Proof: Y (t) = X(t) must have the same exact stabiity as X(t) since they have the same magnitude. dy (t) = dx(t) = r(x(t)) q dt a X(t) p db(t) = r( Y (t)) q dt a Y (t) p db(t) = r(y (t)) q dt a Y (t) p db(t) Hence, the stabiity of X(t) with r, q, a, p must be equivaent to the stabiity with r, q, p, a. Thus, when proving Theorem 3, it suffices to just prove the case with r > 0. The s(x) term takes the form { r x a s(x) = z p qzdz for x > x ( z) p qzdz for x < r a { x S(x) = r a x r a y z p qzdz dy for x > y ( z) p qzdz dy for x <. The m(x) term then takes the form { x p r x a m(x) = a z p qzdz for x > ( x) p r x a a ( z) p qzdz for x < { x a M(x) = y p r y a z p qzdz dy for x > x a ( y) p r y a ( z) p qzdz dy for x <. If x =, then S( ) = M( ) = a r y z p qzdz dy (5) a r y y p z p qzdz dy. (6) a

37 If x =, then S( ) = = NOISE-INDUCED STABILIZATION 37 r a r a y y M( ) = r ( y) p a a = r y p a a ( z) p qzdz dy z p qzdz y y dy (7) ( z) p qzdz dy z p qzdz dy (8) Case : Assume r > 0, q < 0, p > r. Then is a negative constant a since qz is an onentia decay term, the foowing comparison of (5) can be made: r y S( ) z p dz dy a With the assumption that p >, the comparison takes the form r r S( ) a ( p) a ( p) y p dy. Now the integra takes the form cy n dy where c = r is a a ( p) positive constant. Then Lemma indicates that integras of this form diverge so S( ) =. Since r > 0 qy is an onentia a growth term, the foowing comparison hods for (7): r y S( ) z p dz dy a r r = a ( p) a ( p) y p dy The fina comparison integra takes the form cy n dy where c = r is a negative constant n = p is a negative constant. a ( p) Then Lemma indicates that integras of this form diverge so S( ) =. Using the fact that r > 0 qz is an onentia decay a term, the foowing comparison of (6) can be made: M( ) r y y p z p dz dy a a = a r y p r a ( p) a ( p) y p dy.

38 38 ALLEN, GEBHARDT, KLUBALL, KOLBA The onentia term r a ( p) y p must be ess than or equa to on the interva which aows for the foowing comparison of (6): M( ) a r y p dy. a ( p) This fina integra must converge since p must be ess than. Therefore, M( ) <. Once again using the fact that r is a a negative constant qz is an onentia growth term, the foowing comparison of (8) can be made: M( ) r y y p z p dz dy a a = a r y p r a ( p) a ( p) y p dy. Now, the onentia term r a ( p) y p r is bounded between (upper bound) (ower bound) a ( p) due to p being a negative power. This aows for the foowing comparison: M( ) a r y p r dy a ( p) a ( p) = y p dy a >. Once again, this fina integra must converge since p is stricty ess than. Therefore, M( ) <. Appication of the Stochastic Stabiity Theorem indicates that noise-induced stabiization occurs when r > 0, q < 0, p >. Then stabiization aso occurs when r > 0, q < 0, p >. Case : Assume r > 0, q < 0, p r. Since is a positive constant, a qz is an onentia decay term, the foowing comparison of

39 (6) hods: M( ) r y p a a = a r a ( p) NOISE-INDUCED STABILIZATION 39 y z p dz dy y p r a ( p) y p dy. Since p r, the constant must be positive so that the onentia term a ( p) r a ( p) y p is greater than or equa to on the interva. This fact aows for the additiona comparison of (6) to an even smaer magnitude integra: M( ) a r y p dy. a ( p) Since p, this fina integra must diverge. Therefore, M( ) = noise-induced stabiization does not occur when r > 0, q < 0, p. Then it aso does not occur when r > 0, q < 0, p. The onentia decay theorem shows that an onentia noisecoefficient is not required for stabiization of a onentia drift coefficients. In this theorem, a power noise coefficient was used to stabiize the onentia decay term. 6. Logarithmic Function Stabiization In this section we wi investigate the stabiization of the ODE dx(t) = rn( X(t) ) q dt, where r q are rea numbers with q 0. This ODE is unstabe if ony if r 0. Theorem 6. Consider the SDE { dx(t) = rn( X(t) ) q a X(t) m n( X(t) ) p db(t) for x > dt + a( m )n() p db(t) for x where r, a, q, p, m are rea numbers q, p, m 0. Noise-induced stabiization occurs if ony if the ODE is unstabe, one of the foowing conditions is met: m >, m = q+ p >, or m = q+ p = a q > r.

40 40 ALLEN, GEBHARDT, KLUBALL, KOLBA Figure 4. This image depicts an unstabe ogarithmic ODE that is stabiized with the sufficient amount of noise. Figure 4 shows three separate graphs depicting the phenomenon of noise-induced stabiization. The graph on the far eft shows an ODE that diverges off to infinity for a positive initia vaues; therefore, the ODE is unstabe. The midde graph depicts the noise vaue of X(t) 4 n( X(t) +)db(t) added to the previousy unstabe ODE. This specific amount of noise added does not stabiize the system as m <. The fina image depicts X(t) n( X(t) +) added to the unstabe ODE. It can be seen that the SDE converges. This occurs since m >. Proof. For this SDE we are ooking at b(x) = r n( x ) q σ(x) = a x m n( x ) p where r > 0. Assume there exists some bound such that σ(x) is not undefined, then by the Stochastic Stabiity Theorem x s(x) = x After further simpification, x s(x) = x r n( z ) q a z m n( z ) p dz r n( z ) q a z m n( z ) p dz r n( z ) q p dz a z m r n( z ) q p dz a z m for x > for x <. for x > for x <. (9) In order to find where noise-induced stabiization wi occur, we must separate the vaue of m into three different possibiities: m >, m <, m =. Case : m >

41 NOISE-INDUCED STABILIZATION 4 To show that when m > a vaues of r, a, q, p, m there is noiseinduced stabiization we want s(x) to be greater than or equa to some integra that diverges. Since n( z ) n() for z. Thus, x r n() (q p) dz for x > a s(x) z m x for x <. After integrating, s(x) where c = c = c m(x) r n() (q p) dz a z m r n() (q p) z ( m+) a ( m+) x r n() (q p) z ( m+) a ( m+) r n() (q p) x ( m+) a ( m+) r n() (q p) x ( m+) a ( m+) x r n() (q p) x ( m+) a ( m+) c a x m n( x ) p r n() (q p) x ( m+) a ( m+) c a x m n( x ) p r n() (q p) ( m+) a ( m+) c = for x > for x < for x > for x < for x > for x < First assume q p < 0. Using (30) (3) we know r n() (q p) y ( m+) S( ) c dy, a ( m + ) M( ) S( ) M( ) r n() (q p) y ( m+) a ( m+) c a y m n( y ) p dy, (30) (3) r n() (q p) ( m+) a ( m+) r n() (q p) y ( m+) c dy, a ( m + ) r n() (q p) y ( m+) a ( m+) c a y m n( y ) p Looking at S( ) using Lemma aows us to concude that S( ). For M( ) the onentia wi converge since the coefficient is negative; therefore, as a whoe M( ) <. For S( ) the argest vaue dy..

42 4 ALLEN, GEBHARDT, KLUBALL, KOLBA the onentia function can be is ; therefore, S( ) c dy, which equas. From this it can be concuded that S( ) is infinite. Then M( ) < since the onentia function is converging from its coefficient being negative. Now assume q p > 0. To show s(x) is greater than something that diverges we use the comparison n( z ) q p z m. Then, x s(x) x r z (m ) dz a z m r z (m ) dz a z m for x > for x <, which simpifies to x s(x) x r dz a z (m+ ) r dz a z (m+ ) for x > for x <. Then, s(x) r z ( m+ ) a ( m+ ) r z ( m+ ) a ( m+ ) x x for x > for x < c = c r x ( m+ ) a ( m+ ) r x ( m+ ) a ( m+ ) for x > for x < (3) m(x) r x ( m+ ) a ( m+ ) c a x m n( x ) p r x ( m+ ) a ( m+ ) c a x m n( x ) p for x > for x < (33)

43 NOISE-INDUCED STABILIZATION 43 r where c = ( m+ ) r c a ( m+ ) = ( m+ ). Using (3) a ( m+ ) (33), the Stochastic Stabiity Theorem it can be proven that r y ( m+ ) S( ) c a ( m + ) dy, M( ) S( ) M( ) r y ( m+ ) a ( m+ ) dy, c a y m n( y ) p c r y ( m+ ) a ( m + ) dy, r y ( m+ ) a ( m+ ) dy. c a y m n( y ) p When evauating S( ) the coefficient in the onentia is positive; therefore, by the integra comparison test it can be concuded that S( ) diverges. For M( ) the coefficient of the onentia function is negative, which causes it to converge. Since the onentia is converging, m p 0, then the integra as a whoe is aso converging. In concusion, M( ) <. As for S( ), the smaest vaue the onentia function can be is ; therefore S( ) c dy. This integra diverges to, thus S( ). For simiar reasons as M( ), M( ) converges as we. Finay, assume q p = 0 when m >. Then, x s(x) = x r dz a z m r dz a z m After integrating, it can be said that s(x) = r z ( m+) a ( m+) r z ( m+) a ( m+) x x for x > for x < for x > for x <

44 44 ALLEN, GEBHARDT, KLUBALL, KOLBA r x ( m+) c = c a ( m+) r x ( m+) a ( m+) for x > for x < (34) r x ( m+) a ( m+) c m(x) = for x > a x m n( x ) p (35) r x ( m+) a ( m+) c for x <, a x m n( x ) p where c = r ( m+) c a ( m+) = r ( m+). Again, a ( m+) using (34) (35), the Stochastic Stabiity Theorem it can be determined that r y ( m+) S( ) = c dy, a ( m + ) M( ) = S( ) = M( ) = r y ( m+) a ( m+) dy, c a y m n( y ) p r y ( m+) c dy, a ( m + ) r y ( m+) a ( m+) dy. c a y m n( y ) p When evauating the integra of S( ), Lemma says since the coefficient of the onentia function is positive, S( ) is infinite. The coefficient of the onentia function for M( ) is negative; therefore, the onentia function wi converge. Since m p are positive, the onentia function is converging, the integra of those functions wi converge as we. Thus, M( ) <. As for S( ), the ry c ( m+) dy = c ry a ( m+) ( m+) dy, which diverges to negative infinity. Therefore, S( ) =. From M( ) a ( m+) it is aready known that the function in M( ) is converging. Thus, M( ) <. In summary, when m > aways occurs. we found that noise-induced stabiization

45 NOISE-INDUCED STABILIZATION 45 Case : m < Looking back at (9), in order to show that noise-induced stabiization does not occur for m <, s(x) must be ess than or equa to some function that converges. In this case, we use the comparison n() q p n( z ) q p for z to show x r n() (q p) dz for x > a s(x) z m x for x <. After integration, where c = s(x) c c m(x) r n() (q p) dz a z m r n() (q p) z ( m+) a ( m+) r n() (q p) z ( m+) a ( m+) r n() (q p) x ( m+) a ( m+) r n() (q p) x ( m+) a ( m+) r n() (q p) x ( m+) a ( m+) c a x m n( x ) p r n() (q p) x ( m+) a ( m+) c a x m n( x ) p r n() (q p) ( m+) a ( m+) x x for x > for x < for x > for x < for x > for x < c = (36) r n() (q p) ( m+) a ( m+) First suppose q p > 0, then referring back to (36) can be said that r n() (q p) y ( m+) S( ) c dy. a ( m + ) Using Lemma, S( ) is ess than or equa to a converging integra; therefore, noise-induced stabiization does not occur for m < when q p > 0. Now assume q p < 0. Using the comparison n( z ) q p z m, x s(x) x r z (m ) dz a z m r z (m ) dz a z m for x > for x <..