Abstract
We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman-Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.
Introduction

Overview of results
The aim of this paper is to study stability of stationary solutions to a class of non-local semilinear parabolic equations applying the Feynman-Kac formula. Namely, we wish to investigate bounded solutions to the following equation ∂ ∂t u(x, t) = (L J u)(x, t) + V (u(x, t))u(x, t), x ∈ R d , t > 0,
where
, is a generator of a pure jump Markov process, V :
is a bounded locally Lipschitz mapping, and the initial condition u(·, 0) ∈ C b (R d ) belongs to a neighborhood of u ≡ 0. Note that the Feynman-Kac formula for diffusion processes with time-dependent potentials is known (see [8, Theorem 5.7.6] ). However, the corresponding result for general Lévy processes seems to be proved only recently in [12] , where compactly supported smooth initial conditions where assumed. We relax assumptions on the initial conditions, considering bounded continuous functions C b (R d ) and prove the Feynman-Kac formula for the generator L J (see Propositions 2.1 and 2.2). We propose also sufficient conditions for the asymptotic stability of the zero solution to (1.1) uniformly in space (see Theorem 2.3), and apply this to a particular equation ∂ ∂t u(x, t) = κ + (L a + u)(x, t) + κ − θ − (a − * u)(x, t) u(x, t), (1.2) where κ ± , m > 0, θ := κ + −m κ − > 0, and a ± are probability kernels (see [1, 4, 6, 9, 10, 13] ). The equation (1.2) may be considered as a non-local version of the classical logistic equation (see (3. 2) below). There are two constant solution to (1.2), u ≡ 0 and u ≡ θ. Different properties and the long-time behavior of solutions to (1.2), were considered in [5] .
We are interested to find sufficient conditions which ensure that a solution to (1.2) converges to the constant non-zero solution u ≡ θ uniformly in space. Applying Theorem 2.3, we prove (see Theorem 3.6) that a bounded initial condition, which is separated from zero, tends to θ > 0 exponentially fast if only
The condition on J θ may be relaxed under more restrictive assumptions on the initial condition. Namely, introducing a parameter in the initial condition and considering the analytical decomposition of the corresponding solution with respect to the parameter, one can show that if J θ L 1 < κ + and if the initial condition lies in a ball centered at θ, then the solution tends to θ exponentially fast (see Theorem 4.1). An example of a parameter constructed by a stationary random field provides an enhanced asymptotic for the convergence (see Theorem 4.4).
Basic notations
denote the spaces of all bounded continuous, respectively, bounded Borel measurable functions on R d . The functional spaces become Banach ones being equipped with the norm
Consider the following bounded operator (in any of the Banach spaces above)
where µ := R d J(y) dy > 0. Let X t be a jump-process with the state space (R d , B(R d )) and the natural filtration F t = σ(X s | s ≤ t) whose generator is L J (for details see [3] ). It is well known that X t is a Markov process and, for all s, t ≥ 0, f ∈ B b (R d ), the following equation holds,
where p(x, t) is the transition density of X t . Namely, p(x, 0) = δ(x) and p(x, t) satisfies the following equation
For an interval I ⊂ R + := [0, ∞), consider the Banach spaces C b (I → E) of continuous E-valued functions on I, where E is a space above, with the norm
For the simplicity of notations, we set, for any
with the corresponding norms · T1,T2 , · T , · .
The Feynman-Kac Formula and Stability
Let u = u(x, t) describe the local density of a system at the point x ∈ R d , d ≥ 1, at the moment of time t ∈ R + . Prove now a version of the Feynman-Kac formula for the time-dependent potential and operator L J , cf. e.g. [2, Theorem 2.5], [8, Theorem 5.7.6] . Consider a perturbed equation
where W ∈ X T . Then, clearly, (2.1) has a unique solution in X T . The following theorem states that the solution will satisfy the Feynman-Kac formula.
Proof. For f ∈ X T , we denote
By (1.4) and (2.4),
where the last equality holds by the tower rule and Fubini's theorem. One can continue then
In the same manner, we can prove, by the induction, the following equality
By (2.4) and (2.5) (with s = t),
We writeQ for the operator defined by (2.3) with W substituted by |W |. It follows easily that the equation similar to (2.5) holds forQ. In particular, for u 0 ≡ 1 and s = t,
As a result, for n → ∞, (2.6) yields (2.2), that completes the proof.
Consider now a general semi-linear evolution equation with the generator L J :
where V :
is a bounded locally Lipschitz mapping, i.e. 
Note also that since L J is a bounded operator, then the mild solution will be classical one, i.e. u ∈ X T , for any T < T max , and u(x, t) is differentiable in t w.r.t. the norm in C b (R d ). By Proposition 2.1, the following Feynman-Kac-type expression holds for the solution to (2.7). Proposition 2.2. Let (2.8) hold and u be the unique classical solution to (2.7)
The following theorem provides sufficient conditions for the stability of the stationary solution u ≡ 0 to (2.7), Theorem 2.3. Let there exist p :
(2.10)
Suppose that u 0 ∈ E is such that, for some c ≤ 0 and d ≥ 0,
Then, for any T > 0, there exists a unique u ∈ X T , which satisfies the FeynmanKac formula (2.9). Moreover, u ∈ X c,d
∞ , u(·, t) E does not increase in time, and if p(c, d) > 0, then u(·, t) E converges to zero exponentially fast, namely,
Proof. Let us introduce the following operator: we set, for a w ∈ X ∞ ,
Then, for any w ∈ X c,d
T ,
Since p is non-negative, one gets Ψ(X c,d
T , t ∈ [0, T ], the following estimate holds
where M = M max{−c,d} is defined by (2.8). Hence Ψ is a contraction map on
. Therefore, there exists a fixed point u ∈ X c,d
T . By (2.12), the function u satisfies the following estimate
2T , so that the following estimate holds
By induction, u can be extended to X c,d
nT , and for any n ∈ N, ∞ , such that (2.9) and (2.13) hold. Since p is non-negative, {c n } is increasing and {d n } is decreasing. Moreover,
together with (2.10) yield that p(c k , d k ) ≤ p(c n , d n ), for k ≤ n. Therefore, u(·, t) E does not increase in time. Let us prove by induction the following inequalities
14)
The case n = 1 is obvious. Let (2.14) and (2.15) hold for 0 ≤ k ≤ N . We prove them for 0 ≤ k ≤ N + 1. Since c N ≤ 0, we have
Hence (2.14) is proved. Similarly, the following estimate yields (2.15)
where 0 ≤ k ≤ N . By (2.14) and (2.15) with k = 0, both {c n } and {d n } converge to zero exponentially fast if only p(c, d) > 0. Therefore, for t ∈ [nT, (n + 1)T ],
and, by (2.14), (2.15), we have, for k ≥ 0,
As a result, (2.11) holds, because p(c k , d k ) is increasing in k. This proves the theorem.
Spatial logistic equation
We will consider the following equation for a bounded function u(x, t), which describes the (approximate) value of the local density of a system of particles distributed in R d according to the so-called spatial logistic model. More detailed explanation and historical remarks can be found in [5, Subsection 6.1]. Namely, let u t (x) := u(x, t), x ∈ R d , t ∈ I, solves the equation
In particular, u(x, 0) = u 0 (x), x ∈ R d . Here κ + L a + is a generator of the underlying random walk, cf. (1.3):
which spends exponentially distributed random time τ in each particular position x, P {τ > ρ} = e −κ + ρ , and it makes a jump x → x + X, thereafter, where the random variable X has the distribution density a + (x). The constant β = κ + − m > 0 is the difference between the biological rate β of the birth of a new particle and the mortality rate m. The last term in (3.1) describes the competition between particles, the potential κ − a − (x − y) presents the interaction between two particles located at the points x, y ∈ R d . Equation (3.1) is similar to the well-known logistic ordinary differential equation:
whose partial solution is the constant θ := β κ − . All other positive solutions tend exponentially fast to θ. The equation (3.1) has the same solution u(x, t) ≡ θ, x ∈ R d , t ≥ 0 (we suppose R a ± (y)dy = 1). This important particular solution is the exponentially stable attractor for (3.1). We will study the neighborhood of the attractor, using variations of u 0 . Let us denote, for any h ∈ E,
Then (3.1) has the following form
3)
The analysis of the non-linear parabolic equation (3.1) will be based on integral equations. The first of them is given through the standard Duhamel's formula.
Lemma 3.1. Function u solves (3.3) iff it satisfies the following equation
This equation has the Volterra form and can be used for the existenceuniqueness theory (see [5] ). Theorem 3.2. Let u 0 ∈ E be non-negative. Then, for each T > 0, there exists a unique non-negative solution to (3.3) in X T . Now we will estimate solution to (3.3) from below. Let u 0 be a constant function u 0 ≡ q 0 ∈ (0, θ).
Then the corresponding solution to (3.3) is the function
. By Theorem 3.2, this solution is unique. Let us fix κ ∈ [0, θ]. We make the following assumption
where u 0 ∈ E. Then the corresponding to u 0 solution u t to (3.3) satisfies the following inequality
Proof. Let us fix T > 0.
, where K will be defined later. The function v t satisfies the following linear equation
where, for all w ∈ E,
By Theorem 3.2, there exists M > 0, such that
is non-negative for all t ∈ [0, T ] and for all non-negative w ∈ E. Therefore,
Since T is arbitrary, the same holds for any t > 0.
Remark 3.4 (cf. [5, Proposition 3.4] ). In a similar way, it can be shown that if (A1) holds with κ = θ, and u 0 ∈ E is such that 0 ≤ u 0 (x) ≤ θ, x ∈ R d , then the corresponding solution satisfies the following inequality
The following theorem shows, based on the Feynman-Kac formula, that u t satisfies another integral equation.
Theorem 3.5. Let (A1) holds with κ = θ. Suppose that u ∈ X T , T ∈ (0, ∞], is the solution to (3.1) with an initial condition u 0 ∈ E. Then u satisfies the following formula, for all
where X 0 = x.
Proof. Let us denote g t := u t −θ. If u t solves (3.1), then g t satisfies the following equation
where
For such V and the generator L J = L J θ of the jump-process X t , we apply Proposition (2.2) to the solution of (3.5)
Substituting g t = u t −θ into the previous representation completes the proof.
The following theorem shows the asymptotic stability of the positive stationary solution.
Theorem 3.6. Let (A1) holds with κ = θ. Suppose that u 0 ∈ E is an initial condition to (3.1), such that
where 0 ≤ c 1 ≤ θ and c 2 ≥ θ. Then there exists a unique solution u ∈ X ∞ to (3.1). Moreover, u ∈ X c1,c2 ∞ , u t − θ E does not increase in time, and if c 1 > 0, then u t − θ E converges to zero exponentially fast, namely lim sup
Proof. We consider g t = u t − θ. By Theorem 3.2 and 3.5, there exists a unique solution g ∈ X ∞ to (3.5), and this solution satisfies (3.6). The rest of the proof follows from Theorem 2.3 with p(c, d) = p(c) = κ − (θ + c).
Stability on the initial condition
We will be interested in initial conditions of the following form
where ξ :
Since the operator L a + is linear and bounded on E, and F is analytic on E, then the solution u to (3.3) depends analytically on the initial condition u 0 (see e.g. [7, Theorem 3.4.4, Corollary 3.4.5, 3.4.6] ). Hence, by (4.1), the E-valued function λ → u(·, t, λ) is analytic on R for each t ≥ 0. Therefore, for all λ ∈ R, it is given by the following series
where
Hence, the n-th Taylor coefficient satisfies the following equation
Therefore,
where k n,0 (x) = θξ n (x).
Theorem 4.1. Let ξ ∈ E and γ = κ + − J θ L 1 > 0. Then the following estimate holds
Proof. We will estimate k n , n ≥ 0. By (4.3), k 0 ≡ θ. The function k 1 satisfies the following equation
Suppose that
where C l is a positive constant. (Note that, by (4.4), C 1 = 1). Estimate k n . By the mild form of (4.3), the following inequality holds
Therefore, by induction,
where
Put C 0 = 0. Consider the following generating function:
By (4.6), H satisfies the following equation:
Since H(0) = C 0 = 0 and the function z → √ 1 − z is analytic for |z| < 1, one has
Therefore, (4.2), (4.5) and (4.7), we have
This proves the theorem. 
where sup ω∈Ω |λ| ξ(ω) E < ln γ 4β + 1 .
We apply now the general results to the specific case of the random initial data and try to estimate the rate of convergence using L 2 -norm over a probability space. Let us denote, for any f ∈ L 1 (R d ), its Fourier transform by
Letp t (x) be a transition probability density for the jump process with the generator L J for J = J θ (see (1.3) and (A1)). Introduce the following assumption J θ is bounded.
Assumption (A2) is a sufficient condition
The following theorem improves the estimate (4.8), when J θ is nonnegative. Proof. By assumptions of the theorem,p t * B ∈ L
1,t = θ 2 e −2βt E R dp t (y)ξ(y)dy R dp t (z)ξ(z)dz = θ 2 e −2βt
R dp t (y)p t (z)Eξ(z)ξ(y)dydz = θ 2 e −2βt
R dp t (y)p t (z)B(y − z)dydz 

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