Small random perturbations of stable deterministic systems often produce a characteristic metastable behaviour: the process remains close to an attracting state for a long time and then leaves its neighbourhood through a rare fluctuation. In population models this mechanism is related to persistence near a carrying capacity and to the onset of extinction [5–7]. This motivation leads naturally to the study of small-noise autoregressive recursions.

The model considered here is (1.1) X n + 1 ε = h ( X n ε ) + ε ξ n + 1 , X 0 ε = 0 , where the innovations are independent standard Gaussian random variables and ε > 0 is small. We study the first exit time τ ε : = min { n ≥ 1 : | X n ε | ≥ 1 } from ( − 1 , 1 ). The large-deviation method of Klebaner and Liptser [12] (see [17] for the general theory) reduces an exponential upper bound for E 0 τ ε to an infimum, over finite horizons, of quadratic path actions. The resulting constant is a discrete quasipotential: it measures the least noise energy required to drive the deterministic system from the stable fixed point to the boundary. The present paper evaluates this variational constant exactly for a continuous two-slope map.

For the linear autoregression h ( x ) = px, | p | < 1, the variational problem is explicit and gives the sharp value ( 1 − p 2 ) / 2. The same large-deviation framework has been developed for past-dependent recursions, moderate deviations, multivariate autoregressions, ARMA processes, and non-Gaussian noise [12, 13, 11, 14, 9]. Complementary approaches to first-passage and level-crossing questions use martingale and analytic methods [16, 15, 4, 2, 3]. Piecewise-linear and threshold autoregressions are classical nonlinear time-series models [1], but their exit variational problems are much less explicit than in the linear case.

Högnäs and Jung [10] studied several nonlinear and piecewise-linear autoregressive maps and obtained explicit bounds in a number of special cases; see also [8] for exit times and invariant distributions of a related stochastic difference equation. For the continuous two-slope family considered below, their formulas cover the boundary cases in which one of the two slopes vanishes. The case in which both slopes are nonzero was not evaluated. This is the remaining case in which the breakpoint constraints genuinely enter the variational minimization.

We solve this remaining case over the full admissible parameter range. The analysis separates two classes of candidate minimizing paths: paths having a vertex at the breakpoint and paths with no internal vertex at the breakpoint. We compute the least cost in both classes, determine exactly when the stationary candidate in the second class satisfies the required inequalities, and compare it with the best breakpoint path. This yields a complete parameter classification involving an interior stationary value, a breakpoint value, or the linear value ( 1 − p 2 ) / 2. We also treat the remaining boundary regimes q = 1 and p = 0 , q > 1, as well as the limiting case f ( 1 ) = 1, and determine whether the variational infimum is attained at a finite horizon.

The condition studied in this paper is that the graph of the positive branch lies strictly below the diagonal, f ( x ) < x for 0 < x < 1. It guarantees deterministic attraction toward the origin from every interior initial state, but it does not require both slopes to be smaller than one; in particular, an expanding second slope is allowed when the affine branch still remains below the diagonal. The regimes in which

the graph meets or rises above the diagonal inside ( 0 , 1 ) have different deterministic and variational structures and are therefore left for separate treatment.

The paper is organized as follows. Section 2 formulates the exit-time variational problem, introduces the two-slope family and the monotone reduction, recalls the previously known boundary cases, and states the main classification. Section 3 proves the affine one-segment minimization used throughout, and Sections 4 and 5 analyze, respectively, paths that hit the breakpoint and paths with no internal vertex there. Section 6 states the key admissibility and comparison results needed for the constrained classification, Section 7 proves the main theorem, and Section 8 treats the remaining boundary regimes. The detailed proofs of the admissibility and comparison results are given in Section 9.

The analysis of Case I and of the boundary regimes repeatedly reduces to minimizing the action over a single affine piece, and we isolate that minimization here. Let a < b and g ( x ) = αx + γ, where α > 0 and α ≠ 1; for N ≥ 1, denote the constrained and unconstrained minima, respectively, by I N ( a , b ; g ) : = min a = x 0 ≤ ⋯ ≤ x N = b 1 2 ∑ i = 1 N ( x i − g ( x i − 1 ) ) 2 , J N ( a , b ; g ) : = min x 0 = a , x N = b x 1 , … , x N − 1 ∈ R 1 2 ∑ i = 1 N ( x i − g ( x i − 1 ) ) 2 .

Lemma 3.1.

Suppose that g ( x ) ≤ x on [ a , b ], and set d a : = a − g ( a ), d b : = b − g ( b ). Then, for every N ≥ 1, the unrestricted minimum is (3.1) J N ( a , b ; g ) = 1 − α 2 2 ( 1 − α ) 2 ( d b − d a α N ) 2 1 − α 2 N . If d a = 0, then necessarily 0 < α < 1, and (3.2) inf N ≥ 1 I N ( a , b ; g ) = 1 + α 2 ( 1 − α ) d b 2 . If d b = 0, then necessarily α > 1, and (3.3) inf N ≥ 1 I N ( a , b ; g ) = 1 + α 2 ( α − 1 ) d a 2 . Finally, if d a d b ≠ 0, define N 0 : = max { 1 , ⌊ log α ( d a / d b ) ⌋ }. Then (3.4) inf N ≥ 1 I N ( a , b ; g ) = min { J N 0 ( a , b ; g ) , J N 0 + 1 ( a , b ; g ) } .

Proof.

Fix N ≥ 1 and write e i : = x i − g ( x i − 1 ) = x i − α x i − 1 − γ for i = 1 , … , N. Multiplying by α N − i and summing gives the telescoping identity (3.5) ∑ i = 1 N α N − i e i = b − α N a − γ ∑ r = 0 N − 1 α r = d b − d a α N 1 − α , so that, by the Cauchy–Schwarz inequality, ∑ i = 1 N e i 2 ≥ ( ∑ i = 1 N α N − i e i ) 2 ∑ i = 1 N α 2 ( N − i ) = 1 − α 2 ( 1 − α ) 2 ( d b − d a α N ) 2 1 − α 2 N , with equality exactly when the error vector is proportional to ( α N − 1 , … , α , 1 ). The endpoint identity (3.5) fixes the proportionality constant, so the unrestricted minimizer satisfies (3.6) e i = x i − g ( x i − 1 ) = β α N − i , i = 1 , … , N , β = 1 − α 2 1 − α d b − d a α N 1 − α 2 N , and it is unique, the map from the intermediate vertices to the error vector being invertible. Halving the bound proves (3.1).

We next determine when this minimizer is nondecreasing. Since g ( x ) ≤ x on [ a , b ], the endpoint gaps d a and d b are nonnegative, and (3.7) d b − d a = ( 1 − α ) ( b − a ) ; hence d a = 0 forces 0 < α < 1, and d b = 0 forces α > 1. Put L N : = ( d b − d a α N ) / ( 1 − α 2 N ). Iterating (3.6) from x 0 = a and subtracting successive coordinates, (3.8) x i − x i − 1 = − d a α i − 1 + L N ( α N − i + α N + i − 1 ) , i = 1 , … , N . If d a = 0, then 0 < α < 1 and L N = d b / ( 1 − α 2 N ) > 0, so every increment is positive. If d b = 0, then α > 1 and L N = d a α N / ( α 2 N − 1 ) > 0; after division by α i − 1 the right-hand side of (3.8) decreases with i, and its last value x N − x N − 1 = d a α N − 1 ( 1 + α ) / ( α 2 N − 1 ) is positive. Thus in both endpoint cases the minimizer is strictly increasing.

Assume now d a d b > 0. If 0 < α < 1, the normalized increments ( x i − x i − 1 ) / α i − 1 increase with i, so only i = 1 matters: substituting L N into x 1 − x 0 = − d a + L N ( α N − 1 + α N ) ≥ 0 and clearing the positive denominator 1 − α 2 N , the condition becomes (3.9) d b α N − 1 ( 1 + α ) ≥ d a ( 1 + α 2 N − 1 ) . If α > 1, the same normalized increments decrease, so i = N is decisive, and x N − x N − 1 = − d a α N − 1 + L N ( 1 + α 2 N − 1 ) ≥ 0 with L N = ( d a α N − d b ) / ( α 2 N − 1 ) becomes (3.10) d a ( 1 + α ) ≥ d b ( α N + α 1 − N ) . Thus, when d a d b > 0, the unrestricted minimizer is admissible for a given N precisely when the relevant one of (3.9) and (3.10) holds. Neither condition is automatic, and we do not check them yet: we first carry out the minimization over N, which singles out two candidate integers, and then return at the end of the proof to verify the appropriate condition at those two integers only.

We now minimize over N. If d a = 0, then (3.1) reads J N ( a , b ; g ) = 1 + α 2 ( 1 − α ) d b 2 1 − α 2 N , which decreases strictly to the value in (3.2); if d b = 0, then α > 1 and J N ( a , b ; g ) = 1 + α 2 ( α − 1 ) d a 2 α 2 N α 2 N − 1 ↓ 1 + α 2 ( α − 1 ) d a 2 . In both cases the unrestricted minimizer is nondecreasing for every N, so I N ( a , b ; g ) = J N ( a , b ; g ); this proves (3.2) and (3.3) together with the nonattainment statements.

Suppose finally that d a d b > 0. Extend the right-hand side of (3.1) to real x > 0 and set z = α x ; apart from its constant prefactor it becomes ( d b − d a z ) 2 / ( 1 − z 2 ), with (3.11) d dz ( d b − d a z ) 2 1 − z 2 = 2 ( d b − d a z ) ( d b z − d a ) ( 1 − z 2 ) 2 . For 0 < α < 1, (3.7) gives 0 < d a < d b , and z = d a / d b is the only critical point in the admissible range 0 < z < 1; for α > 1 one has d a > d b > 0, and the same point is the only critical point in the admissible range z > 1. Taking into account the direction of z = α x and, for α > 1, the sign of the prefactor in (3.1), the real-variable cost is in both cases strictly decreasing before N r = log α ( d a / d b ) and strictly increasing afterwards. Its integer minimum is therefore among N 0 and N 0 + 1, with N 0 as in Lemma 3.1; if N r < 1, it occurs at N = 1.

It remains to check the ordering constraint at these two integers. Let 0 < α < 1 and N r ≥ 1, so that α N 0 + 1 ≤ d a / d b ≤ α N 0 . For N = N 0 , d a ( 1 + α 2 N − 1 ) ≤ d b α N ( 1 + α 2 N − 1 ) ≤ d b α N − 1 ( 1 + α ) , while for N = N 0 + 1 the same bound, written d a / d b ≤ α N − 1 , gives d a ( 1 + α 2 N − 1 ) ≤ d b α N − 1 ( 1 + α ) directly; thus (3.9) holds at both. Let now α > 1 and N r ≥ 1, so that α N 0 ≤ d a / d b ≤ α N 0 + 1 . For N = N 0 we get d a ( 1 + α ) ≥ d b α N ( 1 + α ) ≥ d b ( α N + α 1 − N ), and for N = N 0 + 1, using d a / d b ≥ α N − 1 , d a ( 1 + α ) ≥ d b α N − 1 ( 1 + α ) ≥ d b ( α N + α 1 − N ); thus (3.10) holds at both. If 0 < N r < 1, the optimal one-step path has no internal vertex and is automatically nondecreasing.

Therefore an unrestricted minimizer realizing the integer minimum is admissible for the monotone problem, and since J N ( a , b ; g ) ≤ I N ( a , b ; g ) for every N, this proves (3.4). □

We keep throughout the distinction just established: for fixed N and d a d b > 0, the unrestricted value (3.1) need not equal the constrained value I N ( a , b ; g ); the two problems agree only after optimization over N, because at least one of the optimal adjacent integers is admissible.

Assume throughout this section that 0 < p < 1, q > 0, q ≠ 1, and Δ > 0. We compute the infimum over nondecreasing paths that hit the breakpoint c; fixing the breakpoint vertex separates the action into two affine one-segment problems.

Lemma 4.1.

Fix N ≥ 2 and j ∈ { 1 , … , N − 1 }. Among paths satisfying y j = c, the two parts of the action can be minimized independently: (4.1) min 0 = y 0 ≤ ⋯ ≤ y N = 1 y j = c I N ( y ; f ) = min 0 = y 0 ≤ ⋯ ≤ y j = c 1 2 ∑ i = 1 j ( y i − f ( y i − 1 ) ) 2 + min c = y j ≤ ⋯ ≤ y N = 1 1 2 ∑ i = j + 1 N ( y i − f ( y i − 1 ) ) 2 .

Proof.

With y 0 = 0, y j = c, and y N = 1 fixed, write A j ( y 1 , … , y j − 1 ) : = 1 2 ∑ i = 1 j ( y i − f ( y i − 1 ) ) 2 and B N − j ( y j + 1 , … , y N − 1 ) : = 1 2 ∑ i = j + 1 N ( y i − f ( y i − 1 ) ) 2 . The variables occurring in A j and B N − j are disjoint, and every admissible full path with y j = c satisfies I N ( y ; f ) = A j + B N − j ; hence its action is at least min A j + min B N − j . Conversely, the two admissible sets are compact and the two actions continuous, so minimizing left and right subpaths exist, and concatenating them at their common endpoint c gives an admissible full path of action exactly min A j + min B N − j . □

Denote by S b ( f ) : = inf N ≥ 2 , 1 ≤ j < N 0 = y 0 ≤ ⋯ ≤ y N = 1 , y j = c I N ( y ; f ) the infimum over the class of paths that hit the breakpoint, put n : = N − j, and set g p ( x ) : = px, g q ( x ) : = qx + c ( p − q ). Since the two positive lengths j and n vary independently, the decomposition lemma yields (4.2) S b ( f ) = inf j ≥ 1 I j ( 0 , c ; g p ) + inf n ≥ 1 I n ( c , 1 ; g q ) . For the left one-segment problem the endpoint gaps are d 0 = 0 and d c = c − g p ( c ) = c ( 1 − p ); the fixed-j minimizer is increasing, and Lemma 3.1 gives (4.3) I j ( 0 , c ; g p ) = 1 − p 2 2 ( 1 − p 2 j ) c 2 ↓ 1 − p 2 2 c 2 ( j → ∞ ) , the limit being strict and not attained at a finite left-hand length. For the right one-segment problem the endpoint gaps d c = c − g q ( c ) = c ( 1 − p ) and d 1 = 1 − g q ( 1 ) = Δ are both positive, and the unconstrained n-step value from Lemma 3.1 is (4.4) J n ( c , 1 ; g q ) = ( 1 + q ) ( Δ − c ( 1 − p ) q n ) 2 2 ( 1 − q ) ( 1 − q 2 n ) . We compare with the sequence S ˆ n of (2.13), whose second term is, at an arbitrary fixed n, an unrestricted algebraic value whose stationary path need not be nondecreasing; thus S ˆ n is not to be read as an admissible path cost for every n. The minimizer of the real-variable extension of the right one-segment problem is x b : = log q ( c ( 1 − p ) / Δ ) > 0, and by Lemma 3.1 the constrained right one-segment minimum is attained at one of the two adjacent integer lengths n ∗ and n ∗ + 1 of (2.14), at which the unrestricted minimizers are themselves nondecreasing. Consequently the algebraic values in (2.13) are attained by admissible paths hitting the breakpoint for the two lengths that determine the minimum.

Proposition 4.2.

The infimum over all paths that hit the breakpoint is (4.5) S b ( f ) = S ˆ , with S ˆ as in (2.14), and it is not attained by any finite path.

Proof.

Combining (4.2) with (4.3) and the integer-minimization part of Lemma 3.1, S b ( f ) = 1 − p 2 2 c 2 + min { J n ∗ ( c , 1 ; g q ) , J n ∗ + 1 ( c , 1 ; g q ) } = min { S ˆ n ∗ , S ˆ n ∗ + 1 } , which is (4.5). The right one-segment problem has an optimal finite length, but the left contribution in (4.3) approaches its infimum strictly as j → ∞; therefore no finite full path attains S ˆ . □

Throughout this section we work under the assumptions of Theorem 2.4: 0 < p < 1, q > 0, q ≠ p , 1, and Δ > 0, and we assume that no internal vertex equals c. Since the path is nondecreasing with y 0 = 0 < c < 1 = y N , there is a unique index k ∈ { 0 , … , N − 1 } such that (5.1) y k < c < y k + 1 ; we write m : = N − k − 1 for the number of steps on the right piece [ c , 1 ]. The following lemmas compute the corresponding algebraic stationary path and then determine when it satisfies (5.1). For m ∈ Z ≥ 0 , put (5.2) N m : = 1 − c ( p − q ) ∑ j = 0 m − 1 q j , D m : = ∑ j = 0 m − 1 q 2 j + q 2 m 1 − p 2 , where an empty sum is zero, so N 0 = 1 and D 0 = ( 1 − p 2 ) − 1 . The quantities S m and Y m of (2.12) then have the equivalent form (5.3) S m = N m 2 2 D m , Y m = N m q m D m , which defines them for every m ≥ 0; the value S m is a Case II action only for indices satisfying the feasibility condition (5.8).