VMSTA Modern Stochastics: Theory and Applications 2351-60542351-60462351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA213 10.15559/22-VMSTA213 Research Article Asymptotic properties of the parabolic equation driven by stochastic measure https://orcid.org/0000-0002-6954-9806 ManikinBorisbmanikin@gmail.com Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 2022 69202294483498204202220820222382022 © 2022 The Author(s). Published by VTeX2022 Open access article under the CC BY license.

A stochastic parabolic equation on [ 0 , T ] × R driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as t → ∞ is studied.

 Stochastic measure mild solution stochastic parabolic equation asymptotic behavior 60G57 Introduction

In this paper we consider the stochastic parabolic equation L u ( t , x ) d t + f ( t , x , u ( t , x ) ) d t + σ ( t , x ) d μ ( x ) = 0 , u ( 0 , x ) = u 0 ( x ) , where ( t , x ) ∈ [ 0 , T ] × R, μ is a general stochastic measure on the Borel σ-algebra on R (see Section 2), f, σ are measurable functions, L is the operator of the kind L u ( t , x ) = a ( t ) ∂ 2 u ( t , x ) ∂ 2 x + b ( t ) ∂ u ( t , x ) ∂ x + c ( t ) u ( t , x ) − ∂ u ( t , x ) ∂ t . Here a, b, c are defined on [ 0 , T ]. We prove that under certain conditions on a, b, c, f, σ the solution of (1), considered in the mild sense, tends to 0 a.s. uniformly on x. Note that regularity of the solution was proved in [2], the solution’s convergence in the case of integrator’s convergence was proved in [18] and the averaging principle for such an equation was established in [12].

The asymptotic behavior of the moments of solutions of a stochastic differential system driven by a Brownian motion was considered in [5]. The problem of the convergence of the solution of a nonautonomous logistic differential equation to zero as time coordinate goes to infinity, with disturbance of coefficients by white noise, centered and noncentered Poisson noises, was studied in [4]. Asymptotics of the solution of the stochastic heat equation with white noise, as time variable goes to infinity for the fixed spatial coordinate, was studied in [10] while asymptotic properties of the solution of the stochastic wave equation driven by a Lévy process were given in [7]. Behavior of solutions of different equations with a general stochastic measure when spatial coordinate goes to infinity was considered in [3] and [1]. In comparison to [15], where asymptotics of the heat equation driven by a general stochastic measure when time coordinate tends to infinity was considered, we study a more general parabolic equation.

The paper is organized as follows. Section 2 contains some general facts about stochastic measures and integrals with respect to them. In Section 3 we prove some technical facts and formulate the main result, which is proved in Section 4 jointly with the auxillary lemma.

 Preliminaries

Let ( Ω , F , P ) be a complete probability space and B be a Borel σ-algebra on R. Denote by L 0 = L 0 ( Ω , F , P ) the set of all real-valued random variables defined on ( Ω , F , P ). Convergence in L 0 means the convergence in probability.

A σ-additive mapping μ : B → L 0 is called stochastic measure (SM).

In other words, μ is a vector measure with values in L 0 . We do not assume any martingale properties or moment existence for SM.

Consider some examples of SMs. If M t is a square integrable martingal then μ ( A ) = ∫ 0 T 1 A ( t ) d M t is an SM. An α-stable random measure on B for α ∈ ( 0 , 1 ) ∪ ( 1 , 2 ], as it is defined in [19, Sections 3.2–3.3], is an SM in sense of Definition 1. For a fractional Brownian motion W t H with Hurst index  H > 1 / 21/2$]]> and a bounded measurable function f : [ 0 , T ] → R we can define an SM μ ( A ) = ∫ 0 T f ( t ) 1 A ( t ) d W t H , see [13, Theorem 1.1]. Ref. [17] contains some other examples.

The integral ∫ A g d μ, where g : R → R is a deterministic measurable function, A ∈ B and μ is an SM, is defined and its basic properties are given in [11, Chapter 7]. In that paper the integral with respect to general stochastic measure was constructed and studied for μ defined on an arbitrary σ-algebra, but here we consider SM on Borel subsets of R. Note that every bounded measurable g is integrable with respect to (w.r.t.) any μ.

In the sequel, μ denotes an SM, C and C ( ω ) denote positive constant and positive random constant, respectively, whose exact values are not important ( C < ∞, C ( ω ) < ∞ a.s.).

We use the following statement.

(Lemma 3.1 in [<xref ref-type="bibr" rid="j_vmsta213_ref_014">14</xref>]).

Let ϕ l : R → R, l ≥ 1, be measurable functions such that ϕ ˜ ( x ) = ∑ l = 1 ∞ | ϕ l ( x ) | is integrable w.r.t. μ on R. Then ∑ l = 1 ∞ ( ∫ R ϕ l d μ ) 2 < ∞ a.s.

We consider the Besov spaces B 22 α ( [ c , d ] ), 0 < α < 1, with a standard norm ‖ g ‖ B 22 α ( [ c , d ] ) = ‖ g ‖ L 2 ( [ c , d ] ) + ( ∫ 0 d − c ( w 2 ( g , r ) ) 2 r − 2 α − 1 d r ) 1 / 2 , where w 2 ( g , r ) = sup 0 ≤ h ≤ r ( ∫ c d − h | g ( y + h ) − g ( y ) | 2 d y ) 1 / 2 .

For any j ∈ Z and all n ≥ 0, put d k n ( j ) = j + k 2 − n , 0 ≤ k ≤ 2 n , Δ k n ( j ) = ( d ( k − 1 ) n ( j ) , d k n ( j ) ] , 1 ≤ k ≤ 2 n .

The following lemma is a key tool for estimates of the stochastic integral.

(Lemma 3 in [<xref ref-type="bibr" rid="j_vmsta213_ref_016">16</xref>]).

Let Z be an arbitrary set, and the function q ( z , s ) : Z × [ j , j + 1 ] → R be such that all paths q ( z , · ) are continuous on [ j , j + 1 ]. Denote q n ( z , s ) = ∑ 1 ≤ k ≤ 2 n q ( z , d ( k − 1 ) n ( j ) ) 1 Δ k n ( j ) ( s ) . Then the random function η ( z ) = ∫ ( j , j + 1 ] q ( z , s ) d μ ( s ) , z ∈ Z , has a version η ˜ ( z ) = ∫ ( j , j + 1 ] q 0 ( z , s ) d μ ( s ) + ∑ n ≥ 1 ( ∫ ( j , j + 1 ] q n ( z , s ) d μ ( s ) − ∫ ( j , j + 1 ] q n − 1 ( z , s ) d μ ( s ) ) such that for all β > 00$]]>, ω ∈ Ω, z ∈ Z | η ˜ ( z ) | ≤ | q ( z , j ) μ ( ( j , j + 1 ] ) | + { ∑ n ≥ 1 2 n β ∑ 1 ≤ k ≤ 2 n | q ( z , d k n ( j ) ) − q ( z , d ( k − 1 ) n ( j ) ) | 2 } 1 / 2 × { ∑ n ≥ 1 2 − n β ∑ 1 ≤ k ≤ 2 n | μ ( Δ k n ( j ) ) | 2 } 1 / 2 .

From Theorem 1.1 [9] it follows that, for α = ( β + 1 ) / 2, { ∑ n ≥ 1 2 n β ∑ 1 ≤ k ≤ 2 n | q ( z , d k n ( j ) ) − q ( z , d ( k − 1 ) n ( j ) ) | 2 } 1 / 2 ≤ C ‖ q ( z , · ) ‖ B 22 α ( [ j , j + 1 ] ) .

Lemma 1 implies that for each β > 00$]]>, j ∈ Z ∑ n ≥ 1 2 − n β ∑ 1 ≤ k ≤ 2 n | μ ( Δ k n ( j ) ) | 2 < + ∞ a . s .

Formulation of the problem and the main result

We refer to the mild solution to (1), i.e. the measurable random function u ( t , x ) = u ( t , x , ω ) : [ 0 , T ] × R × Ω → R that satisfies u ( t , x ) = ∫ R p ( t , x ; 0 , y ) u 0 ( y ) d y + ∫ 0 t d s ∫ R p ( t , x ; s , y ) f ( s , y , u ( s , y ) ) d y + ∫ R d μ ( y ) ∫ 0 t p ( t , x ; s , y ) σ ( s , y ) d s , for each ( t , x ) ∈ [ 0 , + ∞ ) × R a.s. The properties of such solutions are considered in [2]. For example, solution of (7) exists, is unique and can be built as u ( t , x ) = lim n → ∞ u ( n ) ( t , x ) , where u ( 0 ) ( t , x ) = 0 and u ( n ) ( t , x ) = ∫ R p ( t , x ; 0 , y ) u 0 ( y ) d y + ∫ 0 t d s ∫ R p ( t , x ; s , y ) f ( s , y , u ( n − 1 ) ( s , y ) ) d y + ∫ R d μ ( y ) ∫ 0 t p ( t , x ; s , y ) σ ( s , y ) d s . The analogous iteration process for the stochastic heat equation is considered in more detail in [14].

Let the coefficients of operator (2) satisfy the following assumptions.

Functions a ( t ), b ( t ), c ( t ) are continuous and bounded on [ 0 , + ∞ ), and | a ( t 1 ) − a ( t 2 ) | ≤ L t 1 − t 2 λ , a ( t ) ≥ δ , where t , t 1 , t 2 ∈ [ 0 , + ∞ ), L, λ, δ are positive constants.

There exists a constant c 0 > 00$]]> such that c ( t ) ≤ − c 0 ∀ t ≥ 0.

Assumption 1 implies that p ( t , x ; s , y ) = p ( t , x − y ; s , 0 ) for each t , s ∈ [ 0 , + ∞ ), x , y ∈ R. We consider u 0 , f, σ in (7) under the following conditions.

u 0 : R × Ω → R is measurable and for all y , y 1 , y 2 ∈ R | u 0 ( y , ω ) | ≤ C ( ω ) , | u 0 ( y 1 , ω ) − u 0 ( y 2 , ω ) | ≤ L u 0 ( ω ) | y 1 − y 2 | β ( u 0 ) , where C ( ω ), L u 0 ( ω ) are random constants, β ( u 0 ) ≥ 1 / 2.

f : R + × R × R → R is continuous, bounded, and | f ( s , y 1 , v 1 ) − f ( s , y 2 , v 2 ) | ≤ L f ( | y 1 − y 2 | + | v 1 − v 2 | ) for some constant L f and all s ∈ R + , y 1 , y 2 , v 1 , v 2 ∈ R.

σ : R + × R → R is measurable and | σ ( s , y ) | ≤ C σ ( s ) , | σ ( s , y 1 ) − σ ( s , y 2 ) | ≤ L σ ( s ) | y 1 − y 2 | β ( σ ) , for some constant 1 / 2 < β ( σ ) < 1, all s ∈ R + , y 1 , y 2 ∈ R and bounded functions C σ , L σ : R + → R such that C σ ( s ) → 0 , s → ∞ ; L σ ( s ) → 0 , s → ∞ .

To proceed further, we need some statements about L and p. The following lemma [6, Theorem 10 §1] is formulated for our specific L.

Assume that the function v ( t , x ) : [ 0 , T ] × R → R is bounded, Assumptions 1, 2 hold, | v ( 0 , x ) | ≤ M 1 , | L v ( t , x ) | ≤ M 2 . Then | v ( t , x ) | ≤ e − c 0 t ( M 1 + M 2 t ) .

There exist positive constants ν, η, C such that for each x , y ∈ R, t > s > 0s>0$]]> the following estimates hold: | p ( t , x ; s , y ) | ≤ C ( t − s ) − 1 / 2 e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ + 1 ( C ˜ ( t − s ) λ ) , | ∂ p ( t , x ; s , y ) ∂ x | ≤ C ( t − s ) − 1 e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) , | ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ | ≤ C ( ϕ ) | x − x ′ | ϕ ( t − s ) − 3 / 2 × × max { e − ν ( x − y ) 2 t − s , e − ν ( x ′ − y ) 2 t − s } e η ( t − s ) E λ , λ ( C ˜ ( t − s ) λ ) , where E α , β ( z ) = ∑ k = 0 ∞ z k Γ ( α k + β ) is the Mittag-Leffler function, ϕ < 1.

We represent p as p ( t , x ; s , y ) = W ( t , x ; s , y ) + ∫ s t d θ ∫ R W ( t , x ; θ , ζ ) Φ ( θ , ζ ; s , y ) d ζ (see, for example, [6, (4.18)]). Here W ( t , x ; s , y ) = 1 4 π ( t − s ) a ( s ) e − ( x − y ) 2 4 ( t − s ) a ( s ) , the function Φ ( t , x ; s , y ) is a solution of the integral equation Φ ( t , x ; s , y ) = L W ( t , x ; s , y ) + ∫ s t d θ ∫ R L W ( t , x ; θ , ζ ) Φ ( θ , ζ ; s , y ) d ζ . It is easy to calculate that ∂ W ( t , x ; s , y ) ∂ x = 1 4 π ( t − s ) a ( s ) e − ( x − y ) 2 4 ( t − s ) a ( s ) y − x 2 ( t − s ) a ( s ) , ∂ 2 W ( t , x ; s , y ) ∂ x 2 = 1 4 π ( t − s ) a ( s ) e − ( x − y ) 2 4 ( t − s ) a ( s ) ( ( x − y ) 2 4 ( t − s ) 2 a 2 ( s ) − 1 2 ( t − s ) a ( s ) ) , ∂ W ( t , x ; s , y ) ∂ t = 1 4 π ( t − s ) a ( s ) e − ( x − y ) 2 4 ( t − s ) a ( s ) ( ( x − y ) 2 4 ( t − s ) 2 a ( s ) − 1 2 ( t − s ) ) . Using Assumption 1 and boundedness of the function x α e − x on [ 0 , + ∞ ) for arbitrary α > 00$]]>, we easily obtain that a ( s ) ∂ 2 W ∂ x 2 − ∂ W ∂ t = 0 , | W ( t , x ; s , y ) | ≤ C e − ν ( x − y ) 2 t − s ( t − s ) − 1 / 2 , | ∂ W ( t , x ; s , y ) ∂ x | ≤ C e − ν ( x − y ) 2 t − s ( t − s ) − 1 , | ∂ 2 W ( t , x ; s , y ) ∂ x 2 | ≤ C e − ν ( x − y ) 2 t − s ( t − s ) − 3 / 2 , where 0 < ν < ( 4 sup t ∈ R a ( t ) ) − 1 . The solution of (13) can be rewritten as Φ ( t , x ; s , y ) = ∑ k = 1 ∞ Φ k ( t , x ; s , y ) , where Φ 1 ( t , x ; s , y ) = L W ( t , x ; s , y ) , Φ k + 1 ( t , x ; s , y ) = ∫ s t d θ ∫ R L W ( t , x ; θ , ζ ) Φ k ( θ , ζ ; s , y ) d ζ . Using (14)–(17), we obtain that | Φ 1 ( t , x ; s , y ) | ≤ | a ( t ) − a ( s ) | | ∂ 2 W ( t , x ; s , y ) ∂ x 2 | + | b ( t ) | | ∂ W ( t , x ; s , y ) ∂ x | + | c ( t ) | | W ( t , x ; s , y ) | ≤ C e − ν ( x − y ) 2 t − s ( ( t − s ) − 3 / 2 + λ + ( t − s ) − 1 + ( t − s ) − 1 / 2 ) ≤ C e − ν ( x − y ) 2 t − s ( t − s ) − 3 / 2 + λ e η 1 ( t − s ) . Analogously to [6, (4.58)] we show that | Φ k ( t , x ; s , y ) | ≤ C Γ ( ( k − 1 ) λ + λ ) ( C ˜ ( t − s ) λ ) k − 1 e − ν ( x − y ) 2 t − s e η 1 ( t − s ) ( t − s ) − 3 / 2 + λ , where constants C, C ˜ depend on λ. Taking the sum of (18) for each k ≥ 1, we get the inequality | Φ ( t , x ; s , y ) | ≤ C e − ν ( x − y ) 2 t − s e η 1 ( t − s ) ( t − s ) − 3 / 2 + λ E λ , λ ( C ˜ ( t − s ) λ ) . The following inequality plays an important role in further estimates ∫ R ( t − θ ) − 1 / 2 ( θ − s ) − 1 / 2 e − ν ( ( ζ − y ) 2 θ − s + ( x − ζ ) 2 t − θ ) d ζ = π ν ( t − s ) − 1 / 2 e − ν ( x − y ) 2 t − s . Now we use (19) and (20) to obtain (10). | ∫ s t d θ ∫ R W ( t , x ; θ , ζ ) Φ ( θ , ζ ; s , y ) d ζ | ≤ C ∫ s t d θ ∫ R ( t − θ ) − 1 2 e − ν ( x − ζ ) 2 t − θ E λ , λ ( C ˜ ( θ − s ) λ ) ( θ − s ) 3 / 2 − λ e η 1 ( θ − s ) e − ν ( ζ − y ) 2 θ − s d ζ ≤ C e η 1 ( t − s ) ∫ s t E λ , λ ( C ˜ ( θ − s ) λ ) ( θ − s ) 1 − λ d θ ∫ R ( t − θ ) − 1 2 ( θ − s ) − 1 2 e − ν ( ( ζ − y ) 2 θ − s + ( x − ζ ) 2 t − θ ) d ζ = C e η 1 ( t − s ) ( t − s ) − 1 / 2 e − ν ( x − y ) 2 t − s ∫ s t ( θ − s ) − 1 + λ E λ , λ ( C ˜ ( θ − s ) λ ) d θ = C ( t − s ) − 1 / 2 + λ e − ν ( x − y ) 2 t − s e η 1 ( t − s ) E λ , λ + 1 ( C ˜ ( t − s ) λ ) , where the last equality is a consequence of ∫ 0 z E ρ , μ ( λ t ρ ) t μ − 1 d t = z μ E ρ , μ + 1 ( λ z ρ ) [8, chapter III, (1.15)]. From (15), (21) and the inequalities t ≥ s e η ( t − s ) ≥ 1, E λ , λ + 1 ( C ˜ ( t − s ) λ ) ≥ 1 Γ ( λ + 1 ) we obtain (10). We prove (11) analogously, using 1 π ∫ 0 z E ρ , μ ( λ t ρ ) ( z − t ) − 1 / 2 t μ − 1 d t = z μ − 1 / 2 E ρ , μ + 1 / 2 ( λ z ρ ) [8, chapter III, (1.16) with α = 1 / 2] instead of (22).

In the proof of (12) we use | ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ | = | ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ | 1 ( x ′ − x ) 2 < A ( t − s ) + | ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ | 1 ( x ′ − x ) 2 ≥ A ( t − s ) , where A > 00$]]>. Firstly assume that ( x ′ − x ) 2 < A ( t − s ); we prove that for such t, x, s | ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ | ≤ C ( ϕ , A ) | x − x ′ | ϕ ( t − s ) − 3 / 2 × × e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ ( C ˜ ( t − s ) λ ) . Notice that | ∂ W ( t , x ; s , y ) ∂ x − ∂ W ( t , x ′ ; s , y ) ∂ x ′ | ≤ C | ∫ x x ′ 1 ( t − s ) 3 / 2 e − ν ( v − y ) 2 t − s d v | ≤ C 1 ( t − s ) 3 / 2 e − ν 1 ( x ∗ − y ) 2 t − s ∫ 0 | x − x ′ | 2 e − ( ν − ν 1 ) w 2 t − s d w ≤ C 1 ( t − s ) 3 / 2 e − ν 1 ( x ∗ − y ) 2 t − s ∫ 0 | x − x ′ | 2 ( t − s w 2 ) l d w ≤ C 1 ( t − s ) 3 / 2 − l e − ν 1 ( x ∗ − y ) 2 t − s | x − x ′ | 1 − 2 l , where 0 ≤ l < 1 / 2, x ∗ = ϵ x + ( 1 − ϵ ) x ′ , 0 ≤ ϵ ≤ 1, 0 < ν 1 < ν; here we used the fact that the function x α e − x is bounded on [ 0 , + ∞ ) for arbitrary α > 00$]]>. Now we show that ( x ′ − x ) 2 < A ( t − s ) implies e − ν 1 ( x ∗ − y ) 2 t − s ≤ C e − ν 2 ( x − y ) 2 t − s , where ν 1 > ν 2 > 0{\nu _{2}}>0$]]>, C, ν 2 do not depend on y and x ∗ , but C depends on A. We consider two cases.

| x − y | ≤ 3 | x − x ′ |. We have the inequalities: e − ν 1 ( x ∗ − y ) 2 t − s ≤ 1 ≤ e A e − ( x ′ − x ) 2 t − s ≤ e A e − ( x − y ) 2 9 ( t − s ) = C 1 e − ν 21 ( x − y ) 2 t − s .

| x − y | > 3 | x − x ′ |3|x-{x^{\prime }}|$]]>. In this case we have the estimates: e − ν 1 ( x ∗ − y ) 2 t − s = e − ν 1 ( x − y ) 2 t − s e ν 1 ( x − x ∗ ) ( x + x ∗ − 2 y ) t − s ≤ e − ν 1 ( x − y ) 2 t − s e ν 1 | x − x ′ | ( 2 | x − y | + | x − x ∗ | ) t − s < e − ν 1 ( x − y ) 2 t − s e 7 ν 1 ( x − y ) 2 9 ( t − s ) = e − 2 ν 1 ( x − y ) 2 9 ( t − s ) = C 2 e − ν 22 ( x − y ) 2 t − s .

Now we set C = max { C 1 , C 2 }, ν 2 = min { ν 21 , ν 22 } and obtain (25). Therefore, the following estimate holds: | ∂ W ( t , x ; s , y ) ∂ x − ∂ W ( t , x ′ ; s , y ) ∂ x ′ | ≤ C 1 ( t − s ) 3 / 2 − l e − ν 2 ( x − y ) 2 t − s | x − x ′ | 1 − 2 l . Consider the expression ∂ p ( t , x ; s , y ) ∂ x − ∂ p ( t , x ′ ; s , y ) ∂ x ′ = ( ∂ W ( t , x ; s , y ) ∂ x − ∂ W ( t , x ′ ; s , y ) ∂ x ′ ) + ∫ t − | x ′ − x | 2 2 A t d θ ∫ R ∂ W ( t , x ; θ , ζ ) ∂ x Φ ( θ , ζ ; s , y ) d ζ − ∫ t − | x ′ − x | 2 2 A t d θ ∫ R ∂ W ( t , x ′ ; θ , ζ ) ∂ x ′ Φ ( θ , ζ ; s , y ) d ζ + ∫ s t − | x ′ − x | 2 2 A d θ ∫ R ( ∂ W ( t , x ; θ , ζ ) ∂ x − ∂ W ( t , x ′ ; θ , ζ ) ∂ x ′ ) Φ ( θ , ζ ; s , y ) d ζ = J 0 + J 1 + J 2 + J 3 . We estimate J 1 in the following way: | J 1 | = | ∫ t − | x ′ − x | 2 2 A t d θ ∫ R ∂ W ( t , x ; θ , ζ ) ∂ x Φ ( θ , ζ ; s , y ) d ζ | ≤ C ∫ t − | x ′ − x | 2 2 A t d θ ∫ R e − ν ( ( ζ − y ) 2 θ − s + ( x − ζ ) 2 t − θ ) E λ , λ ( C ˜ ( θ − s ) λ ) ( t − θ ) ( θ − s ) 3 / 2 − λ e η 1 ( θ − s ) d ζ ≤ C e η 1 ( t − s ) e − ν ( x − y ) 2 t − s E λ , λ ( C ˜ ( t − s ) λ ) ( t − s ) 1 / 2 ∫ t − | x ′ − x | 2 2 A t ( t − θ ) − 1 / 2 ( θ − s ) λ − 1 d θ ≤ C e η 1 ( t − s ) e − ν ( x − y ) 2 t − s E λ , λ ( C ˜ ( t − s ) λ ) ( t − s ) 1 / 2 ( t − s − | x ′ − x | 2 2 A ) λ − 1 ( | x ′ − x | 2 2 A ) 1 / 2 ≤ C e η 1 ( t − s ) e − ν ( x − y ) 2 t − s ( t − s ) − 3 / 2 + λ E λ , λ ( C ˜ ( t − s ) λ ) | x − x ′ | . Here we used the inequality t − s − | x ′ − x | 2 2 A > t − s 2 \frac{t-s}{2}$]]> and (20). We estimate J 2 in analogous way using (25): | J 2 | = | ∫ t − | x ′ − x | 2 2 A t d θ ∫ R ∂ W ( t , x ′ ; θ , ζ ) ∂ x ′ Φ ( θ , ζ ; s , y ) d ζ | ≤ C e η 1 ( t − s ) e − ν 2 ( x − y ) 2 t − s ( t − s ) − 3 / 2 + λ E λ , λ ( C ˜ ( t − s ) λ ) | x − x ′ | . We apply (20) and (26) with A ˜ = 2 A to prove the estimation for J 3 : | J 3 | ≤ ∫ s t − | x ′ − x | 2 2 A d θ ∫ R | ∂ W ( t , x ; θ , ζ ) ∂ x − ∂ W ( t , x ′ ; θ , ζ ) ∂ x ′ | | Φ ( θ , ζ ; s , y ) | d ζ ≤ C | x − x ′ | 1 − 2 l ∫ s t − | x ′ − x | 2 2 A d θ ∫ R e − ν 2 ( ( ζ − y ) 2 θ − s + ( x − ζ ) 2 t − θ ) E λ , λ ( C ˜ ( θ − s ) λ ) ( t − θ ) 3 / 2 − l ( θ − s ) 3 / 2 − λ e η 1 ( θ − s ) d ζ ≤ C | x − x ′ | 1 − 2 l e η 1 ( t − s ) e − ν 2 ( x − y ) 2 t − s E λ , λ ( C ˜ ( t − s ) λ ) ( t − s ) 1 / 2 ∫ s t ( t − θ ) l − 1 ( θ − s ) λ − 1 d θ ≤ C | x − x ′ | 1 − 2 l e η 1 ( t − s ) e − ν 2 ( x − y ) 2 t − s ( t − s ) − 3 / 2 + l + λ E λ , λ ( C ˜ ( t − s ) λ ) , for arbitrary l ∈ ( 0 , 1 / 2 ). Now (24) follows from (26), (27), (28) and (29).

Let ( x ′ − x ) 2 ≥ A ( t − s ). This implies | ∂ W ( t , x ; s , y ) ∂ x − ∂ W ( t , x ′ ; s , y ) ∂ x ′ | ≤ | ∂ W ( t , x ; s , y ) ∂ x | + | ∂ W ( t , x ′ ; s , y ) ∂ x ′ | ≤ C ( t − s ) − 1 max { e − ν ( x − y ) 2 t − s , e − ν ( x ′ − y ) 2 t − s } ≤ C ( t − s ) − 3 / 2 + l ( | x ′ − x | A ) 1 − 2 l max { e − ν ( x − y ) 2 t − s , e − ν ( x ′ − y ) 2 t − s } , where l ∈ ( 0 , 1 ). On the other hand, | ∫ s t d θ ∫ R ∂ W ( t , x ; θ , ζ ) ∂ x Φ ( θ , ζ ; s , y ) d ζ | ≤ C ∫ s t d θ ∫ R e − ν ( ( ζ − y ) 2 θ − s + ( x − ζ ) 2 t − θ ) E λ , λ ( C ˜ ( θ − s ) λ ) ( t − θ ) ( θ − s ) 3 / 2 − λ e η 1 ( θ − s ) d ζ ≤ C e η 1 ( t − s ) e − ν ( x − y ) 2 t − s E λ , λ ( C ˜ ( t − s ) λ ) ( t − s ) 1 / 2 ( | x ′ − x | A ) 1 − 2 l ∫ s t ( θ − s ) − 1 + λ ( t − θ ) − 1 + l d θ = C e η 1 ( t − s ) ( t − s ) − 3 / 2 + l + λ e − ν ( x − y ) 2 t − s ( | x ′ − x | A ) 1 − 2 l E λ , λ ( C ˜ ( t − s ) λ ) , and the same estimates hold for ∫ s t d θ ∫ R ∂ W ( t , x ′ ; θ , ζ ) ∂ x ′ Φ ( θ , ζ ; s , y ) d ζ. Using (30), (24) and (23), we obtain (12); in (23) we can set, for example, A = 1.  □

The main result of the paper is this theorem.

Let Assumptions 1–5 hold. Then there exists the version of the solution of (7) such that for each ω ∈ Ω: sup x ∈ R | u ( t , x ) | → 0 , t → ∞ .

Proof of the auxillary lemma and the main result

To prove Theorem 1, we consider the integral ∫ R d μ ( y ) ∫ 0 t p ( t , x ; s , y ) σ ( s , y ) d s .

Assume Assumptions 1, 2, 5 hold. Then there exists a version ν 1 ( t , x ) of integral (32) such that for each ω ∈ Ω sup x ∈ R | ν 1 ( t , x ) | → 0 , t → ∞ .

It follows from Lemma 2 and (6) that a version ν 1 ( t , x ) of integral (32) exists such that for each x ∈ R, t ≥ 0, ω ∈ Ω | ν 1 ( t , x ) | ≤ ∑ j ∈ Z | q ( t , x , j ) μ ( ( j , j + 1 ] ) | + C ∑ j ∈ Z ‖ q ( t , x , · ) ‖ B 22 α ( [ j , j + 1 ] ) × ∑ n = 1 ∞ 2 − n ( 2 α − 1 ) ∑ k = 1 2 n | μ ( Δ k n ( j ) ) | 2 1 / 2 ≤ ( ∑ j ∈ Z | q ( t , x , j ) | 2 ) 1 / 2 ( ∑ j ∈ Z | μ ( ( j , j + 1 ] ) | 2 ) 1 / 2 + C ( ∑ j ∈ Z ‖ q ( t , x , · ) ‖ B 22 α ( [ j , j + 1 ] ) 2 ) 1 / 2 ( ∑ j ∈ Z ∑ n = 1 ∞ 2 − n ( 2 α − 1 ) ∑ k = 1 2 n | μ ( Δ k n ( j ) ) | 2 ) 1 / 2 , where q ( t , x , y ) = ∫ 0 t p ( t , x ; s , y ) σ ( s , y ) d s . Next we prove that ν 1 ( t , x ) satisfies (33). In order to estimate the Besov norm on [ j , j + 1 ], we consider | q ( t , x , y + h ) − q ( t , x , y ) | ≤ ∫ 0 t | p ( t , x ; s , y + h ) − p ( t , x ; s , y ) | | σ ( s , y ) | d s + ∫ 0 t | p ( t , x ; s , y + h ) | | σ ( s , y + h ) − σ ( s , y ) | d s = I 1 + I 2 , where y , y + h ∈ [ j , j + 1 ]. Denote Ω M N = ( [ s , + ∞ ) × R ) ∖ ( [ s , s + M ) × ( y − N , y + N ) ) , Ω M N γ = { ( t , x ) : d ( Ω M N , ( t , x ) ) ≤ γ } , η 1 ( v ) = C e 1 | v | 2 − 1 1 { | v | < 1 } ; v ∈ R 2 , ∫ R 2 η 1 ( v ) d v = 1 , η ε ( v ) = ε − 2 η 1 ( v ε − 1 ) , Θ M N γ = ( Ω M N 2 γ ∖ Ω M N ) ∩ { t > s } , s\},\end{aligned}\]]]> where M, N, γ > 00$]]>. To estimate I 2 we introduce the function p ˜ ( t , x ; s , y ) = p ( t , x ; s , y ) 1 { t > s } Ψ ( t , x ; s , y ) , s\}}}\Psi (t,x;s,y),\]]]> where Ψ ( t , x ; s , y ) = ∫ Ω M N γ η γ ( t − v 1 , x − v 2 ) d v , v = ( v 1 , v 2 ) . It is obvious that, for each 0 < γ < min { M / 2 , N / 2 } and arbitrary fixed pair ( s , y ), p ˜ belongs to a class C 1 , 2 ( [ 0 , + ∞ ) × R ) as a function of ( t , x ). It is easy to obtain that p ˜ ( t , x ; s , y ) = p ( t , x ; s , y ) if ( t , x ) ∈ Ω M N and p ˜ ( t , x ; s , y ) = 0 if ( t , x ) ∈ ( [ 0 , + ∞ ) × R ) ∖ Ω M N 2 γ ∪ { s ≥ t }. Moreover, boundedness of p ( t , x ; s , y ) on Ω M N 2 γ ∩ { t ≤ T } for each T > 00$]]> and the fact that | Ψ | ≤ 1 imply boundedness of p ˜ ( t , x ; s , y ) on ( [ 0 , T ] × R ). Now we estimate L p ˜. Taking into consideration properties of p ˜, it is easy to see that L p ˜ = 0 outside the set Θ M N γ . And inside Θ M N γ , L p ˜ = L ( p Ψ ) = Ψ L p + p L Ψ + 2 a ∂ p ∂ x ∂ Ψ ∂ x = L p = 0 p L Ψ + 2 a ∂ p ∂ x ∂ Ψ ∂ x . For the derivatives of η ε ( v ) we have the following well-known inequalities: | ∂ η ε ( v ) ∂ v i | ≤ C ε − 3 , | ∂ 2 η ε ( v ) ∂ v i 2 | ≤ C ε − 4 , i = 1 , 2 , where the constant C does not depend on ε. Let us prove, for example, the first inequality in (36): | ∂ η ε ( v ) ∂ v i | ≤ ε − 3 max | v | ≤ 1 | ∂ η 1 ( v ) ∂ v i | = C ε − 3 , where we used that η 1 ∈ C ∞ ( R 2 ). From (36) and the fact that η ε = 0 outside the ball with radius ε it follows that | ∂ Ψ ∂ x | ≤ C γ − 1 , | ∂ Ψ ∂ t | ≤ C γ − 1 , | ∂ 2 Ψ ∂ x 2 | ≤ C γ − 2 . Using estimates (35), (10) and (11) we obtain the inequality | L p ˜ | ≤ C e − ν ( x − y ) 2 t − s e η ( t − s ) ( γ − 2 E λ , λ + 1 ( C ˜ ( t − s ) λ ) t − s + γ − 1 E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) t − s ) for each ( t , x ) ∈ Θ M N γ . Let γ ∈ ( 0 , 1 / 3 ) be fixed; consider t > 3 γ3\gamma $]]>. Thus, I 2 can be rewritten in following way: I 2 = ∫ 0 t − 3 γ | p ( t , x ; s , y + h ) | | σ ( s , y + h ) − σ ( s , y ) | d s + ∫ t − 3 γ t | p ( t , x ; s , y + h ) | | σ ( s , y + h ) − σ ( s , y ) | d s = I 21 + I 22 . We estimate the first summand using the function p ˜ ( t , x ; s , y ) for M = N = 3 γ. Note that t − s < 3 γ on the set Θ M N γ ; moreover, t − s > γ\gamma $]]> or | x − y | > γ\gamma $]]>. In the first case we have the following consequence of (37): | L p ˜ | ≤ C e 3 η γ γ − 5 / 2 ( E λ , λ + 1 ( C ˜ ( 3 γ ) λ ) + E λ , λ + 1 / 2 ( C ˜ ( 3 γ ) λ ) ) . In the second case, | L p ˜ | ≤ C e 3 η γ γ − 7 / 2 ( E λ , λ + 1 ( C ˜ ( 3 γ ) λ ) + E λ , λ + 1 / 2 ( C ˜ ( 3 γ ) λ ) ) . Anyway, | L p ˜ | ≤ C ( γ ) ∀ ( t , x ) ∈ [ 0 , + ∞ ) × R . Using Lemma 3 for arbitrary T > tt$]]>, we obtain | p ˜ ( t , x ; s , y ) | ≤ C ( γ ) e − c 0 t t , where the constant C does not depend on T. On the other hand, taking into account that p ˜ ( t , x ; s , y ) = p ( t , x ; s , y ) if t − s > 3 γ3\gamma $]]> we obtain | I 21 | ≤ C ( γ ) h β ( σ ) e − c 0 t t 2 → 0 , t → ∞ . Now we estimate I 22 . We get | I 22 | ≤ C h β ( σ ) ∫ t − 3 γ t 1 t − s e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ + 1 ( C ˜ ( t − s ) λ ) L σ ( s ) d s ≤ C h β ( σ ) γ e 3 γ η E λ , λ + 1 ( C ˜ ( 3 γ ) λ ) sup s ∈ [ t − 3 γ , t ] | L σ ( s ) | → 0 , t → ∞ . For further estimates we use the function p ˆ ( t , x ; s , y , h ) = ( p ( t , x ; s , y + h ) − p ( t , x ; s , y ) ) 1 { t > s } Ψ ( t , x ; s , y ) . s\}}}\Psi (t,x;s,y).\]]]> Let M > 2 γ2\gamma $]]>, N > 1 + 2 γ1+2\gamma $]]>. Then function p ˆ has properties, which are analogouos to the properties of p ˜. For example, p ˆ ( t , x ; s , y , h ) = p ( t , x ; s , y + h ) − p ( t , x ; s , y ) when ( t , x ) ∈ Ω M N , p ˆ is bounded on ( [ 0 , T ] × R ), where T > 00$]]>, p ˆ = 0 when ( t , x ) ∈ ( [ 0 , + ∞ ) × R ) ∖ Ω M N 2 γ ∪ { s ≥ t }. Now we estimate L p ˆ. Notice that L p ˜ = 0 outside the set Θ M N γ , and for each ( t , x ) ∈ Θ M N γ the following estimates hold: L p ˆ = L ( ( p y + h − p y ) Ψ ) = Ψ L ( p y + h − p y ) + ( p y + h − p y ) L Ψ + 2 a ∂ ( p y + h − p y ) ∂ x ∂ Ψ ∂ x = L p = 0 ( p y + h − p y ) L Ψ + 2 a ∂ ( p y + h − p y ) ∂ x ∂ Ψ ∂ x , where for convenience we denote p y = p ( t , x ; s , y ). (11) and (12) imply that | ∂ p ( t , x ; s , y + h ) ∂ x − ∂ p ( t , x ; s , y ) ∂ x | ≤ C h ϕ ( t − s ) − 3 / 2 sup y ∈ [ j , j + 1 ] e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ ( C ˜ ( t − s ) λ ) ; | p ( t , x ; s , y + h ) − p ( t , x ; s , y ) | = | ∫ x − y − h x − y ∂ p ( t , v ; s , 0 ) ∂ v d v | ≤ ∫ x − y − h x − y 1 t − s e − ν v 2 t − s e η ( t − s ) E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) d v ≤ h t − s sup y ∈ [ j , j + 1 ] e − ν ( x − y ) 2 t − s e η ( t − s ) E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) . Therefore, for each ( t , x ) ∈ Θ M N γ , | L p ˆ | ≤ ≤ C h ϕ sup y ∈ [ j , j + 1 ] e − ν ( x − y ) 2 t − s e η ( t − s ) × ( γ − 2 E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) t − s + γ − 1 E λ , λ ( C ˜ ( t − s ) λ ) ( t − s ) 3 / 2 ) . We rewrite I 1 as I 1 = ∫ 0 t − 3 γ | p ( t , x ; s , y + h ) − p ( t , x ; s , y ) | | σ ( s , y ) | d s + ∫ t − 3 γ t | p ( t , x ; s , y + h ) − p ( t , x ; s , y ) | | σ ( s , y ) | d s = I 11 + I 12 . Consider the function p ˆ ( t , x ; s , y , h ) for M = 3 γ, N = 3 γ + 1. We estimate I 11 analogouosly to I 21 . It follows from (40) and Lemma 3 that | p ˆ ( t , x ; s , y , h ) | ≤ C ( γ ) h ϕ e − c 0 t t ∀ ( t , x ) ∈ [ 0 , + ∞ ) × R . On the other hand, the equality p ˆ ( t , x ; s , y , h ) = p ( t , x ; s , y + h ) − p ( t , x ; s , y ) for t − s > 3 γ3\gamma $]]> implies that | I 11 | ≤ C ( γ ) h ϕ e − c 0 t t 2 → 0 , t → ∞ . I 12 is estimated in the following way: | I 12 | ≤ C ∫ t − 3 γ t d s ∫ x − y − h x − y C σ ( s ) t − s e − ν v 2 t − s e η ( t − s ) E λ , λ + 1 / 2 ( C ˜ ( t − s ) λ ) d v ≤ C e 3 η γ E λ , λ + 1 / 2 ( C ˜ ( 3 γ ) λ ) sup s ∈ [ t − 3 γ , t ] | C σ ( s ) | ∫ 0 h / 2 v − 2 l d v ∫ t − 3 γ t ( t − s ) 1 − l d s = C e 3 η γ E λ , λ + 1 / 2 ( C ˜ ( 3 γ ) λ ) sup s ∈ [ t − 3 γ , t ] | C σ ( s ) | γ 1 − l h 1 − 2 l → 0 , t → ∞ , where l ∈ ( 0 , 1 / 2 ). We can choose l, ϕ such that 1 − 2 l = β ( σ ), ϕ = β ( σ ).

Now assume that for each y ∈ [ j , j + 1 ] and some m ∈ N the following inequality holds: | x − y | ≥ m + 1 . Then we consider the functions p ˜ ( t , x ; s , y ) and p ˆ ( t , x ; s , y , h ) with M = 3 γ and N = 3 γ + m. For such M and N, provided that (43) holds, ( t , x ) ∈ Ω M N . Moreover, using (37), (40), the fact that (43) implies sup y ∈ [ j , j + 1 ] e − ν ( x − y ) 2 t − s ≤ e − ν 1 m 2 3 γ sup y ∈ [ j , j + 1 ] e − ( 1 − ν 1 ) ( x − y ) 2 t − s ∀ ( t , x ) ∈ Θ M N γ , 0 < ν 1 < ν , and Lemma 3, we obtain: | p ( t , x ; s , y ) | ≤ C ( γ ) e − c 0 t t e − ν 1 m 2 3 γ , | p ( t , x ; s , y + h ) − p ( t , x ; s , y ) | ≤ C ( γ ) h β ( σ ) e − c 0 t t e − ν 1 m 2 3 γ . Now it is easy to estimate I 1 , I 2 : I 1 ≤ C ( γ ) h β ( σ ) e − c 0 t t 2 e − ν 1 m 2 3 γ ≤ C ( γ ) h β ( σ ) e − c 0 t t 2 m − 1 → 0 , t → ∞ , I 2 ≤ C ( γ ) h β ( σ ) e − c 0 t t 2 e − ν 1 m 2 3 γ ≤ C ( γ ) h β ( σ ) e − c 0 t t 2 m − 1 → 0 , t → ∞ . Note that we estimate | q ( t , x , y ) | analogously to I 2 . From (38), (39), (41), (42), (44), (45) it follows that there exists G γ ( t ) : [ 0 , + ∞ ) → [ 0 , + ∞ ) such that G γ ( t ) → 0, t → ∞, and w 2 ( q , r ) ≤ G γ ( t ) r β ( σ ) ∀ t , r ≥ 0 , j ∈ Z , x ∈ R ; | q ( t , x , j ) | ≤ G γ ( t ) ∀ t ≥ 0 , j ∈ Z , x ∈ R ; w 2 ( q , r ) ≤ G γ ( t ) r β ( σ ) m − 1 ∀ t , r ≥ 0 , j ∈ Z , x ∈ R : max y ∈ [ j , j + 1 ] | x − y | ≥ m + 1 ; | q ( t , x , j ) | ≤ G γ ( t ) m − 1 ∀ t ≥ 0 , j ∈ Z , x ∈ R : max y ∈ [ j , j + 1 ] | x − y | ≥ m + 1 . From this it follows that for each α ∈ ( 1 / 2 , β ( σ ) ) the following inequalities hold: ‖ q ( t , x , · ) ‖ B 22 α ( [ j , j + 1 ] ) ≤ C G γ ( t ) ∀ t ≥ 0 , j ∈ Z , x ∈ R ; ‖ q ( t , x , · ) ‖ B 22 α ( [ j , j + 1 ] ) ≤ C G γ ( t ) m − 1 ∀ t ≥ 0 , j ∈ Z , x ∈ R : max y ∈ [ j , j + 1 ] | x − y | ≥ m + 1 . These estimates imply ∑ j ∈ Z | q ( t , x , j ) | 2 ≤ C G γ 2 ( t ) + C G γ 2 ( t ) ∑ m ∈ N 1 m 2 = C G γ 2 ( t ) , ∑ j ∈ Z ‖ q ( t , x , · ) ‖ B 22 α ( [ j , j + 1 ] ) 2 ≤ C G γ 2 ( t ) + C G γ 2 ( t ) ∑ m ∈ N 1 m 2 = C G γ 2 ( t ) . for each x ∈ R. On the other hand, ∑ j ∈ Z | μ ( ( j , j + 1 ] ) | 2 < ∞ a.s., ∑ j ∈ Z ∑ n = 1 ∞ ∑ k = 1 2 n 2 − n ( 2 α − 1 ) | μ ( Δ k n ( j ) ) | 2 < ∞ a.s. Therefore, for each version that satisfies (34) we have | ν 1 ( t , x ) | ≤ C ( ω ) G γ ( t ) . Taking a supremum on x and sending t to infinity, we obtain the statement of the lemma.  □

Now we return to the proof of Theorem 1.

We use the iteration process (9). For each n ∈ N we consider the function ν 2 ( n ) ( t , x ) = ∫ R p ( t , x ; 0 , y ) u 0 ( y ) d y + ∫ 0 t d s ∫ R p ( t , x ; s , y ) f ( s , y , u ( n − 1 ) ( s , y ) ) d y . From [6, Theorem 2 §4] it follows that the function ν 2 ( n ) is a solution, bounded on [ 0 , T ] × R, of the Cauchy problem L v ( t , x ) = − f ( t , x , u ( n − 1 ) ( t , x ) ) , v ( 0 , x ) = u 0 ( x ) , for each ω ∈ Ω, T > 00$]]>. Using Lemma 3, we obtain | ν 2 ( n ) ( t , x ) | ≤ C e − c 0 t ( 1 + t ) . Now from (8) and (46) it follows that | u ( t , x ) | ≤ C e − c 0 t ( 1 + t ) + sup x ∈ R | ν 1 ( t , x ) | . Taking a supremum on x, sending t to infinity and using Lemma 5, we obtain the statement of the theorem.  □

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