Large time probability of failure in diffusive search with resetting for a random target in ℝ^{𝕕}–A functional analytic approach
Authors: Pinsky Ross 
Published: November 2022

We consider a stochastic search model with resetting for an unknown stationary target

a
∈

R

d

,
 
d
≥
1

a\in \mathbb {R}^d,\ d\ge 1

, with known distribution

μ
\mu

. The searcher begins at the origin and performs Brownian motion with diffusion coefficient

D
D

. The searcher is also equipped with an exponential clock with rate

r
>
0

r>0

, so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. In dimension one, the target is considered located when the process hits the point

a
a

, while in dimensions two and higher, one chooses an

ϵ
0

>
0

\epsilon _0>0

and the target is considered located when the process hits the

ϵ
0

\epsilon _0

-ball centered at

a
a

. Denote the position of the searcher at time

t
t

by

X
(
t
)

X(t)

, let

τ
a

\tau _a

denote the time that a target at

a
a

is located, and let

P
0

d
;
(
r
,
0
)

P^{d;(r,0)}_0

denote probabilities for the process starting from 0. Taking a functional analytic point of view, and using the generator of the Markovian search process and its adjoint, we obtain precise estimates, with control on the dependence on

a
a

, for the asymptotic behavior of

P
0

d
;
(
r
,
0
)

(

τ
a

>
t
)

P^{d;(r,0)}_0(\tau _a>t)

for large time, and then use this to obtain large time estimates on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript double-struck upper R Superscript d Endscripts upper P 0 Superscript d semicolon left-parenthesis r comma 0 right-parenthesis Baseline left-parenthesis tau Subscript a Baseline greater-than t right-parenthesis d mu left-parenthesis a right-parenthesis">

∫

R

d

P
0

d
;
(
r
,
0
)

(

τ
a

>
t
)
d
μ
(
a
)

\int _{\mathbb {R}^d}P^{d;(r,0)}_0(\tau _a>t)d\mu (a)

, the probability that the searcher has failed up to time

t
t

to locate the random target, for a variety of families of target distributions

μ
\mu

. Specifically, for

B
,
l
>
0

B,l>0

and

d
∈

N

d\in \mathbb {N}

, let

μ

B
,
l

(
d
)

∈

P

(

R

d

)

\mu ^{(d)}_{B,l}\in \mathcal {P}(\mathbb {R}^d)

denote any target distribution with density

μ

B
,
l

(
d
)

(
a
)

\mu _{B,l}^{(d)}(a)

that satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript StartAbsoluteValue a EndAbsoluteValue right-arrow normal infinity Endscripts StartFraction log mu Subscript upper B comma l Superscript left-parenthesis d right-parenthesis Baseline left-parenthesis a right-parenthesis Over StartAbsoluteValue a EndAbsoluteValue Superscript l Baseline EndFraction equals negative upper B period">

lim

|

a

|

→
∞

log
⁡

μ

B
,
l

(
d
)

(
a
)

|

a

|

l

=
−
B
.

\begin{equation*} \lim _{|a|\to \infty }\frac {\log \mu _{B,l}^{(d)}(a)}{|a|^l}=-B. \end{equation*}

Then we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript t right-arrow normal infinity Endscripts StartFraction 1 Over left-parenthesis log t right-parenthesis Superscript l Baseline EndFraction log integral Underscript double-struck upper R Superscript d Baseline Endscripts upper P 0 Superscript d semicolon left-parenthesis r comma 0 right-parenthesis Baseline left-parenthesis tau Subscript a Baseline greater-than t right-parenthesis mu Subscript upper B comma l Superscript left-parenthesis d right-parenthesis Baseline left-parenthesis d a right-parenthesis equals minus upper B left-parenthesis StartFraction upper D Over 2 r EndFraction right-parenthesis Superscript StartFraction l Over 2 EndFraction Baseline period">

lim

t
→
∞

1

(
log
⁡
t

)
l

log
⁡

∫

R

d

P
0

d
;
(
r
,
0
)

(

τ
a

>
t
)

μ

B
,
l

(
d
)

(
d
a
)
=
−
B
(

D

2
r

)

l
2

.

\begin{equation*} \lim _{t\to \infty }\frac 1{(\log t)^l}\log \int _{\mathbb {R}^d} P_0^{d;(r,0)}(\tau _a>t)\mu _{B,l}^{(d)}(da)=-B(\frac D{2r})^\frac l2. \end{equation*}

The result is independent of the dimension. In particular, for example, if the target distribution is a centered Gaussian of any dimension with variance

σ
2

\sigma ^2

, then for any

δ
>
0

\delta >0

, the probability of not locating the target by time

t
t

falls in the interval
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis e Superscript minus left-parenthesis 1 plus delta right-parenthesis StartFraction upper D Over 4 r sigma squared EndFraction left-parenthesis log t right-parenthesis squared Baseline comma e Superscript minus left-parenthesis 1 minus delta right-parenthesis StartFraction upper D Over 4 r sigma squared EndFraction left-parenthesis log t right-parenthesis squared Baseline right-parenthesis">

(

e

−
(
1
+
δ
)

D

4
r

σ
2

(
log
⁡
t

)
2

,

e

−
(
1
−
δ
)

D

4
r

σ
2

(
log
⁡
t

)
2

)

\big (e^{-(1+\delta )\frac {D}{4r\sigma ^2}(\log t)^2}, e^{-(1-\delta )\frac {D}{4r\sigma ^2}(\log t)^2}\big )

, for sufficiently large

t
t

.