Inverse first-passage problems of a diffusion with resetting

We address some inverse problems for the first-passage place and the first-passage time of a one-dimensional diffusion process with stochastic resetting. This type of diffusion is characterized by the fact that a reset to the position <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript upper R"> <mml:semantics> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">x_R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can occur according to a homogeneous Poisson process with rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As regards the inverse first-passage place problem, for random initial position belonging to an interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (and fixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript upper R"> <mml:semantics> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">x_R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belonging to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma b right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,b))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ</mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the first time at which the process exits the interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 0"> <mml:semantics> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\pi _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the probability of exit from the left end of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given a probability <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the inverse first-passage place problem consists in finding the density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the initial position, if it exists, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 0 equals q"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _0 = q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Concerning the inverse first-passage time problem, for random positive starting point (and fixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript upper R Baseline greater-than 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x_R &gt;0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let again <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ</mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the first-passage time of the process through zero. For a given distribution function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the positive real axis, the inverse first-passage time problem consists in finding the density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the starting point, if it exists, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis tau less-than-or-equal-to t right-parenthesis equals upper F left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(\tau \le t ) = F(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In addition to the case of random initial position, we also study the case when the initial position and the resetting rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are fixed, whereas the reset position <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript upper R"> <mml:semantics> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>R</mml:mi> </