On the Cauchy Problem for Integro-Differential Equations in the Scale of Spaces of Generalized Smoothness

Abstract

Parabolic integro-differential model Cauchy problem is considered in the scale of L_p -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some rough probability density function estimates of the associated Levy process are used as well.

Appendix

We will need the following Levy process moment estimate.

Lemma 17

Let$\pi \in \mathfrak {A}^{\sigma }.$Assume

$$ {\int}_{\left\vert z\right\vert \leq 1}\left\vert z\right\vert^{\alpha_{1}}\pi (dz)+{\int}_{\left\vert z\right\vert >1}\left\vert z\right\vert^{\alpha_{2}}\pi (dz)\leq M, $$

(1)

whereα₁, α₂ ∈ (0, 1] ifσ ∈ (0, 1); α₁, α₂ ∈ (1, 2] ifσ ∈ (1, 2); α₁ ∈ (1, 2] andα₂ ∈ [0, 1) ifσ = 1. Letζ_tbe the Levy process associated toψ^π, that is

$$\mathbf{E}e^{i2\pi \xi \cdot \zeta_{t}}=\exp \{\psi \left( \xi \right) t\},t\geq 0 $$

There is a constantC = C (M) such that

$$\mathbf{E}\left[ \left\vert \zeta_{t}\right\vert^{\alpha_{2}}\right] \leq C\left( 1+t\right) ,t\geq 0. $$

Proof

Recall

$$ \zeta_{t}={{\int}_{0}^{t}}\int \chi_{\sigma }(y)yq(ds,dy)+{{\int}_{0}^{t}}\int (1-\chi_{\sigma }(y))yp(ds,dy),t\geq 0, $$

(2)

p(ds, dy) is Poisson point measure with

$$\mathbf{E}p\left( ds,dy\right) =\pi \left( dy\right) ds,q\left( ds,dy\right) =p\left( ds,dy\right) -\pi \left( dy\right) ds. $$

Now,$\zeta _{t}=\bar {\zeta }_{t}+\tilde {\zeta }_{t}$ with

$$\begin{array}{@{}rcl@{}} \bar{\zeta}_{t} &=&{{\int}_{0}^{t}}{\int}_{\left\vert y\right\vert \leq 1}\chi_{\sigma }(y)yq(ds,dy)+{{\int}_{0}^{t}}{\int}_{\left\vert y\right\vert \leq 1}(1-\chi_{\sigma }(y))yp(ds,dy), \\ \tilde{\zeta}_{t} &=&{{\int}_{0}^{t}}{\int}_{\left\vert y\right\vert >1}\chi_{\sigma }(y)yq(ds,dy)+{{\int}_{0}^{t}}{\int}_{\left\vert y\right\vert >1}(1-\chi_{\sigma }(y))yp(ds,dy),t\geq 0. \end{array} $$

• Case 1: σ ∈ (0, 1). In this case (1) holds with α₁, α₂ ∈ (0, 1]. Then for any t >  0,

  $$\mathbf{E}\left\vert \bar{\zeta}_{t}\right\vert \leq t{\int}_{\left\vert y\right\vert \leq 1}\left\vert y\right\vert^{\alpha_{1}}\pi \left( dy\right) \leq Ct, $$

  and

  $$\left\vert \tilde{\zeta}_{t}\right\vert^{\alpha_{2}}=\sum\limits_{s\leq t}\left[ \left\vert \tilde{\zeta}_{s}\right\vert^{\alpha_{2}}-\left\vert \tilde{ \zeta}_{s-}\right\vert^{\alpha_{2}}\right] \leq {{\int}_{0}^{t}}{\int}_{\left\vert y\right\vert >1}\left\vert y\right\vert^{\alpha_{2}}p\left( ds,dy\right) $$

  implies that $\mathbf {E}\left \vert \tilde {\zeta }_{t}\right \vert ^{\alpha _{2}}\leq Ct.$

• Case 2: σ ∈ (1, 2). In this case, α₁, α₂ ∈ (1, 2]. Then

  $$\mathbf{E[}\left\vert \bar{\zeta}_{t}\right\vert^{2}]={\int}_{\left\vert y\right\vert \leq 1}\left\vert y\right\vert^{2}\pi \left( dy\right) t\leq Ct, $$

  and, by BDG inequality,

  $$\begin{array}{@{}rcl@{}} \mathbf{E[}\left\vert \tilde{\zeta}_{t}\right\vert^{\alpha_{2}}] &\leq &C \mathbf{E}\left[ \left( \sum\limits_{s\leq t}\left( {\Delta} \tilde{\zeta}_{s}\right)^{2}\right)^{\alpha_{2}/2}\right] \\ &\leq &C\mathbf{E}\left[ \sum\limits_{s\leq t}\left( {\Delta} \tilde{\zeta} _{s}\right)^{\alpha_{2}}\right] =Ct{\int}_{\left\vert y\right\vert >1}\left\vert y\right\vert^{\alpha_{2}}d\pi . \end{array} $$

• Case 3: σ = 1. In this case, α₁ ∈ (1, 2] andα₂ ∈ (0, 1). Similarly as above, we find that

  $$\begin{array}{@{}rcl@{}} \mathbf{E[}\left\vert \bar{\zeta}_{t}\right\vert^{2}] &=&t{\int}_{\left\vert y\right\vert \leq 1}\left\vert y\right\vert^{2}\pi \left( dy\right) \leq Ct, \\ \mathbf{E[}\left\vert \tilde{\zeta}_{t}\right\vert^{\alpha_{2}}] &\leq &Ct. \end{array} $$

  The statement is proved.

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We need the following Gaussian moments estimates as well.

Lemma 18

Leta_kj ∈ R,k, j ≥  0, and

$$\left\vert \left\vert a\right\vert \right\vert =\left( \sum\limits_{k,j = 0}^{\infty }a_{kj}^{2}\right)^{1/2}<\infty . $$

and letζ_k, k ≥  0, be a sequence of independent standard normal r.v. Forp ≥ 1 set

$$\xi =\left( \sum\limits_{j = 0}^{\infty }\left( \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right)^{2}\right)^{p/2}. $$

Then there are constants 0 < c₁ < c₂so that

$$c_{1}\left\vert \left\vert a\right\vert \right\vert^{p}\leq \mathbf{E}\xi \leq c_{2}\left\vert \left\vert a\right\vert \right\vert^{p}. $$

Proof

• Case1. Let p ≥ 2. Since ζ_k are independent standard normal, by Minkowski inequality,

  $$\mathbf{E}\xi \leq \left( \sum\limits_{j = 0}^{\infty }\left[ \mathbf{E}\left( \left\vert \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right\vert^{p}\right) \right]^{2/p}\right)^{p/2}\leq C\left( \sum\limits_{j = 0}^{\infty }\sum\limits_{k = 0}^{\infty }a_{kj}^{2}\right)^{p/2}. $$

  On the other hand, by Hölder inequality,

  $$\mathbf{E}\xi \geq \left( \mathbf{E}\sum\limits_{j = 0}^{\infty }\left( \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right)^{2}\right)^{p/2}\geq c\left( \sum\limits_{j,k = 0}^{\infty }a_{kj}^{2}\right)^{p/2}. $$

• Case 2. Let p ∈ [1, 2). Then, by Hölder inequality,

  $$\begin{array}{@{}rcl@{}} &&\mathbf{E}\left[ \left( \sum\limits_{j = 0}^{\infty }\left( \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right)^{2}\right)^{p/2}\right] \\ &\leq &\left( \mathbf{E}\sum\limits_{j = 0}^{\infty }\left( \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right)^{2}\right)^{p/2}=\left( \sum\limits_{j,k = 0}^{\infty }a_{kj}^{2}\right)^{p/2}. \end{array} $$

  On the other hand, by Hölder and Minkowski inequality (recall ζ_k are independent standard normal),

  $$\begin{array}{@{}rcl@{}} &&\mathbf{E}\xi \geq \left[ \mathbf{E}\left( \sum\limits_{j = 0}^{\infty }\left\vert \sum\limits_{k = 0}^{\infty }\zeta_{k}a_{kj}\right\vert^{2}\right)^{1/2}\right]^{p} \\ &\geq &\left( \sum\limits_{j,k = 0}^{\infty }\left( \mathbf{E}\left\vert \zeta_{k}a_{kj}\right\vert \right)^{2}\right)^{p/2}\geq c\left( \sum\limits_{j,k = 0}^{\infty }a_{kj}^{2}\right)^{p/2}. \end{array} $$

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