On Diffusion Processes with Drift in L_d+ 1

Abstract

This paper is a natural continuation of Krylov (2020 and 2021) where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb {R}^{d+1})$ and some properties of their Green’s functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

Appendix:

Here we present without proofs some results from [12] frequently used in the main text.

Set

$$ \tau^{\prime}_{R}(x)=\inf\{t\geq0:x+x_{t}\not\in B_{R}\},\quad \gamma_{R}(x)=\inf\{t\geq0:x+x_{t} \in \bar B_{R}\}. $$

(A.1)

Theorem A.1 (Theorem 2.3)

There are constants $\bar \xi =\bar \xi (d,\delta )\in (0,1)$ and $\bar N=\bar N(d,p_{0}, \delta )$ continuously depending on δ such that if, for an $R\in (0,\infty )$, we have

$$ \bar N \bar b_{R}\leq 1, $$

(A.2)

then for |x|≤ R

$$ P(\tau_{R}(x) = R^{2} )\leq 1-\bar\xi, \quad P(\tau_{R} = R^{2} )\geq \bar\xi . $$

(A.3)

Moreover for n = 1, 2,... and |x|≤ R

$$ P(\tau^{\prime}_{R}(x)\geq nR^{2}) = P(\tau_{nR^{2},R}(x) = n R^{2} )\leq (1-\bar\xi)^{n}, $$

(A.4)

so that $E\tau ^{\prime }_{R}(x)\leq N(d,\delta )R^{2}$.

Furthermore, for any $x\in \bar B_{9R/16}$

$$ P(\tau^{\prime}_{R}(x)>\gamma_{R/16}(x))\geq\bar\xi. $$

(A.5)

Theorem A.2 (Theorem 2.6)

For any λ,R > 0 we have

$$ Ee^{-\lambda \tau_{R} }\leq e^{\bar\xi/2}e^{- \sqrt{\dot\lambda} R \bar\xi/2}= \begin{cases} e^{\bar\xi/2}e^{-\sqrt\lambda R\bar\xi/2} \quad\text{if}\quad \lambda \geq \underline{\lambda}\\ e^{\bar\xi/2}e^{-\lambda R\underline{R} \bar\xi/2}\quad\text{if} \quad \lambda \leq \underline{\lambda}, \end{cases} $$

(A.6)

where

$$ \dot\lambda= \lambda\min(1, \lambda/\underline{\lambda} ),\quad \underline{\lambda}=\underline{R}^{-2}. $$

In particular, for any R > 0 and $t\leq R\underline {R} \bar \xi /4$ we have

$$ P(\tau_{R} \leq t )\leq e^{\bar\xi/2}\exp\Big(-\frac{{\bar \xi}^{2}R^{2}}{16 t}\Big). $$

(A.7)

Theorem A.3 (Theorem 2.9)

Let $R\in (0,\underline {R}]$, $x,y\in \mathbb {R}^{d}$ and 16|x − y|≥ 3R. For r > 0 denote by S_r(x,y) the open convex hull of B_r(x) ∪ B_r(y). Then there exist T₀,T₁, depending only on $\bar \xi$, such that $0<T_{0}<T_{1}<\infty$ and the probability π that x + x_t will reach $\bar B_{R/16}(y)$ before exiting from S_R(x,y) and this will happen on the time interval [nT₀R²,nT₁R²] is greater than ${\pi _{0}^{n}}$, where

$$ n= \Big\lfloor \frac{16|x-y|+R}{4R}\Big\rfloor $$

and $\pi _{0}=\bar \xi /3$.

Theorem A.4 (Theorem 4.3)

There exists d₀ ∈ (1,d), depending only on δ, d, $\underline {R}$, p₀, such that for any p ≥ d₀ + 1 and λ > 0

$$ {\int}_{0}^{\infty}{\int}_{\mathbb{R}^{d}}G_{\lambda}^{p/(p-1)}(t,x) dxdt\leq N(\delta,d,\underline{R},p_{0},\lambda,p). $$

Furthermore, the above constant $N(\delta ,d,\underline {R},p_{0},\lambda ,p)$ can be taken in the form

$$ N(\delta,d,\underline{R},p_{0},p)\ddot{\lambda}_{p}^{(d+2)/(2p)-1}, $$

where

$$ \ddot{\lambda}_{p}=\lambda(1\wedge\lambda)^{d/(2p-d-2)}. $$

Theorem A.5 (Theorem 4.8)

Suppose

$$ p ,q \in [1,\infty],\quad \nu:=1-\frac{d_{0}}{p }-\frac{1}{q }\geq 0. $$

(A.8)

Then there is $N=N(\delta ,d,\underline {R},p,q,p_{0},\bar b_{\infty } )$ such that for any λ > 0 and Borel nonnegative f we have

$$ E{\int}_{0}^{\infty}e^{- \lambda t} f(t,x_{t}) dt\leq N\ddot{\lambda}_{d_{0}+1}^{-\nu+(d-2d_{0})/(2p )}\|{\Psi}_{\lambda}^{1-\nu} f\|_ {L_{p ,q }(\mathbb{R}^{d+1}_{+})}, $$

(A.9)

where ${\Psi }_{\lambda }(t,x)=\exp (- \sqrt {\dot \lambda } (|x|+ \sqrt t)\bar \xi /16)$. In particular, if f is independent of t, p ≥ d₀, and $q=\infty$

$$ E{\int}_{0}^{\infty}e^{- \lambda t} f(x_{t}) dt\leq N\ddot{\lambda}_{d_{0}+1}^{-1+d/(2p)}\|\bar {\Psi}_{\lambda}^{d_{0}/p} f\|_ {L_{p }(\mathbb{R}^{d } )}, $$

where $\bar {\Psi }_{\lambda }(x)=\exp (- \sqrt {\dot \lambda } |x| \bar \xi /16)$.

Theorem A.6 (Theorem 4.9)

Assume that Eq. A.8 holds. Then

(ii) for any n = 1, 2,..., nonnegative Borel f on $\mathbb {R}^{d+1}_{+}$, and T ≤ 1 we have

$$ E{\Big[{\int}_{0}^{T}} f(t,x_{t}) dt\Big]^{n}\leq n!N^{n} T^{n\chi }\| {\Psi}^{(1-\nu)/n}_{1/T} f\|^{n}_{L_{p,q}(\mathbb{R}^{d+1}_{+}) }, $$

(A.10)

where $N=N(\delta ,d,\underline {R},p,q,p_{0},\bar b_{\infty } )$ and χ = ν + (2d₀ − d)/(2p);

(ii) for any nonnegative Borel f on $\mathbb {R}^{d+1}_{+}$, and T ≥ 1 we have

$$ I:=E {{\int}_{0}^{T}} f(t,x_{t}) dt \leq N T^{1-1/q} \| {\Psi}^{1-\nu }_{1} f\|_{L_{p,q}(\mathbb{R}^{d+1}_{+}) }, $$

(A.11)

where $N=N(\delta ,d,\underline {R},p,q,p_{0},\bar b_{\infty } )$.

Theorem A.7 (Theorem 4.10)

Assume that Eq. A.8 holds with ν = 0. Then for any $R\in (0,\bar R]$, x, and Borel nonnegative f given on C_R, we have

$$ E{\int}_{0}^{\tau_{R}(x)}f(t,x+x_{t}) dt\leq NR^{(2d_{0}-d)/p}\|f\|_{L_{p,q}(C_{R})}, $$

(A.12)

where $N=N(\delta ,d,\underline {R},p, p_{0},\bar b_{\bar R},\bar R )$.

Theorem A.8 (Theorem 4.11)

Assume that Eq. A.8 holds with ν = 0 and $p<\infty$, $q<\infty$. Let Q be a bounded domain in $\mathbb {R}^{d+1}$, 0 ∈ Q, b be bounded, and $u\in W^{1,2}_{p,q}(Q)\cap C(\bar Q)$. Then, for τ defined as the first exit time of (t,x_t) from Q and for all t ≥ 0,

$$ u(t\wedge\tau,x_{t\wedge\tau}) =u(0,0)+{\int}_{0}^{t\wedge\tau}D_{i}u(s,x_{s}) d{m^{i}_{s}} $$

$$ +{\int}_{0}^{t\wedge\tau}[ \partial_{t}u(s,x_{s})+ a^{ij}_{s}D_{ij}u(s,x_{s}) +{b^{i}_{s}}D_{i}u(s,x_{s})] ds $$

(A.13)

and the stochastic integral above is a square-integrable martingale.

Theorem A.9 (Theorem 5.1)

Let $0<R\leq \bar R$, domain Q ⊂ C_R, and assume that Eq. ?? holds with ν = 0, $p<\infty$, $q<\infty$, and that we are given a function $u\in W^{1,2}_{p,q,\text{loc}}(Q)\cap C(\bar Q)$. Take a function c ≥ 0 on Q. Then on Q

$$ u \leq NR^{(2d_{0}-d)/p} \|I_{Q,u>0}(\partial_{t}u+Lu-cu)_{-}\|_{L_{p,q} } +\sup_{\partial^{\prime}Q}u_{+}, $$

(A.14)

where $N=N(\delta ,d,\underline {R},\bar R,p, p_{0},\bar b_{\bar R})$ and $\partial ^{\prime }Q$ is the parabolic boundary of Q. In particular (the maximum principle), if ∂_tu + Lu − cu ≥ 0 in Q and u ≤ 0 on $\partial ^{\prime }Q$, then u ≤ 0 in Q.

Lemma A.10 (Lemma 2.2)

We have

$$ A:= E \tau_{R} (x)\leq R^{2}, $$

(A.15)

and, assuming that Eq. A.8 holds with ν = 0, for any Borel nonnegative f we have

$$ E{\int}_{0}^{\tau_{R}(x)} f(t,x_{t}) dt\leq N(d,p_{0}, \delta)(1+\bar b_{R})^{d/ p } R^{d/ p } \|f\|_{L_{p ,q }}. $$

(A.16)

Corollary A.11 (Corollary 2.10)

Let $R\leq \underline {R}$, κ ∈ [0, 1), and |x|≤ κR. Then for any T > 0

$$ NP(\tau^{\prime}_{R}(x)> T)\geq e^{-\nu T/[(1-\kappa)R]^{2}}, $$

(A.17)

where N and ν > 0 depend only on $\bar \xi$.

Corollary A.12 (Corollary 2.7)

Let ${\Lambda }\in (0,\infty )$. Then there is a constant $N =N (\underline {R},\bar R,{\Lambda },\bar \xi )$ such that for any $R\in (0,\bar R]$, λ ∈ [0,Λ]

$$ N E \tau_{R} \geq R^{2},\quad N E{\int}_{0}^{\tau_{R}} e^{-\lambda t} dt \geq R^{2} . $$

(A.18)