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❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❑②✐✈ ◆❛t✐♦♥❛❧ ❙❝❤❡✈❝❤❡♥❦♦ ❯♥✐✈❡rs✐t②✱ ❯❦r❛✐♥❡✳ ✸ ãðóäíÿ ✷✵✶✷ ð✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶✳ ❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠✳ ✷✳ ▼♦t✐✈❛t✐♦♥ ❛♥❞ r❡✈✐❡✇ ♦❢ t❤❡ ❡①✐st✐♥❣ r❡s✉❧ts✳ ✸✳ ▼❛✐♥ r❡s✉❧ts✳ ✹✳ ❆♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠② ♦❢ st♦❝❤❛st✐❝ s②st❡♠s ❛♥❞ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ♦❢ t❤❡ s❡❝♦♥❞ ♦r❞❡r ❧✐♥❡❛r st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ✺✳ ■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s✳ ✻✳ ❆♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ ❛ st❛❜✐❧✐t②✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥tr♦❞✉❝t✐♦♥

■♥ t❤✐s ♣❛♣❡r ■ ✇✐❧❧ ❞✐s❝✉ss ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ t❤❡ q✉❛❧✐t❛t✐✈❡ ♣r♦♣❡rt✐❡s ♦❢ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ◆❛♠❡❧② t❤❡ st♦❝❤❛st✐❝ ❝❛s❡ ✇✐❧❧ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡✳ ▼② t❛❧❦ ✐s ❞❡❞✐❝❛t❡❞ t♦ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥s ✉♥❞❡r ✇❤✐❝❤ s✉❝❤ ❛♥ ❛♣♣r♦❛❝❤ ✐s ♣♦ss✐❜❧❡✳ ❚❤✐s ♣❛♣❡r ❝♦♥s✐sts ♦❢ t✇♦ ♣❛rts✳ ❚❤❡ ✜rst ♦♥❡ ❞❡❛❧s ✇✐t❤ ❛ st✉❞② ♦❢ ❛♥ ❛s②♠♣t♦t✐❝❛❧ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ st♦❝❤❛st✐❝ s②st❡♠s ❢♦r t → ∞ ❜② st✉❞②✐♥❣ ❛♥ ❛s②♠♣t♦t✐❝❛❧ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ s♣❡❝✐✜❝ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ■ ✇✐❧❧ t❛❧❦ ❛❜♦✉t ❛♥ ❡①✐st❡♥❝❡ ♦❢ st❛❜❧❡✱ ✐♥✈❛r✐❛♥t ❞❡t❡r♠✐♥✐st✐❝ ♠❛♥✐❢♦❧❞s ❢♦r st♦❝❤❛st✐❝ s②st❡♠s✱ ✇❤✐❝❤ ❛❧❧♦✇ t♦ tr❛♥s❢♦r♠ t❤❡ ♦r✐❣✐♥❛❧ st♦❝❤❛st✐❝ s②st❡♠ ✐♥t♦ ❛ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠ ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❛t ✐s t♦ ✐♥✈❡st✐❣❛t❡ ❛ st♦❝❤❛st✐❝ s②st❡♠ ❜② r❡❞✉❝✐♥❣ ✐t t♦ ❞❡t❡r♠✐♥✐st✐❝ ♦♥❡✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠

❈♦♥s✐❞❡r t❤❡ st♦❝❤❛st✐❝ ■t♦ s②st❡♠ dy = g(t, y)dt + σ(t, y)dW (t), ✭✶✮ ✇❤❡r❡

◮ g(t, y), σ(t, y) ❛r❡ ❝♦♥t✐♥✉♦✉s ✐♥ t ≥ 0, y ∈ Rn ❛♥❞ s❛t✐s❢② t❤❡

❣❧♦❜❛❧ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ ✐♥ y✱

◮ W (t) ✐s ❛ ✉s✉❛❧ s❝❛❧❛r ❲✐❡♥❡r ♣r♦❝❡ss ❞❡✜♥❡❞ ❢♦r t ≥ 0 ♦♥ t❤❡

♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, ❋, P)✱

◮ {Ft, t ≥ 0} ✐s ❛ ❢❛♠✐❧② ♦❢ σ✲❛❧❣❡❜r❛s s✳t✳ W (t) ✐s ❝♦♥s✐st❡♥t

✇✐t❤ Ft✳ ❚❤❡ s②st❡♠ ✭✶✮ s✉❜❥❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ y(t0) = y0✱ E|y0|2 < ∞ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ y(t) ❢♦r t ≥ t0 ≥ 0.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠

❲❡ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✭✶✮ ❢♦r t → ∞✳ ❚❤❡ ❛♥❛❧②s✐s ✇✐❧❧ ❜❡ ❝❛rr✐❡❞ ♦✉t ✉s✐♥❣ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ✐♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❖❉❊✮ ♠❡t❤♦❞ ♦❢ ❛s②♠♣t♦t✐❝ ❡q✉✐✈❛❧❡♥❝❡✱ ✇❤❡♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ❢♦r t → ∞ ❜❡❤❛✈❡ s✐♠✐❧❛r❧② t♦ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ s✐♠♣❧❡r s②st❡♠✳ ■♥ ♦✉r ❝❛s❡ ✇❡ ✇✐❧❧ ❜❡ ❝♦♠♣❛r✐♥❣ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ st♦❝❤❛st✐❝ s②st❡♠ ✇✐t❤ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ s♣❡❝✐❛❧❧② ❝♦♥str✉❝t❡❞ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠✳ ❆❧♦♥❣ ✇✐t❤ ✭✶✮ ❝♦♥s✐❞❡r t❤❡ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠ dx = f (t, x)dt ✭✷✮

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠

❉❡❢✳✶✳■❢ ❢♦r ❡✈❡r② s♦❧✉t✐♦♥ y(t) ♦❢ ✭✶✮ ♦♥❡ ❝❛♥ ✜♥❞ ❛ s♦❧✉t✐♦♥ x(t) ♦❢ ✭✷✮ s✳t✳ lim

t→∞ E|x(t) − y(t)|2 = 0,

t❤❡♥ t❤❡ s②st❡♠ ✭✷✮ ✐s ❝❛❧❧❡❞ ❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②st❡♠ ✭✶✮ ✐♥ sq✉❛r❡ ♠❡❛♥✳ ❉❡❢✳✷✳■❢ ❢♦r ❡✈❡r② s♦❧✉t✐♦♥ y(t) ♦❢ ✭✶✮ ♦♥❡ ❝❛♥ ✜♥❞ ❛ s♦❧✉t✐♦♥ x(t) ♦❢ ✭✷✮ s✳t✳✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡ lim

t→∞ |x(t) − y(t)| = 0,

t❤❡♥ t❤❡ s②st❡♠ ✭✷✮ ✐s ❝❛❧❧❡❞ ❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②st❡♠ ✭✶✮ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳ ❖✉r ♠❛✐♥ q✉❡st✐♦♥ ♦❢ ✐♥t❡r❡st ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ✉♥❞❡r ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ❝❛♥ ♦♥❡ ❝♦♥str✉❝t ❛♥ ❖❉❊ s②st❡♠ ✭✷✮ ✇❤✐❝❤ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ st♦❝❤❛st✐❝ s②st❡♠ ✭✶✮ ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❉❡✜♥✐t✐♦♥s ✶ ❛♥❞ ✷❄

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❊①✐st✐♥❣ r❡s✉❧ts ❛♥❞ ♠♦t✐✈❛t✐♦♥✳ ▲❡✈✐♥s♦♥ ❚❍▼

❚❤❡ ❛♣♣r♦❛❝❤ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡ ✐s ✇❡❧❧ ❦♥♦✇♥ ✐♥ ❖❉❊✳ ❚❤❡ ❝❧❛ss✐❝ r❡s✉❧ts ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ❛r❡ ❞✉❡ t♦ ❲✐♥t♥❡r✱ ▲❡✈✐♥s♦♥ ❛♥❞ ❨❛❦✉❜♦✈✐❝❤✳ ▲❡✈✐♥s♦♥ ❚❤❡♦r❡♠ ✭✶✾✹✽✮ ❣✐✈❡s t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ❧✐♥❡❛r s②st❡♠s dy dt = (A + B(t))y ✭✸✮ dx dt = Ax ✭✹✮ ❚❍▼✳✭▲❡✈✐♥s♦♥✮ ■❢ ❛❧❧ s♦❧✉t✐♦♥s ♦❢ ✭✹✮ ❛r❡ ❜♦✉♥❞❡❞ ❢♦r t ≥ 0 ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥

∞

• ||B(t)||dt < ∞,

✭✺✮ ❤♦❧❞s✱ t❤❡♥ t❤❡ s②st❡♠s ✭✸✮ ❛♥❞ ✭✹✮ ❛r❡ ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t✱ ✐✳❡✳ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r s♦❧✉t✐♦♥s y(t) ❛♥❞ x(t) s✳t✳ lim

t→∞ |x(t) − y(t)| = 0.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❲✐♥t♥❡r ❚❍▼

❆ r❡s✉❧t s✐♠✐❧❛r t♦ ▲❡✈✐♥s♦♥ ❚❤❡♦r❡♠ ✇❛s ♦❜t❛✐♥❡❞ ❜② ❲✐♥t♥❡r ✭✶✾✹✻✮✳ ❍❡ ❝♦♥s✐❞❡r❡❞ t❤❡ s②st❡♠s dx dt = A1(t)x, dy dt = A2(t)y. ❚❍▼✳✭❲✐♥t♥❡r✮ ▲❡t A1(t)✱ A2(t) ❜❡ ❝♦♥t✐♥✉♦✉s ❢♦r t ≥ 0 ❛♥❞ ✶✳ ❡✈❡r② s♦❧✉t✐♦♥ y(t) ✐s ❜♦✉♥❞❡❞ ❢♦r t ≥ 0❀ ✷✳ lim inft→∞

t

• SpA2(s)ds > −∞❀

✸✳

∞

• ||A1(s) − A2(s)||ds < ∞.

❚❤❡♥ ✇❡ ❝❛♥ ✜♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ s②st❡♠s s✉❝❤ t❤❛t |x(t) − y(t)| = O(

∞

• t

||A1(t) − A2(t)||ds) → 0, t → ∞.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❨❛❦✉❜♦✈✐❝❤ ❚❍▼

❇♦t❤ ▲❡✈✐♥s♦♥ ❛♥❞ ❲✐♥t♥❡r t❤❡♦r❡♠s ❛r❡ ❜❛s❡❞ ♦♥ ♦♥❡ ✐♠♣♦rt❛♥t ❝♦♥❞✐t✐♦♥✿ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✉♥♣❡rt✉r❜❡❞ s②st❡♠ ❛r❡ ❜♦✉♥❞❡❞ ❢♦r t → ∞✳ ❨❛❦✉❜♦✈✐❝❤ ✭✶✾✺✶✮ ❣❡♥❡r❛❧✐③❡❞ t❤❡ r❡s✉❧ts ♦❢ ▲❡✈✐♥s♦♥ ❛♥❞ ❲✐♥t♥❡r ❢♦r t❤❡ ❝❛s❡ ♦❢ ✉♥❜♦✉♥❞❡❞ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✉♥♣❡rt✉r❜❡❞ s②st❡♠✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠s ✇❡r❡ ❝♦♥s✐❞❡r❡❞ dy dt = Ay + f (t, y) ✭✻✮ dx dt = Ax. ✭✼✮ ✇✐t❤ |f (t, y)| ≤ g(t)|y| ❛♥❞ s❛t✐s❢②✐♥❣ t❤❡ ▲✐♣s❝❤✐t③ ❝♦♥❞✐t✐♦♥ |f (t, y1) − f (t, y2)| ≤ g(t)|y1 − y2|, f (t, 0) = 0.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❨❛❦✉❜♦✈✐❝❤ ❚❍▼

❚❍▼✳✭❨❛❦✉❜♦✈✐❝❤✮ ▲❡t f s❛t✐s❢② t❤❡ ❝♦♥❞✐t✐♦♥s ❛❜♦✈❡ ❛♥❞

∞

• tm+p−2eλtg(t)dt < ∞.

❚❤❡♥ t❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✭✻✮ ❛♥❞ ✭✼✮ s✳t✳ |x(t) − y(t)| → 0, t → ∞, ✭✽✮ ▼♦r❡♦✈❡r✱ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ✭✽✮ ✐s s♣❡❝✐✜❡❞✳ ❍❡r❡ λ = max Re(λi)✱ ✇❤❡r❡ {λi} ❛r❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ A✱ m ✐s t❤❡ ♠❛①✐♠❛❧ s✐③❡ ♦❢ t❤❡ ❏♦r❞❛♥ ❝❡❧❧ ❢♦r ✇❤✐❝❤ Re(λi) = λ✱ p ✐s t❤❡ ♠❛①✐♠❛❧ s✐③❡ ♦❢ t❤❡ ❏♦r❞❛♥ ❝❡❧❧ ❢♦r ✇❤✐❝❤ Re(λi) = 0 ✱ p = 1 ✐❢ t❤❡r❡ ❛r❡ ♥♦ s✉❝❤ λi✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❆❤♠❡t♦✈ ❚❍▼

❆♠♦♥❣ t❤❡ r❡❝❡♥t r❡s✉❧ts ✇❡ ♠❡♥t✐♦♥ t❤❡ r❡s✉❧ts ♦❢ ❆❤♠❡t♦✈ ✭✷✵✵✼✮✳ ❍❡ st✉❞✐❡❞ t❤❡ ❝❛s❡ A = A(t)✳ ❈♦♥s✐❞❡r t❤❡ s②st❡♠s dy dt = (A(t) + B(t))y ✭✾✮ ❛♥❞ dx dt = A(t)x ✭✶✵✮ ▲❡t X(t) ❜❡ t❤❡ ♠❛tr✐❝✐❛♥t ♦❢ t❤❡ s②st❡♠ ✭✶✵✮✱ X(0) = E✳ ❙❡t u := X −1(t)y ✇✐t❤ y s♦❧✈✐♥❣ ✭✾✮✳ ❚❤❡♥✱ ❝❧❡❛r❧②✱ u s❛t✐s✜❡s ˙ u = P(t)u, P(t) = X −1(t)B(t)X(t).

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❆❤♠❡t♦✈ ❚❍▼

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ C1)

∞

• P(t)dt < ∞.

C2) lim

t→∞ X(t)Φ(t) = 0✱ ✇❤❡r❡ Φ(t) ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢

˙ φ = P(t)(φ + E) ✭✶✶✮ s✳t✳ Φ(t) → 0, t → ∞ ✭t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ Φ ✇❛s ❡st❛❜❧✐s❤❡❞✮✳ ❚❍▼✳✭❆❤♠❡t♦✈✮ ❯♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s C1 ❛♥❞ C2 t❤❡ s②st❡♠s ✭✾✮ ❛♥❞ ✭✶✵✮ ❛r❡ ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▼♦t✐✈❛t✐♦♥

❆❢t❡r t❤❡ ♣✐♦♥❡❡r✐♥❣ ✇♦r❦s ♦❢ ▲❡✈✐♥s♦♥✱ ❲✐♥t♥❡r ❛♥❞ ❨❛❦✉❜♦✈✐❝❤✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛s②♠♣t♦t✐❝ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ❞✐✛❡r❡♥t✐❛❧ s②st❡♠s ✐♥❝❧✉❞✐♥❣ ❧✐♥❡❛r✱ ♥♦♥❧✐♥❡❛r ❛♥❞ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s ❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✳ ❖✉r ❣♦❛❧ ✐s t♦ ❡st❛❜❧✐s❤ s✐♠✐❧❛r r❡s✉❧ts ❢♦r st♦❝❤❛st✐❝ s②st❡♠s✱ ✐✳❡✳ t♦ ❝♦♠♣❛r❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ st♦❝❤❛st✐❝ s②st❡♠s ✇✐t❤ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s✱ ❛♥❞ t❤✉s✱ t♦ r❡❞✉❝❡ t❤❡ ❛♥❛❧②s✐s ♦❢ st♦❝❤❛st✐❝ s②st❡♠s t♦ t❤❡ ❛♥❛❧②s✐s ♦❢ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s✳ ❚❤✐s ✇✐❧❧ ❡♥❛❜❧❡ ✉s t♦ ❛❞❞r❡ss ♠❛♥② ✐ss✉❡s ♦❢ t❤❡ q✉❛❧✐t❛t✐✈❡ t❤❡♦r② ♦❢ st♦❝❤❛st✐❝ s②st❡♠s✱ ✐♥❝❧✉❞✐♥❣ st❛❜✐❧✐t②✱ ❞✐ss✐♣❛t✐✈✐t②✱ ❞✐❝❤♦t♦♠②✱ t❤❡♦r② ♦❢ ▲②❛♣✉♥♦✈ ❡①♣♦♥❡♥ts ❡t❝✳✱ s✐♥❝❡ t❤♦s❡ ✐ss✉❡s ❛r❡ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ❢♦r ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ♦♥❡ ♦❢ t❤❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤✐s ♠❡t❤♦❞ t♦ t❤❡ ♦s❝✐❧❧❛t✐♦♥ t❤❡♦r②✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▼❛✐♥ r❡s✉❧ts✳ ❆♥❛❧♦❣ ♦❢ ▲❡✈✐♥s♦♥ ❚❤❡♦r❡♠

❈♦♥s✐❞❡r t✇♦ s②st❡♠s✱ ♦♥❡ ♦❢ ✇❤✐❝❤ ✐s ❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ t❤❡ ♦t❤❡r ✐s st♦❝❤❛st✐❝✿ dx = Axdt ✭✶✷✮ dy = (A + B(t))ydt + D(t)y dW (t), ✭✶✸✮ ✇❤❡r❡ B(t) ❛♥❞ D(t) ❛r❡ ♠❡❛s✉r❛❜❧❡ ♠❛tr✐❝❡s ❢♦r t ≥ 0✳ ❚❍▼✳ ▲❡t t❤❡ s②st❡♠ ✭✶✷✮ ❜❡ st❛❜❧❡ ❢♦r t ≥ 0✱ ✐✳❡✳ ❛❧❧ ✐ts s♦❧✉t✐♦♥s ❛r❡ ❜♦✉♥❞❡❞ ❢♦r t ≥ 0✳ ■❢

∞

• B(t)dt < ∞ ❛♥❞

∞

• D(t)2dt < ∞,

t❤❡♥ t❤❡ s②st❡♠ ✭✶✷✮ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②st❡♠ ✭✶✸✮ ❜♦t❤ ✐♥ sq✉❛r❡ ♠❡❛♥ ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳ ❆ s✐♠✐❧❛r r❡s✉❧t ✐s ❛❧s♦ tr✉❡ ❢♦r ✇❡❛❦❧② ♥♦♥❧✐♥❡❛r st♦❝❤❛st✐❝ s②st❡♠s✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▼❛✐♥ r❡s✉❧ts✳ ▼♦r❡ ❣❡♥❡r❛❧ s❡tt✐♥❣

❲❡ ♥♦✇ ❣✐✈❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❧✐♥❡❛r ♣❛rt ✐s ♥♦♥❝♦♥st❛♥t✱ ❛♥❞ ✉♥♣❡rt✉r❜❡❞ s②st❡♠ ✭✶✷✮ ♠❛② ❤❛✈❡ ✉♥❜♦✉♥❞❡❞ s♦❧✉t✐♦♥s✳ ❚❤✐s r❡s✉❧t ✐s ♥♦✈❡❧ ❡✈❡♥ ❢♦r ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s✳ ❈♦♥s✐❞❡r ♥♦♥❧✐♥❡❛r ■t♦ s②st❡♠ dy = (A(t)y + f (t, y))dt + σ(t, y)dW (t) ✭✶✹✮ ❛♥❞ ✉♥♣❡rt✉r❜❡❞ s②st❡♠ ♦❢ ❖❉❊s dx = A(t)xdt. ✭✶✺✮ ❚❤❡ ♠❛tr✐① A(t) ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞ ♦♥ R1✱ a := sup

t∈R1 A(t). ❚❤❡ ❢✉♥❝t✐♦♥s f (t, y), σ(t, y) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r

t ≥ 0, y ∈ Rn ❛♥❞ ▲✐♣s❝❤✐t③ ✐♥ y✳ ❆s ✇❡ ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ t❤✐s ❣✉❛r❛♥t❡❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ y(t0) = y0✱ E|y0|2 < ∞, ❢♦r t ≥ t0 ≥ 0.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▼❛✐♥ r❡s✉❧t

❆❞❞✐t✐♦♥❛❧❧②✱ f ❛♥❞ σ ❛r❡ s♠❛❧❧ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ t❤❡r❡ ❡①✐st ♥♦♥♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥s ν, ρ s✳t✳ ❢♦r t ≥ 0 ❛♥❞ x ∈ Rn✱ |f (t, x)| ≤ ν(t)|x|✱ |σ(t, x)| ≤ ρ(t)|x|. ▲❡t X(t) ❜❡ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♠❛tr✐① ♦❢ ✭✶✺✮✱ X(0) = E✳ ▼❛✐♥ ❝♦♥❞✐t✐♦♥ ♦♥ ✉♥♣❡rt✉r❜❡❞ s②st❡♠✿ ✭✶✺✮ ✐s ❡①♣♦♥❡♥t✐❛❧❧② ❞✐❝❤♦t♦♠✐❝ ♦♥ R✱ ✐✳❡✳ t❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts N1, N2, ν1, ν2 ❛♥❞ ❝♦♠♣❧❡♠❡♥t✐♥❣ ♣r♦❥❡❝t♦rs P1✱ P2 s✳t✳ X(t)P1X −1(s) ≤ N1e−ν1(t−s), t ≥ s X(t)P2X −1(s) ≤ N2e−ν2(t−s), s ≥ t. ❚❍▼✳ ■❢

∞

• eatν(t)dt < ∞ ❛♥❞

∞

• e2atρ2(t)dt < ∞✱ t❤❡♥ ✭✶✺✮ ✐s

❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✭✶✹✮ ✐♥ sq✉❛r❡ ♠❡❛♥✳ ■❢ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤

∞

• te2atρ2(t)dt < ∞✱ t❤❡♥ ✭✶✺✮ ✐s

❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✭✶✹✮ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▲✐♥❡❛r ❝❛s❡

❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ ✭✶✹✮ ✐s ❧✐♥❡❛r✿ dy = (A(t) + B(t))ydt + D(t)ydW (t), ✭✶✻✮ ✇❤❡r❡ B(t), D(t) ❛r❡ ❝♦♥t✐♥✉♦✉s ❞❡t❡r♠✐♥✐st✐❝ ♠❛tr✐❝❡s✳ ■♥ t❤✐s ❝❛s❡ ✇❡ ♦❜t❛✐♥ ❛ str♦♥❣❡r r❡s✉❧t✱ ♥❛♠❡❧②✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ s②st❡♠s ✭✶✻✮ ❛♥❞ ✭✶✺✮ s✳t✳ ❡❛❝❤ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✶✻✮ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ✭✶✺✮✳ ❍❡r❡✱ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ✐s t❤❡ s♦❧✉t✐♦♥ t❤❛t ❜❡❝♦♠❡s ✐❞❡♥t✐❝❛❧❧② ③❡r♦ ✇✐t❤ ③❡r♦ ♣r♦❜❛❜✐❧✐t②✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

▼❛✐♥ r❡s✉❧t ✐♥ ❧✐♥❡❛r ❝❛s❡

❚❍▼✳ ▲❡t t❤❡ s②st❡♠ ✭✶✺✮ ❜❡ ❡①♣♦♥❡♥t✐❛❧❧② ❞✐❝❤♦t♦♠✐❝ ♦♥ R✳ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞✿ ✶✮

∞

• e2atB(t)2dt < ∞;

✷✮

∞

• te2atD(t)2dt < ∞.

❚❤❡♥ t❤❡ s②st❡♠ ✭✶✺✮ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②st❡♠ ✭✶✻✮ ❜♦t❤ ✐♥ sq✉❛r❡ ♠❡❛♥ ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r s♦❧✉t✐♦♥s ✐♥ s✉❝❤ ✇❛② t❤❛t ❡✈❡r② ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✶✻✮ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ✭✶✺✮✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❈♦r♦❧❧❛r② ❛❜♦✉t st❛❜✐❧✐t② ✐♥ ❧✐♥❡❛r ❝❛s❡

■♥ t❤❡ ❝♦✉rs❡ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤✐s ❚❤❡♦r❡♠✱ ✇❡ ♦❜t❛✐♥ ❛♥ ✐♠♣♦rt❛♥t ❝♦r♦❧❧❛r② ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ ✭✶✻✮✳ ❈♦r♦❧❧❛r②✳

◮ ■❢ t❤❡ s②st❡♠ ✭✶✺✮ ✐s ❡①♣♦♥❡♥t✐❛❧❧② st❛❜❧❡✱ t❤❛t ✐s ✐♥ t❤❡

❞❡✜♥✐t✐♦♥ ♦❢ ❞✐❝❤♦t♦♠② t❤❡ ♣r♦❥❡❝t♦r P2 ✐s ③❡r♦✱ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✭✶✻✮ ❝♦♥✈❡r❣❡ t♦ ③❡r♦ ❢♦r t → ∞ ❜♦t❤ ✐♥ sq✉❛r❡ ♠❡❛♥ ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✳

◮ ■❢ ✭✶✺✮ ✐s ❡①♣♦♥❡♥t✐❛❧❧② ✉♥st❛❜❧❡✱ t❤❛t ✐s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢

❞✐❝❤♦t♦♠② t❤❡ ♣r♦❥❡❝t♦r P1 ✐s ③❡r♦✱ ❛❧❧ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥s ♦❢ ✭✶✻✮ ❣♦ t♦ ✐♥✜♥✐t② ❢♦r t → ∞ ❜♦t❤ ✐♥ sq✉❛r❡ ♠❡❛♥ ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❊①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠②

◆❡①t ✇❡ ✇✐❧❧ t❛❧❦ ❛❜♦✉t s♣❡❝✐✜❝ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ r❡s✉❧ts✳ ❊❛r❧✐❡r ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠② ❢♦r ❧✐♥❡❛r s②st❡♠ ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s dx = A(t)xdx. ✭✶✼✮ ❉❡❢✳✸✳❲❡ s❛② t❤❛t t❤❡ s②st❡♠ ✭✶✼✮ ✐s ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠♦✉s ♦♥ t❤❡ ❛①✐s ✐❢ t❤❡ s♣❛❝❡ Rn ❝♦✉❧❞ ❜❡ ♣r❡s❡♥t❡❞ ❛s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ s✉❜s♣❛❝❡s R−, R+✿ Rn = R− ⊕ R+ s✳t✳ ❛♥② s♦❧✉t✐♦♥ x(t, x0) ♦❢ t❤❡ s②st❡♠ ✭✶✼✮ ✇✐t❤ x(0, x0) = x0 ∈ R− s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② |x(t, x0)| ≤ K exp −j(t − τ)|x(τ, x0)| ❢♦r t ≥ τ✱ ❛♥❞ ❛♥② s♦❧✉t✐♦♥ x(t, x0) ♦❢ t❤❡ s②st❡♠ ✭✶✼✮ ✇✐t❤ x(0, x0) = x0 ∈ R+ s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② |x(t, x0)| ≥ K1 exp j1(t − τ)|x(τ, x0)| ❢♦r t ≥ τ✱ ✇❤❡r❡ τ ∈ R✱ s♦♠❡ ❝♦♥st❛♥ts K, K1, j, j1 ❛r❡ ✉♥❞❡♣❡♥❞❛❜❧❡ ♦❢ τ ❛♥❞ x0✳ ❚❤❛t ✐s t❤❡ s♦❧✉t✐♦♥ ✇❤✐❝❤ ❤❛s ❛ st❛rt ❢r♦♠ s✉❜s♣❛❝❡ R− ✭ R+✮ ✴st❛❜❧❡ ✭✉♥st❛❜❧❡✮ s✉❜s♣❛❝❡✴ ❞❡❝r❡❛s❡ ✭✐♥❝r❡❛s❡✮ ❡①♣♦♥❡♥t✐❛❧❧②✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❊①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠②

❙✐♠✐❧❛r ♣r♦❜❧❡♠s ❢♦r st♦❝❤❛st✐❝ s②st❡♠s dy = A(t)ydy + B(t)ydW (t) ✭✶✽✮ ✇❡r❡ st✉❞✐❡❞ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❆✳▼✳❙❛♠♦❥❧❡♥❦♦✳ ❉❡❢✳✹✳❲❡ s❛② t❤❛t t❤❡ s②st❡♠ ✭✶✽✮ ✐s ❡①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠♦✉s ✐♥ sq✉❛r❡ ♠❡❛♥ ❢♦r t ≥ 0 ✐❢ t❤❡ s♣❛❝❡ Rn ❝♦✉❧❞ ❜❡ ♣r❡s❡♥t❡❞ ❛s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ s✉❜s♣❛❝❡s R−, R+ s✳t✳ ❛♥② s♦❧✉t✐♦♥ x(t, x0) ♦❢ t❤❡ s②st❡♠ ✭✶✽✮ ✇✐t❤ x(0, x0) = x0 ∈ R− s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② E|x(t, x0)|2 ≤ K exp −j(t − τ)E|x(τ, x0)|2 ✭✶✾✮ ❢♦r t ≥ τ ≥ 0✱ ❛♥❞ ❛♥② s♦❧✉t✐♦♥ x(t, x0) ♦❢ t❤❡ s②st❡♠ ✭✶✽✮ ✇✐t❤ x(0, x0) = x0 ∈ R+ s❛t✐s✜❡s t❤❡ ✐♥❡q✉❛❧✐t② E|x(t, x0)|2 ≥ K1 exp j1(t − τ)E|x(τ, x0)|2 ✭✷✵✮ ❢♦r t ≥ τ ≥ 0✱ ✇❤❡r❡ τ ∈ R✱ s♦♠❡ ❝♦♥st❛♥ts K, K1, j, j1 ❛r❡ ✉♥❞❡♣❡♥❞❛❜❧❡ ♦❢ τ ❛♥❞ x0✳ ❚❤❛t ✐s t❤❡ s❡❝♦♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦♠❡♥ts ♦❢ s♦❧✉t✐♦♥s ❞❡❝r❡❛s❡ ♦r ✐♥❝r❡❛s❡ ❡①♣♦♥❡♥t✐❛❧❧②✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ ❛ s✉❜s♣❛❝❡ ✇❤❡r❡ t❤❡ s♦❧✉t✐♦♥s ❤❛✈❡ ❛ st❛rt✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❊①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠②

◆❡①t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✇✐t❤ t❤❡ ❜❡❤❛✈✐♦r ♦❢ tr❛❥❡❝t♦r✐❡s ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳ ❖❜t❛✐♥❡❞ ❛❜♦✈❡ r❡s✉❧ts ❛❜♦✉t ❛s②♠♣t♦t✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❣✐✈❡ ✉s ❛♥ ♦♣♣♦rt✉♥✐t② t♦ st✉❞② t❤✐s ✐ss✉❡✳ ❚❍▼✳ ▲❡t t❤❡ ♠❛tr✐❝❡s B(t) ❛♥❞ D(t) ❢r♦♠ t❤❡ s②st❡♠ ✭✶✻✮ t❡♥❞ t♦ ③❡r♦ ❛s t → ∞ ❛♥❞ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛r❡ s❛t✐s✜❡❞✳ ❚❤❡♥ ∃t0 > 0 s✳t✳ ❢♦r t ≥ t0 t❤❡ s②st❡♠ ✭✶✻✮ ✐s ❡①♣♦♥❡♥t✐❛❧❧② ❞✐❝❤♦t♦♠♦✉s✳ ▼♦r❡♦✈❡r s♦❧✉t✐♦♥s ✇❤✐❝❤ ❤❛✈❡ ❛ st❛rt ❢r♦♠ R− ✭❢r♦♠ ❉❡❢✳✹✮ t❡♥❞ t♦ ③❡r♦ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ❛s t → ∞✱ ❛♥❞ s♦❧✉t✐♦♥s ✇❤✐❝❤ ❞♦ ♥♦t ❤❛✈❡ ❛ st❛rt ❢r♦♠ R− t❡♥❞ ♦✈❡r ♥♦r♠ t♦ ✐♥✜♥✐t② ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ❛s t → ∞✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❆♣♣❧✐❝❛t✐♦♥ t♦ ❛ st✉❞② ♦❢ t❤❡ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ❢♦r st♦❝❤❛st✐❝ ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ❡q✉❛t✐♦♥s

❲❡ ♥♦✇ ✐❧❧✉str❛t❡ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ t❤❡♦r② ❛❜♦✈❡ t♦ t❤❡ ♣r❛❝t✐❝❛❧ ❡①❛♠♣❧❡✳ ❇❛s❡❞ ♦♥ t❤❡ t❤❡♦r❡♠ ♦♥ ❛s②♠♣t♦t✐❝ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇❡ ❜✉✐❧t t❤❡ ❛♥❛❧♦❣ ♦❢ ❙t✉r♠ ♦s❝✐❧❧❛t✐♦♥ t❤❡♦r② ❢♦r ❧✐♥❡❛r st♦❝❤❛st✐❝ s❡❝♦♥❞ ♦r❞❡r ❡q✉❛t✐♦♥s ♦❢ t❤❡ t②♣❡ ¨ x + (p(t) + q(t) ˙ W (t))x = 0, ✭✷✶✮ ✇❤❡r❡ x ∈ R✱ t ≥ 0❀ p(t), q(t) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❛♥❞ ˙ W (t) ✐s t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛♥❞❛r❞ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❚❤❡ ❡q✉❛t✐♦♥ ✭✷✶✮ ✐s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ♦❢ ✈❛r✐♦✉s r❡❛❧✲❧✐❢❡ ♣r♦❝❡ss❡s ✐♥ ♠❡❝❤❛♥✐❝s✱ ✇❤✐❝❤ ❛r❡ ✐♥✢✉❡♥❝❡❞ ❜② r❛♥❞♦♠ ❢❛❝t♦rs✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♥♣❡rt✉r❜❡❞ ❖❉❊ ¨ x + p(t)x = 0 ✭✷✷✮ ❞❡s❝r✐❜❡s t❤❡ ♠♦t✐♦♥ ♦❢ ❛ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠✱ ✇❤✐❝❤ ✐s ✐♥✢✉❡♥❝❡❞ ❜② t❤❡ ❡❧❛st✐❝ ❢♦r❝❡ ✇✐t❤ ❡❧❛st✐❝✐t② ❝♦❡✣❝✐❡♥t p(t)✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❆♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ❖s❝✐❧❧❛t✐♦♥s ❚❤❡♦r②

❍♦✇❡✈❡r✱ t❤❡ ❝♦❡✣❝✐❡♥t p(t) ✐s ♦♥❧② t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ t❤❡ tr✉❡ ❡❧❛st✐❝✐t② ❝♦❡✣❝✐❡♥t✳ ■ts tr✉❡ ✈❛❧✉❡ ✐s ❛ r❛♥❞♦♠ ♣r♦❝❡ss ✇✐t❤ ❛ s♠❛❧❧ ❝♦rr❡❧❛t✐♦♥ ✐♥t❡r✈❛❧✳ ❚❤❡r❡❢♦r❡ t❤❡ ♠♦❞❡❧ ✭✷✶✮ ✐s ♠♦r❡ ❛❝❝✉r❛t❡✳ ❲❡ ♥♦✇ s❡t ❛s✐❞❡ t❤❡ ♠❡❝❤❛♥✐❝❛❧ ❛s♣❡❝t ❛♥❞ ❢♦❝✉s ♦♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✳ ❚❤❡ ❡q✉❛t✐♦♥ ✭✷✶✮ ✐s ♥♦t r✐❣♦r♦✉s ❛s ✐s ❜❡❝❛✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❲✐❡♥❡r ♣r♦❝❡ss ❞♦❡s ♥♦t ❡①✐st✳ ❚❤❡r❡❢♦r❡ ✇❡ ✇✐❧❧ ✉♥❞❡rst❛♥❞ ✭✷✶✮ ❛s ❛ s②st❡♠ ♦❢ st♦❝❤❛st✐❝ ■t♦ ❡q✉❛t✐♦♥s

• dx1 = x2dt

dx2 = −p(t)x1dt − q(t)x1dW (t). ✭✷✸✮ ✇❤✐❝❤ ✐s ❛❜s♦❧✉t❡❧② r✐❣♦r♦✉s ♥♦✇✳ ■♥ t❤✐s ♥♦t❛t✐♦♥✱ x(t) = x1(t)✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❈♦♥❝❡♣t ♦❢ ♦s❝✐❧❧❛t✐♦♥ ❢♦r r❛♥❞♦♠ ♣r♦❝❡ss❡s

❲❡ ♥♦✇ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ t❤❡ ✜rst ③❡r♦ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ ✭✷✸✮✱ t❤❡♥ ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥❝❡♣ts ♦❢ ♦s❝✐❧❧❛t✐♥❣ ✴ ♥♦♥✲♦s❝✐❧❧❛t✐♥❣ s♦❧✉t✐♦♥s ♦♥ s❡♠✐✲❛①✐s✳ ❙✐♥❝❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✭✷✸✮ ❛r❡ r❛♥❞♦♠ ♣r♦❝❡ss❡s ✇✐t❤ ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s✱ ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❝♦♥❝❡♣t ♦❢ ③❡r♦ r❡q✉✐r❡s ❛ s✉❜t❧❡ ❝♦♥str✉❝t✐♦♥✱ ✉♥❧✐❦❡ ✐♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡✳ ❈❧❡❛r❧② t❤❡ ③❡r♦s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛r❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♥❡❡❞ t♦ ✐♥tr♦❞✉❝❡ ❛ ③❡r♦ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐t ✐s ❛ ▼❛r❦♦✈✬s ♠♦♠❡♥t r❡❧❛t✐✈❡❧② t♦ t❤❡ ❢❛♠✐❧② ♦❢ σ✲ ❛❧❣❡❜r❛s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❩❡r♦s ♦❢ t❤❡ s♦❧✉t✐♦♥

❉❡✜♥❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ τ1 ❛s τ1 := inf{t > t0|x1(t) = 0}, ✐❢ t❤❡ s❡t ♦✈❡r ✇❤✐❝❤ ✐♥✜♠✉♠ ✐s t❛❦❡♥ ✐♥ ♥♦♥✲❡♠♣t② ❛♥❞ τ1 = +∞ ♦t❤❡r✇✐s❡✳ ❉❡❢✳ ❚❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ τ1 ✐s ❝❛❧❧❡❞ t❤❡ ✜rst ③❡r♦ ♦❢ x(t) ♦♥ t❤❡ ✐♥t❡r✈❛❧ t > t0 ✐❢ τ1 < +∞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳ ❙✐♥❝❡ ✭✷✸✮ ✐s ❧✐♥❡❛r ❛♥❞ x1(t) ✐s s♠♦♦t❤✱ ✐t ✐s ♥♦t ❞✐✣❝✉❧t t♦ s❤♦✇ t❤❛t ✐♥ s♦♠❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ ✜rst ③❡r♦ τ1 t❤❡ ❝♦♠♣♦♥❡♥t x1(t) ✐s ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦✳ ❚❤✉s ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ τ2 ❛s τ2 := inf{t > τ1 : x1(t) = 0}, ✐❢ t❤❡ s❡t ♦✈❡r ✇❤✐❝❤ ✐♥✜♠✉♠ ✐s t❛❦❡♥ ✐♥ ♥♦♥✲❡♠♣t② ❛♥❞ τ2 = +∞ ♦t❤❡r✇✐s❡✳ ■❢ τ2 < +∞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✱ ✐t ✐s ❝❛❧❧❡❞ t❤❡ s❡❝♦♥❞ ③❡r♦ ♦❢ x(t) ♦♥ t❤❡ ✐♥t❡r✈❛❧ t > t0✳ ❇② ✐♥❞✉❝t✐♦♥ ✇❡ ❝❛♥ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡ ♦❢ ③❡r♦s {τn} ♦❢ t❤❡ s♦❧✉t✐♦♥ x(t) ♦♥ t❤❡ ✐♥t❡r✈❛❧ t > t0✳ ■❢ t0 = 0✱ t❤❡♥ ✇❡ ❤❛✈❡ ③❡r♦s ♦♥ t❤❡ s❡♠✐✲❛①✐s t ≥ 0✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❖s❝✐❧❧❛t✐♥❣ s♦❧✉t✐♦♥s

❚❤✐s s❡q✉❡♥❝❡ ♦❢ ③❡r♦s ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ▼❛r❦♦✈✬s ♠♦♠❡♥ts✱ ✇❤✐❝❤ ♦❢t❡♥ ❡♥❛❜❧❡s ✉s t♦ ✇♦r❦ ✇✐t❤ t❤❡♠ ❛s ✇✐t❤ ❞❡t❡r♠✐♥✐st✐❝ ✭❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ♣❡r❢♦r♠ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ♦♥ t❤❡ ✐♥t❡r✈❛❧ (τn−1, τn).✮ ❉❡❢✳ ❆ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ x(t) ♦❢ ✭✷✶✮ ✐s ❝❛❧❧❡❞ ♦s❝✐❧❧❛t✐♥❣ ♦♥ t❤❡ s❡♠✐✲❛①✐s t > 0 ✐❢ ✐t ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② ③❡r♦s ♦♥ t❤✐s ✐♥t❡r✈❛❧✳ ▲❡t I = (t0, t1) ❜❡ ❛ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✱ x(t) ❜❡ ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ✭✷✶✮ ❢♦r t ≥ t0✱ τ1 ✐s t❤❡ ✜rst ③❡r♦ ♦♥ t❤❡ ✐♥t❡r✈❛❧ t > t0 ✭❛ss✉♠✐♥❣ t❤❛t ✐t ❡①✐sts✮✳ ❉❡❢✳ ❲❡ s❛② t❤❛t t❤❡ s♦❧✉t✐♦♥ x(t) ❤❛s t❤❡ ✜rst ③❡r♦ ♦♥ t❤❡ ✐♥t❡r✈❛❧ I ✐❢ t0 < τ1 < t1 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳ ❉❡❢✳❚❤❡ s♦❧✉t✐♦♥ x(t) ✐s ❝❛❧❧❡❞ ♦s❝✐❧❧❛t✐♥❣ ♦♥ I ✐❢ ✐t ❤❛s ❛t ❧❡❛st t✇♦ ③❡r♦s τ1, τ2 ♦♥ t❤❡ ✐♥t❡r✈❛❧ t > t0 ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✱ τ1 ∈ I ❛♥❞ τ2 ∈ I✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❙t✉r♠ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ❢♦r ❙❉❊

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥❛❧♦❣ ♦❢ t❤❡ t❤❡♦r❡♠ ❢r♦♠ t❤❡ ❝❧❛ss✐❝ ❙t✉r♠ ♦s❝✐❧❧❛t✐♦♥ t❤❡♦r② ❤♦❧❞s✿ ▲❡♠♠❛✳ ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ❛♥② ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ♦❢ ✭✷✶✮ ❤❛s ❛t ♠♦st ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ③❡r♦s ♦♥ ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧✳ ❲❡ ♥♦✇ ❣✐✈❡ t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠✳ ❆❧♦♥❣ ✇✐t❤ ✭✷✶✮ ❝♦♥s✐❞❡r ❛ s✐♠✐❧❛r ❡q✉❛t✐♦♥ ¨ y + (˜ p(t) + q(t) ˙ W (t))y = 0. ✭✷✹✮ ❋r♦♠ ♥♦✇ ♦♥✱ I = (t0, t1) ✐s ❛ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳ ❚❍▼✳✭❈♦♠♣❛r✐s♦♥✮ ❆ss✉♠❡ ˜ p(t) ≥ p(t) ♦♥ I✳ ❚❤❡♥✱ ✐❢ τ1, τ2 ❛r❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s ♦❢ ❛ s♦❧✉t✐♦♥ ♦❢ ✭✷✶✮ ♦♥ I✱ t❤❡♥ ❛♥② s♦❧✉t✐♦♥ y(t) ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✷✸✮ ❤❛s ❛t ❧❡❛st ♦♥❡ ③❡r♦ τ ♦♥ t❤❡ ✐♥t❡r✈❛❧ I ❛♥❞✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✱ τ1 ≤ τ ≤ τ2✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❙t✉r♠ ❖s❝✐❧❧❛t✐♦♥ ❚❤❡♦r② ❢♦r ❙❉❊

❆s ❛ ❝♦r♦❧❧❛r② ♦❢ t❤❡ ❈♦♠♣❛r✐s♦♥ ❚❤❡♦r❡♠✱ ✇❡ ❣❡t ❚❍▼✳ ▲❡t τ1, τ2 ❜❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s ♦❢ ❛ s♦❧✉t✐♦♥ x(t) ♦❢ ✭✷✶✮ ♦♥ I✳ ❚❤❡♥ ❛♥② ♦t❤❡r s♦❧✉t✐♦♥ ˜ x(t) ♦❢ ✭✷✶✮✱ ✇❤✐❝❤ ✐s ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ✇✐t❤ x(t)✱ ❤❛s ❡①❛❝t❧② ♦♥❡ ③❡r♦ ♦♥ t❤❡ ♦♣❡♥ st♦❝❤❛st✐❝ ✐♥t❡r✈❛❧ (τ1, τ2)✳ ❚❍▼✳✭◆♦♥✲♦s❝✐❧❧❛t✐♦♥✮ ■❢ p(t) ≤ 0 ❢♦r t ≥ t0 ❛♥❞

∞

• t0

q2(t)dt < ∞, t❤❡♥ ❛❧❧ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥s ♦❢ ✭✷✶✮ ❛r❡ ♥♦t ♦s❝✐❧❧❛t✐♥❣ ♦♥ t❤❡ s❡♠✐✲❛①✐s✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❆✉①✐❧✐❛r② ❡q✉❛t✐♦♥

❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ ¨ x + (a2 + q(t) ˙ W (t))x = 0, ✭✷✺✮ ✇❤✐❝❤ ♣❧❛②s t❤❡ r♦❧❡ ♦❢ ❛ st❛♥❞❛r❞ ❡q✉❛t✐♦♥ ¨ x + a2x = 0 ✐♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♦s❝✐❧❧❛t✐♦♥ t❤❡♦r②✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s♦❧✈❡ t❤❡ ❧❛st ❡q✉❛t✐♦♥✱ ✇❡ ❦♥♦✇ ❡✈❡r②t❤✐♥❣ ❛❜♦✉t ✐ts s♦❧✉t✐♦♥s✱ ❡✳❣✳ ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❡①❛❝t ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s ✐s π

a .

❯♥❢♦rt✉♥❛t❡❧②✱ ✐ts st♦❝❤❛st✐❝ ❛♥❛❧♦❣ ✭✷✺✮ ❝❛♥♥♦t ❜❡ s♦❧✈❡❞ ❡①♣❧✐❝✐t❧②✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦❜t❛✐♥ s♦♠❡ ❛s②♠♣t♦t✐❝ r❡s✉❧ts ❢♦r ✐t✳ ❚❍▼✳ ▲❡t

∞

• q2(t)dt < ∞. ❚❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s x(t) ♦❢ t❤❡

❡q✉❛t✐♦♥ ✭✷✹✮ ❛r❡ ♦s❝✐❧❧❛t✐♥❣ ♦♥ t❤❡ s❡♠✐✲❛①✐s ❛♥❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ ξn = τn+1 − τn → π a , n → ∞, ✇❤❡r❡ τn ❛♥❞ τn+1 ❛r❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❖s❝✐❧❧❛t✐♦♥ ❛♥❞ ❛s②♠♣t♦t✐❝s ♦❢ t❤❡ ③❡r♦s ❢♦r t → ∞.

❉❡♥♦t❡ m = inf

t≥0 p(t) ❛♥❞ M = sup t≥0

p(t). ❚❍▼✳ ▲❡t p(t) ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❛♥❞ s✉❝❤ t❤❛t ♦♥ ❛♥② s✉❜✐♥t❡r✈❛❧ ♦❢ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s t❤❡ ❝♦♥❞✐t✐♦♥ 0 < m < p(t) < M < ∞ ❤♦❧❞s✳ ■❢✱ ❛❞❞✐t✐♦♥❛❧❧②✱

∞

• q2(t)dt < ∞,

t❤❡♥ ❛❧❧ t❤❡ s♦❧✉t✐♦♥s ♦❢ ✭✷✶✮ ❛r❡ ♦s❝✐❧❧❛t✐♥❣ ♦♥ t❤❡ s❡♠✐❛①✐s [0, ∞)✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② ε > 0 s✳t✳ ε <

2 √ M t❤❡r❡ ❡①✐sts ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡

T(ω) > 0✱ s♦ t❤❛t ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ x(t) ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✷✶✮ ❢♦r τn ≥ T(ω) ✇❡ ❝❛♥ ❡st✐♠❛t❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✿ π √ M − ε ≤ τn+1 − τn ≤ π √m + ε. ✭✷✻✮

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❊①❛♠♣❧❡s

❊①❛♠♣❧❡ ✶✳ ❇❡ss❡❧ ❡q✉❛t✐♦♥ ♣❡rt✉r❜❡❞ ✇✐t❤ ✇❤✐t❡ ♥♦✐s❡✿ ¨ x(t) + (1 − ν2 − 1

4

t2 + q(t) ˙ w(t))x(t) = 0, ✭✷✼✮ ✇❤❡r❡

∞

• q2(t)dt < ∞. ❚❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t ✇✐t❤

♣r♦❜❛❜✐❧✐t② ✶ ✇❡ ❝❛♥ ❡st✐♠❛t❡ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ③❡r♦s ♦❢ ✭✷✼✮✿ lim

n→∞(τn+1 − τn) = π. ❚❤✐s r❡s✉❧t ✐s t❤❡ s❛♠❡ ❛s ❢♦r

t❤❡ ♦r❞✐♥❛r② ❡q✉❛t✐♦♥s✳ ❊①❛♠♣❧❡ ✷✳ ❆✐r② ❡q✉❛t✐♦♥ ♣❡rt✉r❜❡❞ ✇✐t❤ ✇❤✐t❡ ♥♦✐s❡✿ ¨ x(t) + (t + q(t) ˙ w(t))x(t) = 0 ✭✷✽✮ ✇❤❡r❡✱ ❛s ❜❡❢♦r❡✱

∞

• q2(t)dt < ∞. ❋r♦♠ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱

lim

n→∞(τn+1 − τn) = 0✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

❲❡ ❝♦♥s✐❞❡r t❤❡ st♦❝❤❛st✐❝ s②st❡♠ dx = a(t, x)dt +

k

• r=1

br(t, x)dWr(t), ✭✷✾✮ ✇❤❡r❡ t ≥ 0✱ x ∈ Rn✱ a✱ br✱ r = 1, k ❛r❡ ✈❡❝t♦rs ✐♥ Rn✱ W1, ..., Wr ❛r❡ ✉♥❞❡♣❡♥❞❛❜❧❡✱ s❝❛❧❛r st❛♥❞❛r❞ ❲✐❡♥❡r ♣r♦❝❡ss❡s✳ ❲❡ ❛ss✉♠❡ t❤❛t a✱ br ❛r❡ ♥♦♥ r❛♥❞♦♠ ❛♥❞ s✳t✳ t❤❡ ❡q✉❛t✐♦♥ ✭✷✾✮ ❤❛s ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡s x(t0) = x0 ∈ Rn ❢♦r t ≥ t0✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

❲❡ ❞❡♥♦t❡ S ✐s s♦♠❡ ❇♦rr❡❧ s❡t ✐♥ {t ≥ 0} × Rn✳ ❆❧s♦ ❧❡t St ❜❡ ❛ s❡t ✐♥ Rn s✳t✳ St = {x ∈ Rn : (t, x) ∈ S} ❛♥❞ ❧❡t St = ∅ ❢♦r ❛❧❧ t ≥ 0✳ ❉❡❢✳✺✳❲❡ s❛② t❤❛t ❛ s❡t S ✐s ♣♦s✐t✐✈❡❧② ✐♥✈❛r✐❛♥t ❢♦r t❤❡ s②st❡♠ ✭✷✾✮ ✐❢ P{(t, x(t, t0, x0)) ∈ S, ∀t ≥ t0} = 1 ✭✸✵✮ ❢♦r (t0, xo(ω)) ∈ S ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✱ ✇❤❡r❡ x(t, t0, x0) ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✷✾✮ s✳t✳ x(t0, t0, x0) = x0 ❢♦r t0 ≥ 0

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ ♦✈❡r ❛♥ ✐♥✈❛r✐❛♥t s❡t t❤❡ ♦r✐❣✐♥❛❧ st♦❝❤❛st✐❝ s②st❡♠ ❝♦✉❧❞ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥ ❞❡t❡r♠✐♥✐st✐❝ ♦♥❡✳ ❚❤❡♥ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ st♦❝❤❛st✐❝ s②st❡♠ st❛❜✐❧✐t② ❝♦✉❧❞ ❜❡ r❡❞✉❝❡❞ t♦ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠ st❛❜✐❧✐t②✳ ❲❡ ❝♦♥s✐❞❡r ❢♦r x ∈ Rn✱ y ∈ Rm✱ t ≥ 0 t❤❡ ■t♦ t②♣❡ st♦❝❤❛st✐❝ s②st❡♠✿ dx = X(x, y)dt dy = A(t)ydy + σ(t, x, y)dW (t) ✭✸✶✮ ✇❤❡r❡ σ(t, x, y) ✐s m × r✲❞✐♠❡♥s✐♦♥❛❧ ♠❛tr✐①✱ W (t) ✐s r✲❞✐♠❡♥s✐♦♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ✇✐t❤ ✉♥❞❡♣❡♥❞❛❜❧❡ ❝♦♠♣♦♥❡♥ts✳ ❲❡ ❜❡❧✐❡✈❡ X(x, y)✱ σ(t, x, y) ❛r❡ ▲✐♣s❤✐t③ ♦✈❡r x, y ❢♦r t ≥ 0✱ x ∈ Rn✱ y ∈ Rm ✇✐t❤ ❝♦♥st❛♥ts L1 ❛♥❞ L2 ❝♦rr❡s♣♦♥❞✐♥❣❧②✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

▲❡t t❤❡ ♠❛tr✐❝✐❛♥t Φ(t, τ) ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠ dy dx = A(t)y ♣❡r♠✐ts ❛♥ ❡st✐♠❛t✐♦♥ Φ(t, τ) ≤ Re−ρ(t−τ), ✭✸✷✮ ✇❤❡r❡ R, ρ ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✱ ✉♥❞❡♣❡♥❞❛❜❧❡ ♦❢ t ❛♥❞ τ✳ ❆❧s♦ ✇❡ s✉♣♣♦s❡ t❤❛t σ(t, x, 0) ≡ 0 ✭✸✸✮ ❚❤❡ ❧❛st ❡q✉✐✈❛❧❡♥❝❡ ♠❡❛♥s t❤❛t t❤❡ s②st❡♠ ✭✸✶✮ ❤❛s t❤❡ ✐♥✈❛r✐❛♥t s❡t {y = 0}✱ ♦✈❡r ✇❤✐❝❤ t❤✐s s②st❡♠♠ ❝♦✉❧❞ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ ❞❡t❡r♠✐♥✐st✐❝ ♦♥❡ dx dt = X(x, 0) ✭✸✹✮ ❆s ✇❡❧❧ ❧❡t X(0, 0) = 0✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

❍❡♥❝❡ t❤❡ st♦❝❤❛st✐❝ s②st❡♠ ✭✸✶✮ ❤❛s ③❡r♦ s♦❧✉t✐♦♥ x = 0, y = 0✱ s♦ ✇❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤✐s s♦❧✉t✐♦♥✳ ❋♦r ❝♦♥✈❡♥✐❡♥❝❡ ✇❡ ❞❡✜♥❡ z = (x, y)✱ z0 = (x0, y0)✳ ❚❤❡ st❛❜✐❧✐t② ✇❡ ✉♥❞❡rst❛♥❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ sq✉❛r❡ ♠❡❛♥✱ t❤❛t ✐s ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ♥❡①t ❞❡✜♥✐t✐♦♥✳ ❉❡❢✳✻✳❲❡ s❛② t❤❛t ③❡r♦ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✸✶✮ ✐s sq✉❛r❡ ♠❡❛♥ st❛❜❧❡ ✐❢ ❢♦r ❛❧❧ ε > 0 ❡①✐sts δ s✳t✳ E|z(t, t0, z0)|2 < ε, ❢♦r t ≥ 0 ❛♥❞ E|z0|2 < δ.

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

■♥✈❛r✐❛♥t s❡ts ♦❢ st♦❝❤❛st✐❝ s②st❡♠s

❚❍▼✳ ▲❡t t❤❡ ③❡r♦ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✸✹✮ ❜❡ ✉♥✐❢♦r♠❧② ❛s②♠♣t♦t✐❝❛❧❧② st❛❜❧❡ ❛♥❞ L2 < √2ρ R . ❚❤❡♥ t❤❡ ③❡r♦ s♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✸✶✮ ✐s sq✉❛r❡ ♠❡❛♥ st❛❜❧❡ ✉♥✐❢♦r♠❧② ♦♥ t0 ≥ 0✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡♠♣❤❛s✐③❡ t❤❛t t❤✐s ♣❛♣❡r ✐♥tr♦❞✉❝❡❞ ♦♥❧② ♦♥❡ ♦❢ t❤❡ r❡s✉❧ts ♦❢ s✉❝❤ t②♣❡✱ ✇❤❡♥ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ st♦❝❤❛st✐❝ s②st❡♠ st❛❜✐❧✐t② ✇❛s r❡❞✉❝❡❞ t♦ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❞❡t❡r♠✐♥✐st✐❝ s②st❡♠ st❛❜✐❧✐t②✳ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❝♦✉❧❞ ❜❡ ♦❜t❛✐♥❡❞ ❢♦r ♦t❤❡r t②♣❡ s②st❡♠s✳ ❙✐♠✐❧❛r ♠❡t❤♦❞s ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ st✉❞② t❤❡ st❛❜✐❧✐t② ♦❢ ♠♦r❡ ❝♦♠♣❧❡① ✐♥✈❛r✐❛♥t s❡ts t❤❛♥ ♣♦✐♥t ♦♥❡s✳ ■♥ t❤✐s ❝❛s❡ t❤❡r❡ ✐s ♥❡❡❞ t♦ ✉s❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥s✳

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦✦✦

❆♥♥♦✉♥❝❡♠❡♥t✿ ❝♦♥❢❡r❡♥❝❡ ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❛♥❞ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✐♥ ❤♦♥♦r ♦❢ ✼✺t❤ ❛♥♥✐✈❡rs❛r② ♦❢ ❆✳▼✳ ❙❛♠♦✐❧❡♥❦♦ ❏✉♥❡ ✷✸✲✸✵✱ ✷✵✶✷✱ ❙❡✈❛st♦♣♦❧✱ ❯❦r❛✐♥❡ ▼♦r❡ ✐♥❢♦r♠❛t✐♦♥✿ ✇✇✇✳✐♠❛t❤✳♦r❣✳✉❛

❆✳ ❙t❛♥③❤②ts❦②✐ ❆ st✉❞② ♦❢ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥s ❜② r❡❞✉❝✐♥❣ t❤❡♠ t♦ ❖❉❊